Explicit multipeakon solutions of Novikovs cubically nonlinear integrable Camassa-Holm type equation

Expliit m ultip eak on solutions of No vik o v's ubially nonlinear in tegrable CamassaHolm t yp e equation Andrew N. W. Hone ∗ Hans Lundmark † Jaek Szmigielski ‡ Mar h 20, 2009 Abstrat Reen tly Vladimir No vik o v found a new in tegrable analogue of the CamassaHolm equation, admitting p eak ed soliton ( p e akon ) solutions, whi h has nonlinear terms that are ubi, rather than quadrati. In this pap er, the expliit form ulas for m ultip eak on solutions of No vik o v's ubi- ally nonlinear equation are alulated, using the matrix Lax pair found b y Hone and W ang. By a transformation of Liouville t yp e, the asso iated sp etral problem is related to a ubi string equation, whi h is dual to the ubi string that w as previously found in the w ork of Lundmark and Szmigielski on the m ultip eak ons of the Degasp erisPro esi equation. 1 In tro dution In tegrable PDEs with nonsmo oth solutions ha v e attrated m u h atten tion in reen t y ears, sine the diso v ery of the CamassaHolm shallo w w ater w a v e equation and its p eak-shap ed soliton solutions alled p e akons [5 ℄. Our pur- p ose in this pap er is to expliitly ompute the m ultip eak on solutions of a new in tegrable PDE, equation (3.1 ) b elo w, whi h is of the CamassaHolm form u t − u xxt = F ( u, u x , u xx , . . . ) , but has ubially nonlinear terms instead of quadrati. This equation w as found b y Vladimir No vik o v, and published in a reen t pap er b y Hone and W ang [19 ℄. W e will apply in v erse sp etral metho ds. The spatial equation in the Lax pair for No vik o v's equation turns out to b e equiv alen t to what w e all the dual ubi string , a sp etral problem losely related to the ubi string that w as used for nding the m ultip eak on solutions to the Degasp erisPro esi equation ∗ Institute of Mathematis, Statistis & A tuarial Siene, Univ ersit y of Ken t, Can terbury CT2 7NF, United Kingdom; an whk en t.a.uk † Departmen t of Mathematis, Linköping Univ ersit y , SE-581 83 Linköping, Sw eden; halunmai.liu.se ‡ Departmen t of Mathematis and Statistis, Univ ersit y of Sask at hew an, 106 Wiggins Road, Sask ato on, Sask at hew an, S7N 5E6, Canada; szmigielmath.usask.a 1 [27 , 28 , 23 ℄. One this relation is established, the No vik o v p eak on solution an b e deriv ed in a straigh tforw ard w a y using the results obtained in [23 ℄. The onstan ts of motion ha v e a more ompliated struture than in the Camassa Holm and Degasp erisPro esi ases, and the study of this giv es as an in teresting b y-pro dut a om binatorial iden tit y onerning the sum of all minors in a sym- metri matrix, whi h w e ha v e dubb ed the Canada Day The or em (Theorem 4.1 , pro v ed in App endix A). The p eak on problem for No vik o v's equation presen ts in addition one imp or- tan t  hallenge. Unlik e its CamassaHolm or Degasp erisPro esi oun terparts, the Lax pair for the No vik o v equation is originally ill-dened in the p eak on se- tor. The problem is aused b y terms whi h in v olv e m ultipliation of a singular measure b y a dison tin uous funtion. W e pro v e in App endix B that there ex- ists a regularization of the Lax pair whi h preserv es in tegrabilit y of the p eak on setor, th us allo wing us to use sp etral and in v erse sp etral metho ds to obtain the m ultip eak on solutions to the No vik o v equation. This regularization problem has a subtle but nev ertheless real impat on the form ulas. In general, the use of Lax pairs to onstrut distributional solutions to nonlinear equations whi h are Lax in tegrable in the smo oth setor but ma y not b e so in the whole non-smo oth setor is relativ ely un harted territory , and the ase of No vik o v's equation ma y pro vide some relev an t insigh t in this regard. 2 Ba kground The main example of a PDE admitting p eak ed solitons is the family u t − u xxt + ( b + 1 ) uu x = bu x u xx + u u xxx , (2.1) often written as m t + m x u + bmu x = 0 , m = u − u xx , (2.2) whi h w as in tro dued b y Degasp eris, Holm and Hone [ 10 ℄, and is Hamiltonian for all v alues of b [ 17 ℄. It inludes the CamassaHolm equation as the ase b = 2 , and another in tegrable PDE alled the Degasp erisPro esi equation [11 , 10 ℄ as the ase b = 3 . These are the only v alues of b for whi h the equation is in tegrable, aording to a v ariet y of in tegrabilit y tests [ 11 , 30 , 18 , 20 ℄. (Ho w ev er, w e note that the ase b = 0 is exluded from the aforemen tioned in tegrabilit y tests; y et this ase pro vides a regularization of the in visid Burgers equation that is Hamiltonian and has lassial solutions globally in time [ 4℄.) Multip e akons are w eak solutions of the form u ( x, t ) = n X i =1 m i ( t ) e −| x − x i ( t ) | , (2.3) formed through sup erp osition of n p e akons (p eak ed solitons of the shap e e −| x | ). This ansatz satises the PDE (2.2 ) if and only if the p ositions ( x 1 , . . . , x n ) and 2 momen ta ( m 1 , . . . , m n ) of the p eak ons ob ey the follo wing system of 2 n ODEs: ˙ x k = n X i =1 m i e −| x k − x i | , ˙ m k = ( b − 1) m k n X i =1 m i sgn( x k − x i ) e −| x k − x i | . (2.4) Here, sgn x denotes the sign um funtion, whi h is +1 , − 1 or 0 dep ending on whether x is p ositiv e, negativ e or zero. In shorthand notation, with  f ( x )  denoting the a v erage of the left and righ t limits,  f ( x )  = 1 2  f ( x − ) + f ( x + )  , (2.5) the ODEs an b e written as ˙ x k = u ( x k ) , ˙ m k = − ( b − 1) m k  u x ( x k )  . (2.6) In the CamassaHolm ase b = 2 , this is a anonial Hamiltonian system gen- erated b y h = 1 2 P n j,k =1 m j m k e −| x j − x k | . Expliit form ulas for the n -p eak on solution of the CamassaHolm equation w ere deriv ed b y Beals, Sattinger and Szmigielski [1 , 2 ℄ using in v erse sp etral metho ds, and the same thing for the Degasp erisPro esi equation w as aomplished b y Lundmark and Szmigielski [27 , 28 ℄. It requires some are to sp eify the exat sense in whi h the p eak on solu- tions satisfy the PDE. The form ulation (2.2 ) suers from the problem that the pro dut mu x is ill-dened in the p eak on ase, sine the quan tit y m = u − u xx = 2 P n i =1 m i δ x i is a disrete measure, and it is m ultiplied b y a funtion u x whi h has jump dison tin uities exatly at the p oin ts x k where the Dira deltas in the measure m are situated. T o a v oid this problem, one an instead rewrite ( 2.1 ) as (1 − ∂ 2 x ) u t + ( b + 1 − ∂ 2 x ) ∂ x  1 2 u 2  + ∂ x  3 − b 2 u 2 x  = 0 . (2.7) Then a funtion u ( x, t ) is said to b e a solution if • u ( · , t ) ∈ W 1 , 2 lo c ( R ) for ea h xed t , whi h means that u ( · , t ) 2 and u x ( · , t ) 2 are lo ally in tegrable funtions, and therefore dene distributions of lass D ′ ( R ) (i.e., on tin uous linear funtionals ating on ompatly supp orted C ∞ test funtions on the real line R ), • the time deriv ativ e u t ( · , t ) , dened as the limit of a dierene quotien t, exists as a distribution in D ′ ( R ) for all t , • equation ( 2.7), with ∂ x tak en to mean the usual distributional deriv ativ e, is satised for all t in the sense of distributions in D ′ ( R ) . It is w orth men tioning that funtions in the spae W 1 , 2 lo c ( R ) are on tin uous, b y the Sob olev em b edding theorem. Ho w ev er, the term u 2 x is absen t from equation (2.7 ) if b = 3 , so in that partiular ase one requires only that u ( · , t ) ∈ L 2 lo c ( R ) ; this means that the Degasp erisPro esi an admit solutions u that are not on tin uous [6, 7, 26 ℄. It is often appropriate to rewrite equation (2.7 ) as a nonlo al ev olution equa- tion for u b y in v erting the op erator (1 − ∂ 2 x ) , as w as done in [8, 9℄ for the CamassaHolm equation. Ho w ev er, the distributional form ulation used here is v ery on v enien t when w orking with p eak on solution. 3 3 No vik o v's equation The new in tegrable equation found b y Vladimir No vik o v is u t − u xxt + 4 u 2 u x = 3 uu x u xx + u 2 u xxx , (3.1) whi h an b e written as m t + ( m x u + 3 mu x ) u = 0 , m = u − u xx , (3.2) to highligh t the similarit y in form to the Degasp erisPro esi equation, or as (1 − ∂ 2 x ) u t + (4 − ∂ 2 x ) ∂ x  1 3 u 3  + ∂ x  3 2 uu 2 x  + 1 2 u 3 x = 0 (3.3) in order to rigorously dene w eak solutions as ab o v e, exept that here one re- quires that u ( · , t ) ∈ W 1 , 3 lo c ( R ) for all t , so that u 3 and u 3 x are lo ally in tegrable and therefore dene distributions in D ′ ( R ) ; it then follo ws from Hölder's in- equalit y with the onjugate indies 3 and 3 / 2 that uu 2 x is lo ally in tegrable as w ell, and (3.3) an th us b e in terpreted as a distributional equation. Sine fun- tions in W 1 ,p lo c ( R ) with p ≥ 1 are automatially on tin uous, No vik o v's equation is similar to the CamassaHolm equation in that it only admits on tin uous dis- tributional solutions (as opp osed to the Degasp erisPro esi equation, whi h has dison tin uous solutions as w ell). Lik e the equations in the b -family (2.1), No vik o v's equation admits (in the w eak sense just dened) m ultip eak on solutions of the form (2.3 ), but in this ase the ODEs for the p ositions and momen ta are ˙ x k = u ( x k ) 2 = n X i =1 m i e −| x k − x i | ! 2 , ˙ m k = − m k u ( x k )  u x ( x k )  = m k n X i =1 m i e −| x k − x i | !   n X j =1 m j sgn( x k − x j ) e −| x k − x j |   . (3.4) These equations w ere stated in [ 19 ℄, where it w as also sho wn that they onstitute a Hamiltonian system ˙ x k = { x k , h } , ˙ m k = { m k , h } , generated b y the same Hamiltonian h = 1 2 P n j,k =1 m j m k e −| x j − x k | as the CamassaHolm p eak ons, but with resp et to a dieren t, non-anonial, P oisson struture giv en b y { x j , x k } = sgn( x j − x k )  1 − E 2 j k  , { x j , m k } = m k E 2 j k , { m j , m k } = sgn( x j − x k ) m j m k E 2 j k , where E j k = e −| x j − x k | . (3.5) As will b e sho wn b elo w, ( 3.4) is a Liouville in tegrable system (Theorem 4.7 ); in fat, it is ev en expliitly solv able in terms of elemen tary funtions (Theorem 9.1 ). 4 4 F orw ard sp etral problem In order to in tegrate the No vik o v p eak on ODEs, w e are going to mak e use of the matrix Lax pair found b y Hone and W ang [19 ℄, sp eied b y the follo wing matrix linear system: ∂ ∂ x   ψ 1 ψ 2 ψ 3   =   0 z m 1 0 0 z m 1 0 0     ψ 1 ψ 2 ψ 3   , (4.1) ∂ ∂ t   ψ 1 ψ 2 ψ 3   =   − uu x u x z − 1 − u 2 mz u 2 x uz − 1 − z − 2 − u x z − 1 − u 2 mz − u 2 uz − 1 uu x     ψ 1 ψ 2 ψ 3   . (4.2) (Compared with referene [ 19 ℄ w e ha v e added a onstan t m ultiple of the iden tit y to the matrix on the righ t hand side of (4.2 ), and used z in plae of λ .) In the p eak on ase, when u = P n i =1 m i e −| x − x i | , the quan tit y m = u − u xx = 2 P n i =1 m i δ x i is a disrete measure. W e assume that x 1 < x 2 < · · · < x n (whi h at least remains true for a while if it is true for t = 0 ). These p oin ts divide the x axis in to n + 1 in terv als whi h w e n um b er from 0 to n , so that the k th in terv al runs from x k to x k +1 , with the on v en tion that x 0 = −∞ and x n +1 = + ∞ . Sine m v anishes b et w een the p oin t masses, equation ( 4.1 ) redues to ∂ x ψ 1 = ψ 3 , ∂ x ψ 2 = 0 and ∂ x ψ 3 = ψ 1 in ea h in terv al, so that in the k th in terv al w e ha v e   ψ 1 ψ 2 ψ 3   =   A k e x + z 2 C k e − x 2 z B k A k e x − z 2 C k e − x   for x k < x < x k +1 , (4.3) where the fators on taining z ha v e b een inserted for later on v eniene. These pieewise solutions are then glued together at the p oin ts x k . The prop er in ter- pretation of (4.1) at these p oin ts turns out to b e that ψ 3 m ust b e on tin uous, while ψ 1 and ψ 2 are allo w ed to ha v e jump dison tin uities; moreo v er, in the term z mψ 2 , one should tak e ψ 2 ( x ) δ x k to mean  ψ 2 ( x k )  δ x k . This p oin t is fully explained in App endix B. This leads to   A k B k C k   =   1 − λ m 2 k − 2 λ m k e − x k − λ 2 m 2 k e − 2 x k m k e x k 1 λ m k e − x k m 2 k e 2 x k 2 m k e x k 1 + λm 2 k     A k − 1 B k − 1 C k − 1   =: S k ( λ )   A k − 1 B k − 1 C k − 1   , where λ = − z 2 . (4.4) W e imp ose the b oundary ondition ( A 0 , B 0 , C 0 ) = (1 , 0 , 0) , whi h is onsisten t with the time ev olution giv en b y (4.2) for x < x 1 . Then all ( A k , B k , C k ) are determined b y suessiv e appliation of the jump matries S k ( λ ) as in (4.4). F or x > x n , equation (4.2 ) implies that ( A, B , C ) := ( A n , B n , C n ) ev olv es as ˙ A = 0 , ˙ B = B − AM + λ , ˙ C = 2 M + ( B − AM + ) λ , (4.5) 5 where M + = P N k =1 m k e x k . Th us A is in v arian t. It is the (1 , 1) en try of the total jump matrix S ( λ ) = S n ( λ ) . . . S 2 ( λ ) S 1 ( λ ) , (4.6) and therefore it is a p olynomial in λ of degree n , A ( λ ) = n X k =0 H k ( − λ ) k =  1 − λ λ 1  . . .  1 − λ λ n  , (4.7) where H 0 = 1 (sine S (0) = I , the iden tit y matrix), and where the other o e- ien ts H 1 , . . . , H n are P oisson omm uting onstan ts of motion (see Theorems 4.2 and 4.7 b elo w). The rst linear equation (4.1), together with the b oundary onditions ex- pressed b y the requiremen ts that B 0 = C 0 = 0 and A n ( λ ) = 0 , is a sp etral problem whi h has the zeros λ 1 , . . . , λ n of A ( λ ) as its eigenv alues. (T o b e pre- ise, one should p erhaps sa y that it is the orresp onding v alues of z = ± √ − λ that are the eigen v alues, but w e will so on sho w that the λ k are p ositiv e, at least in the pure p eak on ase, and therefore more on v enien t to w ork with than the purely imaginary v alues of z ; see ( 4.19 ) b elo w.) Elimination of ψ 1 from (4.1) giv es ∂ x ψ 2 = z mψ 3 and ( ∂ 2 x − 1) ψ 3 = z mψ 2 , and the b oundary onditions ab o v e imply that ( ψ 2 , ψ 3 ) → (0 , 0) as x → −∞ and ψ 3 → 0 as x → + ∞ . Using the Green's funtion − e −| x | / 2 for the op erator ∂ 2 x − 1 with v anishing b oundary onditions, w e an rephrase the problem as a system of in tegral equations, ψ 2 ( x ) = z Z x −∞ ψ 3 ( y ) dm ( y ) , ψ 3 ( x ) = − z Z ∞ −∞ 1 2 e −| x − y | ψ 2 ( y ) dm ( y ) , (4.8) with in tegrals tak en with resp et to the disrete measure m = 2 P n i =1 m i δ x i . Here, there is again the problem of Dira deltas m ultiplying a funtion ψ 2 with jump dison tin uities, and w e tak e ψ 2 ( x ) δ x k to mean the a v erage  ψ 2 ( x k )  δ x k , in full agreemen t with the earlier denition of the singular term app earing in the sp etral problem. Then  ψ 2 ( x j )  = z 2 j − 1 X k =1 ψ 3 ( x k ) m k + ψ 3 ( x j ) m j ! , ψ 3 ( x j ) = − z n X k =1 e −| x j − x k |  ψ 2 ( x k )  m k , (4.9) whi h an b e written in blo  k matrix notation as  Ψ 2  Ψ 3  = z  0 T P − E P 0    Ψ 2  Ψ 3  , (4.10) 6 where Ψ 3 =  ψ 3 ( x 1 ) , . . . , ψ 3 ( x n )  t ,  Ψ 2  =  ψ 2 ( x 1 )  , . . . ,  ψ 2 ( x n )  t , P = diag ( m 1 , . . . , m n ) , E = ( E j k ) n j,k =1 , where E j k = e −| x j − x k | , T = ( T j k ) n j,k =1 , where T j k = 1 + sgn( j − k ) . (4.11) (In w ords, T is the lo w er triangular n × n matrix that has 1 on the main diagonal and 2 ev erywhere b elo w it.) In terms of  Ψ 2  alone, w e ha v e  Ψ 2  = − z 2 T P E P  Ψ 2  , (4.12) so the eigen v alues are giv en b y 0 = det( I + z 2 T P E P ) = det( I − λT P E P ) , where of ourse I denotes the n × n iden tit y matrix. Sine the eigen v alues are the zeros of A ( λ ) , and sine A (0) = 1 , it follo ws that A ( λ ) = det( I − λT P E P ) . (4.13) This giv es us a fairly onrete represen tation of the onstan ts of motion H k , whi h b y denition are the o eien ts of A ( λ ) (see (4.7 )), and it an b e made ev en more expliit thanks to the urious om binatorial result in Theorem 4.1 . W e remind the reader that a k × k minor of an n × n matrix X is, b y denition, the determinan t of a submatrix X I J = ( X ij ) i ∈ I , J ∈ J whose ro ws and olumns are seleted among those of X b y t w o index sets I , J ⊆ { 1 , . . . , n } with k elemen ts ea h, and a prinip al minor is one for whi h I = J . Compare the result of the theorem with the w ell-kno wn fat that the o eien t of s k in det( I + sX ) equals the sum of all prinip al k × k minors of X , regardless of whether X is symmetri or not. Theorem 4.1 (The Canada Da y Theorem) . L et the matrix T b e dene d as in (4.11 ) ab ove. Then, for any symmetri n × n matrix X , the  o eient of s k in the p olynomial det( I + s T X ) e quals the sum of al l k × k minors (prinip al and non-prinip al) of X . Pr o of. The pro of is presen ted in App endix A. It relies on the Cau h yBinet form ula, Lindström's Lemma, and some rather in triate dep endenies among the minors of X due to the symmetry of the matrix. Theorem 4.1 is named after the date when w e started trying to pro v e it: July 1, 2008, Canada's national da y . (It turned out that the pro of w as more diult than w e exp eted, so w e didn't nish it un til a few da ys later.) Summa- rizing the results so far, w e no w ha v e the follo wing desription of the onstan ts of motion: Theorem 4.2. The Novikov p e akon ODEs (3.4 ) admit n  onstants of motion H 1 , . . . , H n , wher e H k e quals the sum of al l k × k minors (prinip al and non- prinip al) of the n × n symmetri matrix P E P = ( m j m k E j k ) n j,k =1 . (Se e (4.11 ) for notation.) 7 Pr o of. This follo ws at one from (4.7 ), (4.13 ), and Theorem 4.1 . Example 4.3. The sum of all 1 × 1 minors of P E P is of ourse just the sum of all en tries, H 1 = n X j,k =1 m j m k E j k = n X j,k =1 m j m k e −| x j − x k | , (4.14) and the Hamiltonian of the p eak on ODEs (3.4 ) is h = 1 2 H 1 . Higher order minors of P E P are easily omputed using Lindström's Lemma, as explained in Setion A.3 in the app endix. In partiular, the onstan t of motion of highest degree in the m k is H n = det( P E P ) = n − 1 Y j =1 (1 − E 2 j,j +1 ) n Y j =1 m 2 j . (4.15) Example 4.4. W ritten out in full, the onstan ts of motion in the ase n = 3 are H 1 = m 2 1 + m 2 2 + m 2 3 + 2 m 1 m 2 E 12 + 2 m 1 m 3 E 13 + 2 m 2 m 3 E 23 , H 2 = (1 − E 2 12 ) m 2 1 m 2 2 + (1 − E 2 13 ) m 2 1 m 2 3 + (1 − E 2 23 ) m 2 2 m 2 3 + 2 ( E 23 − E 12 E 13 ) m 2 1 m 2 m 3 + 2 ( E 12 − E 13 E 23 ) m 1 m 2 m 2 3 , H 3 = (1 − E 2 12 )(1 − E 2 23 ) m 2 1 m 2 2 m 2 3 . (4.16) F rom no w on w e mainly restrit ourselv es to the pure p eak on ase when m k > 0 for all k (no an tip eak ons). Our rst reason for this is that w e an then use the p ositivit y of H 1 and H n to sho w global existene of p eak on solutions. Theorem 4.5. L et P = { x 1 < · · · < x n , al l m k > 0 } (4.17) b e the phase sp a e for the Novikov p e akon system (3.4 ) in the pur e p e akon  ase. If the initial data ar e in P , then the solution ( x ( t ) , m ( t )) exists for al l t ∈ R , and r emains in P . Pr o of. Lo al existene in P is automati in view of the smo othness of the ODEs there. By (4.14 ) and (4.15 ) , b oth H 1 and H n are stritly p ositiv e on P . Sine m 2 k < H 1 , all m k remain b ounded from ab o v e. The p ositivit y of H n ensures that the m k are b ounded a w a y from zero, and that the p ositions remain ordered. The v elo ities ˙ x k are all b ounded b y ( P m k ) 2 , hene 0 < ˙ x k ≤ C for some onstan t C , and the p ositions x k ( t ) are therefore nite for all t ∈ R . Sine neither x k nor m k an blo w up in nite time, the solution exists globally in time. Remark 4.6. The p eak on ODEs ( 3.4 ) are in v arian t under the transformation ( m 1 , . . . , m n ) 7→ ( − m 1 , . . . , − m n ) , so the analogous result holds also when all m k are negativ e. 8 Theorem 4.7. The  onstants of motion H 1 , . . . , H n of The or em 4.2 ar e fun- tional ly indep endent and  ommute with r esp e t to the Poisson br aket (3.5 ) , so the Novikov p e akon system ( 3.4) is Liouvil le inte gr able on the phase sp a e P . Pr o of. T o pro v e funtional indep endene, one should  he k that J := dH 1 ∧ dH 2 ∧ . . . ∧ dH n do es not v anish on an y op en set in P . Sine J is rational in the v ariables { m k , e x k } n k =1 , it v anishes iden tially if it v anishes on an op en set, so it is suien t to sho w that J is not iden tially zero. T o see this, note that H k = e k ( m 2 1 , . . . , m 2 n ) + O ( E pq ) , (4.18) where e k denotes the k th elemen tary symmetri funtion in n v ariables, and O ( E pq ) denotes terms in v olving exp onen tials of the p ositions x j . It is w ell kno wn that the rst n elemen tary symmetri funtions are indep enden t (they pro vide a basis for symmetri funtions of n v ariables [29℄), and therefore the leading part of J (negleting the O ( E pq ) terms) do es not v anish. Sine the O ( E pq ) terms an b e made arbitrarily small b y taking the x k far apart, w e see that there is a region in P where J do es not v anish, and w e are done. T o pro v e that the quan tities H k P oisson omm ute with resp et to the bra k et (3.5 ), it is on v enien t to adapt some argumen ts of Moser that he applied to the sattering of partiles in the T o da lattie and the rational CalogeroMoser sys- tem [31℄. The P oisson bra k et of t w o onstan ts of motion is itself a onstan t of motion, so { H j , H k } is indep enden t of time. Consider no w this bra k et at a xed p oin t ( x 0 , m 0 ) := ( x 0 1 , x 0 2 , . . . , x 0 n , m 0 1 , m 0 2 , . . . m 0 n ) ∈ P whi h w e onsider as an initial ondition for the p eak on o w ( x ( t ) , m ( t )) , whi h exists globally in time b y Theorem 4.5 . Theorem 9.4 , whi h will b e pro v ed later without using what w e are pro ving here, sho ws that the p eak ons satter as t → −∞ ; more preisely , m 2 k tends to 1 /λ k , while the x k mo v e apart, so that the terms O ( E pq ) tend to zero. (It should also b e p ossible to pro v e these sattering prop erties diretly from the p eak on ODEs, along the lines of what w as done for the Degasp erisPro esi equation in [28 , Theorem 2.4℄, but w e ha v e not done that.) Th us, from ( 4.18 ), { H j , H k } ( x 0 , m 0 ) = { H j , H k } ( x ( t ) , m ( t )) = lim t →−∞ { H j , H k } ( x ( t ) , m ( t )) = lim t →−∞ { e j , e k } ( x ( t ) , m ( t )) . No w the P oisson bra k ets of these symmetri funtions are giv en b y linear om binations of the P oisson bra k ets { m j , m k } with o eien ts dep enden t only on the amplitudes. Ho w ev er, from (3.5) it is lear that { m j , m k } ( x ( t ) , m ( t )) = O ( E pq ) → 0 , from whi h it follo ws that { e j , e k } ( x ( t ) , m ( t )) → 0 as t → −∞ , and hene { H j , H k } ( x 0 , m 0 ) = 0 as re- quired. Remark 4.8. Sine the v anishing of the P oisson bra k et is a purely algebrai relation, the H k P oisson omm ute at ea h p oin t of R 2 n , not just in the region P . The λ k , whi h are dened as the zeros of A ( λ ) , are the eigen v alues of the in v erse of the matrix T P E P , sine A ( λ ) = det( I − λT P E P ) . Another reason wh y w e restrit our atten tion to the ase with all m k > 0 is that the matrix T P E P an then b e sho wn to b e osillatory (see Setion A.2 in the app endix), whi h implies that its eigen v alues are p ositiv e and simple. Consequen tly , the 9 λ k are also p ositiv e and simple, and for deniteness w e will n um b er them su h that 0 < λ 1 < · · · < λ n . (4.19) (F or another pro of that the sp etrum is p ositiv e and simple, see Theorem 6.1 .) T urning no w to B = S ( λ ) 21 and C = S ( λ ) 31 , w e nd from (4.6 ) and (4.4) that they are p olynomials in λ of degree n − 1 , with B (0) = M + and C (0) = M 2 + , where M + = P N k =1 m k e x k as b efore. This means that the t w o W eyl funtions ω ( λ ) = − B ( λ ) A ( λ ) and ζ ( λ ) = − C ( λ ) 2 A ( λ ) (4.20) are rational funtions of order O (1 / λ ) as λ → ∞ , ha ving p oles at the eigen- v alues λ k . Let b k and c k denote the residues, ω ( λ ) = n X k =1 b k λ − λ k , ζ ( λ ) = n X k =1 c k λ − λ k . (4.21) The time ev olution of ( A, B , C ) , giv en b y ( 4.5 ), translates in to ˙ ω ( λ ) = ω ( λ ) − ω (0) λ , ˙ ζ ( λ ) = − ω (0) ˙ ω ( λ ) . (4.22) Comparing residues on b oth sides in (4.22 ) giv es ˙ b k = b k λ k , ˙ c k = − ω (0) b k λ k = n X m =1 b m b k λ m λ k . (4.23) This at one implies b k ( t ) = b k (0) e t/λ k , and in tegrating ˙ c k ( τ ) from τ = −∞ (assuming that c k v anishes there) to τ = t then giv es c k = n X m =1 b m b k λ m + λ k . (4.24) A purely algebrai pro of of this relation b et w een the W eyl funtions, not relying on time dep endene and in tegration, will b e giv en b elo w; see Theorem 6.1 . W e note the iden tities P n 1 c k /λ k = 1 2 ( P n 1 b k /λ k ) 2 and P n 1 λ k c k = 1 2 ( P n 1 b k ) 2 . The m ultip eak on solution is obtained as follo ws. The initial data x k (0) , m k (0) (for k = 1 , . . . , n ) determine initial sp etral data λ k (0) , b k (0) , whi h after time t ha v e ev olv ed to λ k ( t ) = λ k (0) , b k ( t ) = b k (0) e t/λ k (sine the λ k are the zeros of the time-in v arian t p olynomial A ( λ ) , and sine the b k satisfy (4.23 ) ). Solving the in v erse sp etral problem for these sp etral data at time t giv es the solution x k ( t ) , m k ( t ) . The remainder of the pap er is dev oted to this in v erse sp etral problem. 10 5 The dual ubi string Just lik e for the CamassaHolm and Degasp erisPro esi equations, some terms in the Lax pair's spatial equation (equation (4.1) in this ase, rep eated as (5.1) b elo w) an b e remo v ed b y a  hange of b oth dep enden t and indep enden t v ari- ables. W e refer to this as a Liouville transformation, sine it is reminisen t of the transformation used for bringing a seond-order SturmLiouville op erator to a simple normal form. This simpliation rev eals an in teresting onnetion b et w een the No vik o v equation and the Degasp erisPro esi equation, and allo ws us to solv e the in v erse sp etral problem b y making use of the to ols dev elop ed in the study of the latter. Theorem 5.1. The sp e tr al pr oblem ∂ ∂ x   ψ 1 ψ 2 ψ 3   =   0 z m ( x ) 1 0 0 z m ( x ) 1 0 0     ψ 1 ψ 2 ψ 3   (5.1) on the r e al line x ∈ R , with b oundary  onditions ψ 2 ( x ) → 0 , as x → −∞ , e x ψ 3 ( x ) → 0 , as x → −∞ , e − x ψ 3 ( x ) → 0 , as x → + ∞ , (5.2) is e quivalent (for z 6 = 0 ), under the hange of variables y = tanh x, φ 1 ( y ) = ψ 1 ( x ) c osh x − ψ 3 ( x ) s inh x, φ 2 ( y ) = z ψ 2 ( x ) , φ 3 ( y ) = z 2 ψ 3 ( x ) / cosh x, g ( y ) = m ( x ) cosh 3 x, λ = − z 2 , (5.3) to the dual ubi string pr oblem ∂ ∂ y   φ 1 φ 2 φ 3   =   0 g ( y ) 0 0 0 g ( y ) − λ 0 0     φ 1 φ 2 φ 3   (5.4) on the nite interval − 1 < y < 1 , with b oundary  onditions φ 2 ( − 1) = φ 3 ( − 1) = 0 φ 3 (1) = 0 . (5.5) In the disr ete  ase m = 2 P n k =1 m k δ x k , the r elation b etwe en the me asur es m and g should b e interpr ete d as g ( y ) = n X k =1 g k δ y k , y k = tanh x k , g k = 2 m k cosh x k . (5.6) 11 Pr o of. Straigh tforw ard omputation using the  hain rule and, for the disrete ase, δ x k = dy dx ( x k ) δ y k . Remark 5.2. The ubi string equation, whi h pla ys a ruial role in the deriv ation of the Degasp erisPro esi m ultip eak on solution [28 ℄, is ∂ 3 y φ = − λg φ, (5.7) whi h an b e written as a system b y letting Φ = ( φ 1 , φ 2 , φ 3 ) = ( φ, φ y , φ y y ) : ∂ ∂ y   φ 1 φ 2 φ 3   =   0 1 0 0 0 1 − λg ( y ) 0 0     φ 1 φ 2 φ 3   . (5.8) The dualit y b et w een (5.4 ) and (5.8 ) manifests itself in the disrete ase as an in ter hange of the roles of masses g k and distanes l k = y k +1 − y k ; see Setion 6. When the mass distribution is giv en b y a on tin uous funtion g ( y ) > 0 , the systems are instead related via the  hange of v ariables dened b y d ˜ y dy = g ( y ) = 1 ˜ g ( ˜ y ) , (5.9) where y and g ( y ) refer to the primal ubi string (5.8), and ˜ y and ˜ g ( ˜ y ) to the dual ubi string (5.4 ) (or the other w a y around; the transformation (5.9 ) is ob viously symmetri in y and ˜ y , so that the dual of the dual is the original ubi string again). Remark 5.3. The onept of a dual string gures prominen tly in the w ork of Krein on the ordinary string equation ∂ 2 y φ = − λg φ (as opp osed to the ubi string). F or a omprehensiv e aoun t of Krein's theory , see [12 ℄. Remark 5.4. As a motiv ation for the transformation (5.3 ), w e note that one an eliminate ψ 1 from (5.1), whi h giv es ∂ x ψ 2 = z mψ 3 , ( ∂ 2 x − 1) ψ 3 = z mψ 2 . F rom the study of CamassaHolm p eak ons [2℄ it is kno wn that the transformation y = ta nh x , φ ( y ) = ψ ( x ) / cosh x tak es the expression ( ∂ 2 x − 1) ψ to a m ultiple of φ y y , so it is not far-fet hed to try something similar on ψ 3 while lea ving ψ 2 essen tially un hanged. F rom no w on w e onen trate on the disrete ase. The Liouville transfor- mation maps the pieewise dened ( ψ 1 , ψ 2 , ψ 3 ) giv en b y (4.3) to   φ 1 φ 2 φ 3   =   A k ( λ ) − λ C k ( λ ) − 2 λ B k ( λ ) − λ A k ( λ ) (1 + y ) − λ 2 C k ( λ ) (1 − y )   for y k < y < y k +1 . (5.10) The initial v alues ( A 0 , B 0 , C 0 ) = (1 , 0 , 0) th us orresp ond to Φ( − 1; λ ) = (1 , 0 , 0) t , where Φ( y ; λ ) =  φ 1 , φ 2 , φ 3  t , and at the righ t endp oin t y = 1 w e ha v e Φ(1; λ ) =   A n ( λ ) − λ C n ( λ ) − 2 λ B n ( λ ) − 2 λ A n ( λ )   . (5.11) 12 In partiular, the ondition A n ( λ ) = 0 dening the sp etrum orresp onds to φ 3 (1; λ ) = 0 , exept that the latter ondition giv es an additional eigen v alue λ 0 = 0 whi h is only presen t on the nite in terv al. (This is not a on tradition, sine the Liouville transformation from the line to the in terv al is not in v ertible when z = − λ 2 = 0 .) The omp onen t φ 3 is on tin uous and pieewise linear, while φ 1 and φ 2 are pieewise onstan t with jumps at the p oin ts y k where the measure g is supp orted. More preisely , at p oin t mass n um b er k w e ha v e φ 1 ( y + k ) − φ 1 ( y − k ) = g k  φ 2 ( y k )  , φ 2 ( y + k ) − φ 2 ( y − k ) = g k φ 3 ( y k ) , (5.12) and in in terv al n um b er k , with length l k = y k +1 − y k , φ 3 ( y − k +1 ) − φ 3 ( y + k ) = l k ∂ y φ 3 ( y + k ) = − λ l k φ 1 ( y + k ) . (5.13) In terms of the v etor Φ these relations tak e the form Φ( y + k ) =   1 g k 1 2 g 2 k 0 1 g k 0 0 1   Φ( y − k ) , (5.14) and Φ( y − k +1 ) =   1 0 0 0 1 0 − λl k 0 1   Φ( y + k ) , (5.15) resp etiv ely . If w e in tro due the notation G ( x, λ ) =   1 0 0 0 1 0 − λx 0 1   , L ( x ) =   1 x 1 2 x 2 0 1 x 0 0 1   , (5.16) it follo ws immediately that Φ(1; λ ) = G ( l n , λ ) L ( g n ) . . . G ( l 2 , λ ) L ( g 2 ) G ( l 1 , λ ) L ( g 1 ) G ( l 0 , λ )  1 0 0  . (5.17) W e dene the W eyl funtions W and Z of the dual ubi string to b e W ( λ ) = − φ 2 (1; λ ) φ 3 (1; λ ) , Z ( λ ) = − φ 1 (1; λ ) φ 3 (1; λ ) . (5.18) It is lear from (5.11 ) that they are related to the W eyl funtions ω and ζ previously dened on the real line (see ( 4.20 )) as follo ws: W ( λ ) = − B n ( λ ) A n ( λ ) = ω ( λ ) = n X k =1 b k λ − λ k , Z ( λ ) = A n ( λ ) − λC n ( λ ) 2 λA n ( λ ) = 1 2 λ + ζ ( λ ) = 1 2 λ + n X k =1 c k λ − λ k . (5.19) 13 6 Relation to the Neumann-lik e ubi string K ohlen b erg, Lundmark and Szmigielski [23℄ studied the disrete ubi string with Neumann-lik e b oundary onditions. W e will briey reall some results from that pap er, with notation and sign on v en tions sligh tly altered to suit our needs here. The sp etral problem in question is φ y yy ( y ) = − λg ( y ) φ ( y ) for y ∈ R , φ y ( −∞ ) = φ y y ( −∞ ) = 0 , φ y y ( ∞ ) = 0 , (6.1) where g = P n k =0 g k δ y k is a disrete measure with n + 1 p oin t masses g 0 , . . . , g n at p ositions y 0 < y 1 < · · · < y n ; b et w een these p oin ts are n nite in terv als of length l 1 , . . . , l n (where l k = y k − y k − 1 ). Sine φ y yy = 0 a w a y from the p oin t masses, the b oundary onditions an equally w ell b e written as φ y ( y − 0 ) = φ y y ( y − 0 ) = 0 , φ y y ( y + n ) = 0 . Using the normalization φ ( −∞ ) = 1 (or φ ( y − 0 ) = 1 ) and the notation Φ = ( φ, φ y , φ y y ) t , one nds Φ( y + n ; λ ) = G ( g n , λ ) L ( l n ) . . . G ( g 2 , λ ) L ( l 2 ) G ( g 1 , λ ) L ( l 1 ) G ( g 0 , λ )  1 0 0  , (6.2) with matries G and L as in (5.16 ) ab o v e. Under the assumption that all g k > 0 , the zeros of φ y y ( y + n ; λ ) , whi h onstitute the sp etrum, are 0 = λ 0 < λ 1 < · · · < λ n , and the W eyl funtions are W ( λ ) = − φ y ( y + n ; λ ) φ y y ( y + n ; λ ) = n X k =1 b k λ − λ k , Z ( λ ) = − φ ( y + n ; λ ) φ y y ( y + n ; λ ) = 1 γ λ + n X k =1 c k λ − λ k , γ = n X k =0 g k , (6.3) with all b k > 0 . They satisfy the iden tit y Z ( λ ) + Z ( − λ ) + W ( λ ) W ( − λ ) = 0 , (6.4) from whi h it follo ws, b y taking the residue at λ = λ k , that c k = n X m =1 b m b k λ m + λ k . (6.5) Th us Z ( λ ) is uniquely determined b y the funtion W ( λ ) and the onstan t γ . No w note that (6.2 ) is exatly the same kind of relation as (5.17 ), exept that the roles of g k and l k are in ter hanged, and the righ t endp oin t is alled y = y + n 14 instead of y = 1 . The denitions of the W eyl funtions (6.3 ) also orresp ond p erfetly to the W eyl funtions ( 5.18 ) for the dual ubi string. Therefore, all the results ab o v e are also true in the setting of the dual ubi string. The assumption that the n distanes l k and the n + 1 p oin t masses g k are all p ositiv e for the Neumann ubi string orresp onds of ourse to the requiremen t that the n p oin t masses g k and the n + 1 distanes l k are p ositiv e for the dual ubi string. The onstan t γ = P n k =0 g k in the term 1 /γ λ in ( 6.3) orresp onds to the onstan t 2 in the term 1 / 2 λ in ( 5.19 ), sine P n k =0 l k = 2 is the length of the in terv al − 1 < y < 1 . In summary: Theorem 6.1. Assume that al l p oint masses g k ar e p ositive. Then the disr ete dual ubi string of The or em 5.1 has nonne gative and simple sp e trum, with eigenvalues 0 = λ 0 < λ 1 < · · · < λ n , and its W eyl funtions (5.18 ) have p ositive r esidues and satisfy (6.4) and (6.5) . In p artiular, the se  ond W eyl funtion Z ( λ ) is uniquely determine d by the rst W eyl funtion W ( λ ) . 7 In v erse sp etral problem The in v erse sp etral problem for the disrete dual ubi string onsists in re- o v ering the p ositions and masses { y k , g k } n k =1 giv en the sp etral data onsisting of eigen v alues and residues { λ k , b k } n k =1 (or, equiv alen tly , giv en the rst W eyl funtion W ( λ ) ). The orresp onding problem for the Neumann-lik e ubi string w as solv ed in [23 ℄, and w e need only translate the results, as in Setion 6. See also [28 ℄ for more information ab out in v erse problems of this kind and [ 3℄ for the underlying theory of Cau h y biorthogonal p olynomials. T o b egin with, w e state the result in terms of the bimomen t determinan ts D ( ab ) m and D ′ m dened b elo w. Things will b eome more expliit in the next setion (Corollary 8.4 ), where the determinan ts are expressed diretly in terms of the λ k and b k . Denition 7.1. Supp ose µ is a measure on R + (the p ositiv e part of the real line) su h that its momen ts, β a = Z κ a dµ ( κ ) , (7.1) and its bimomen ts with resp et to the Cau h y k ernel K ( x, y ) = 1 / ( x + y ) , I ab = I ba = Z Z κ a λ b κ + λ dµ ( κ ) dµ ( λ ) , (7.2) are nite. F or m ≥ 1 , let D ( ab ) m denote the determinan t of the m × m bimomen t matrix whi h starts with I ab in the upp er left orner: D ( ab ) m =            I ab I a,b +1 . . . I a,b + m − 1 I a +1 ,b I a +1 ,b +1 . . . I a +1 ,b + m − 1 I a +2 ,b I a +2 ,b +1 . . . I a +2 ,b + m − 1 . . . . . . I a + m − 1 ,b I a + m − 1 ,b +1 . . . I a + m − 1 ,b + m − 1            = D ( ba ) m . (7.3) 15 Let D ( ab ) 0 = 1 , and D ( ab ) m = 0 for m < 0 . Similarly , for m ≥ 2 , let D ′ m denote the m × m determinan t D ′ m =            β 0 I 10 I 11 . . . I 1 ,m − 2 β 1 I 20 I 21 . . . I 2 ,m − 2 β 2 I 30 I 31 . . . I 3 ,m − 2 . . . . . . β m − 1 I m 0 I m 1 . . . I m,m − 2            , (7.4) and dene D ′ 1 = β 0 and D ′ m = 0 for m < 1 . Theorem 7.2. Given  onstants 0 < λ 1 < · · · < λ n and b 1 , . . . , b n > 0 , dene the sp e tr al me asur e µ = n X i =1 b i δ λ i , (7.5) and let I ab b e its bimoments, I ab = Z Z κ a λ b κ + λ dµ ( κ ) dµ ( λ ) = n X i =1 n X j =1 λ a i λ b j λ i + λ j b i b j . (7.6) Then the unique disr ete dual ubi string (with p ositive masses g k ) having the W eyl funtion W ( λ ) = n X k =1 b k λ − λ k = Z dµ ( κ ) λ − κ is given by y k ′ = D (00) k − 1 2 D (11) k − 1 D (00) k + 1 2 D (11) k − 1 , g k ′ = 2 D (00) k + 1 2 D (11) k − 1 D ′ k , (7.7) wher e k ′ = n + 1 − k for k = 0 , . . . , n + 1 . The distan es b etwe en the masses ar e given by l k ′ − 1 = y k ′ − y k ′ − 1 =  D (10) k  2  D (00) k + 1 2 D (11) k − 1  D (00) k +1 + 1 2 D (11) k  . (7.8) Pr o of. F or 0 ≤ k ≤ n , let a (2 k +1) ( λ ) b e the pro dut of the rst 2 k + 1 fators in (5.17 ) , a (2 k +1) ( λ ) = G ( l n , λ ) L ( g n ) G ( l n − 1 , λ ) L ( g n − 1 ) . . . . . . G ( l k ′ , λ ) L ( g k ′ ) G ( l k ′ − 1 , λ ) , (7.9) 16 where k ′ = n + 1 − k . By Lemma 4.1 and Theorem 4.2 in [ 23℄, the en tries in the rst olumn of a = a (2 k +1) ( λ ) ,   a 11 a 21 a 31   =:   b P P Q   , satisfy what in [23 ℄ w as alled a T yp e I appro ximation problem. This means that ( b P ( λ ) , P ( λ ) , Q ( λ )) are p olynomials in λ of degree k , k , k + 1 , resp etiv ely , satisfying the normalization onditions b P (0) = 1 , P (0) = 0 , Q (0) = 0 , the appro ximation onditions Q ( λ ) W ( λ ) + P ( λ ) = O (1) , Q ( λ ) Z ( λ ) + b P ( λ ) = O ( λ − 1 ) , as λ → ∞ , and the symmetry ondition Q ( λ ) Z ( − λ ) − P ( λ ) W ( − λ ) − b P ( λ ) = O ( λ − k − 1 ) , as λ → ∞ . A ording to Theorem 4.15 in [23 ℄, this determines ( b P , P, Q ) uniquely; in par- tiular, the o eien ts of a (2 k +1) 31 ( λ ) = Q ( λ ) = P k +1 i =1 q i λ i are giv en b y the nonsingular linear system        I 00 + 1 2 I 01 · · · I 0 k I 10 I 11 · · · I 1 k I 20 I 21 · · · I 2 k . . . . . . I k 0 I k 1 · · · I kk               q 1 q 2 q 3 . . . q k +1        = −        1 0 0 . . . 0        . (7.10) F rom (7.9 ) one nds that a (2 k +1) 31 ( λ ) = ( − λ )( l n + l n − 1 + · · · + l k ′ − 1 ) + . . . + ( − λ ) k +1  g 2 n 2 g 2 n − 1 2 . . . g 2 k ′ 2 l n l n − 1 . . . l k ′ − 1  , (7.11) and the lo w est and highest o eien ts are then extrated from ( 7.10 ) using Cramer's rule: − q 1 = D (11) k D (00) k +1 + 1 2 D (11) k = n X j = k ′ − 1 l j = 1 − y k ′ − 1 , ( − 1) k +1 q k +1 = D (10) k D (00) k +1 + 1 2 D (11) k =   n Y j = k ′ g 2 j l j 2   l k ′ − 1 . (7.12) The rst equation giv es a form ula for y k ′ − 1 righ t a w a y , and of ourse also for y k ′ (with 1 ≤ k ≤ n + 1 ) after ren um b ering. This form ula ( 7.7) for y k ′ holds also 17 for k = 0 , sine it giv es y 0 ′ = y n +1 = +1 b eause of the w a y D ( ab ) m is dened for m ≤ 0 . (That it indeed giv es y ( n +1) ′ = y 0 = − 1 when k = n + 1 is not as ob vious; this dep ends on D (00) n +1 b eing zero when the measure µ is supp orted on only n p oin ts. See [23 , App endix B℄.) Subtration giv es a form ula for l k ′ − 1 whi h simplies to (7.8 ) with the help of Lewis Carroll's iden tit y [24, Prop. 10℄ applied to the determinan t D (00) k +1 : D (00) k +1 D (11) k − 1 = D (00) k D (11) k − D (10) k D (01) k . (7.13) Finally , the seond form ula in ( 7.12 ) , divided b y the orresp onding form ula with k replaed b y k − 1 , giv es an expression for 1 2 g 2 k ′ l k ′ − 1 from whi h one obtains g k ′ =  D (00) k + 1 2 D (11) k − 1  s 2 D (10) k D (10) k − 1 . The form ula for g k ′ presen ted in (7.7 ) no w follo ws from the iden tit y ( D ′ k ) 2 = 2 D (10) k D (10) k − 1 and the p ositivit y of D ′ k , whi h are immediate onsequenes of (8.6) b elo w. (The determinan t iden tit y an also b e pro v ed diretly b y expanding D ′ k along the rst olumn, squaring, and using β i β j = I i +1 ,j + I i,j +1 .) Remark 7.3. W e tak e this opp ortunit y to orret a ouple of mistak es in [23 ℄: the form ula in Corollary 4.17 should read [ Q 3 k +2 ] = ( − 1 ) k +1 D k / A k +1 , and onsequen tly it should b e m n − k = D 2 k 2 A k +1 A k in (4.54). 8 Ev aluation of bimomen t determinan ts The aim of this setion is just to state some form ulas for the bimomen t determi- nan ts D ( ab ) m and D ′ m , tak en from [28 , Lemma 4.10℄ and [23 , App endix B℄. Quite a lot of notation is needed. Denition 8.1. F or k ≥ 1 , let t k = 1 k ! Z R k ∆( x ) 2 Γ( x ) dµ k ( x ) x 1 x 2 . . . x k , u k = 1 k ! Z R k ∆( x ) 2 Γ( x ) dµ k ( x ) , v k = 1 k ! Z R k ∆( x ) 2 Γ( x ) x 1 x 2 . . . x k dµ k ( x ) , (8.1) where ∆( x ) = ∆( x 1 , . . . , x k ) = Y i n .) W e an no w nally state the promised form ulas for the bimomen t determi- nan ts. Lemma 8.3. F or al l m , D (00) m =     t m u m − 1 t m +1 u m     2 m , D (11) m =     u m v m − 1 u m +1 v m     2 m , D (10) m = ( u m ) 2 2 m , D ′ m = u m u m − 1 2 m − 1 . (8.6) In the disr ete  ase when µ = n X k =1 b k δ λ k , this r e du es to D (00) m = Z m 2 m , D (11) m = W m 2 m , D (10) m = ( U m ) 2 2 m , D ′ m = U m U m − 1 2 m − 1 . (8.7) 19 Corollary 8.4. The solution to the inverse sp e tr al pr oblem for the disr ete dual ubi string (The or em 7.2 )  an b e expr esse d as y k ′ = Z k − W k − 1 Z k + W k − 1 , g k ′ = Z k + W k − 1 U k U k − 1 , (8.8) l k ′ − 1 = y k ′ − y k ′ − 1 = 2 ( U k ) 4 ( Z k + W k − 1 )( Z k +1 + W k ) . (8.9) The expression W k an b e ev aluated expliitly in terms of λ k and b k , al- though the form ula is somewhat in v olv ed [28 , Lemma 2.20℄: W k = X I ∈ ( [1 ,n ] k ) ∆ 4 I Γ 2 I λ I b 2 I + k X m =1 X I ∈ ( [1 ,n ] k − m ) J ∈ ( [1 ,n ] 2 m ) I ∩ J = ∅ b 2 I b J ( 2 m +1 ∆ 4 I ∆ 2 I ,J λ I ∪ J Γ I Γ I ∪ J X C ∪ D = J | C | = | D | = m min( C ) < min( D ) ∆ 2 C ∆ 2 D Γ C Γ D !) , (8.10) where ∆ 2 I ,J = Y i ∈ I ,j ∈ J ( λ i − λ j ) 2 . The orresp onding form ula for Z k is obtained b y replaing b i with b i /λ i ev erywhere. 9 The m ultip eak on solution In order to obtain the solution to the in v erse sp etral problem on the real line, whi h pro vides the m ultip eak on solution, w e merely ha v e to map the form ulas for the in terv al (Corollary 8.4 ) ba k to the line via the Liouville transformation (5.6 ). W e remind the reader that in this pap er w e primarily study the pure p eak on ase where it is assumed that all m k > 0 and also that x 1 < · · · < x n . This assumption guaran tees that the solutions are globally dened in time (Theo- rem 4.5 ) and, regarding the sp etral data, that all b k > 0 and 0 < λ 1 < · · · < λ n (Theorem 6.1 ). Details regarding mixed p eak on-an tip eak on solutions are left for future resear h, but w e p oin t out that sine the v elo it y ˙ x k = u ( x k ) 2 is alw a ys nonnegativ e, No vik o v an tip eak ons mo v e to the right just lik e p eak ons (unlik e the b -family (2.1 ), where pure p eak ons mo v e to the righ t and an tip eak ons to the left, if they are suien tly far apart). Nev ertheless, p eak ons and an tip eak ons ma y ollide after nite time also for the No vik o v equation, ausing division b y zero in the solution form ula for m k in (9.1) b elo w, and this breakdo wn leads to the usual subtle questions regarding on tin uation of the solution b ey ond the ollision. 20 Theorem 9.1. In the notation of Se tion 8 , the n -p e akon solution of Novikov's e quation is given by x k ′ = 1 2 ln Z k W k − 1 , m k ′ = p Z k W k − 1 U k U k − 1 , (9.1) wher e k ′ = n + 1 − k for k = 1 , . . . , n , and wher e the time evolution is given by b k ( t ) = b k (0) e t/λ k . (9.2) Pr o of. The in v erse of the o ordinate transformation ( 5.6 ) is x k = 1 2 ln 1 + y k 1 − y k , m k = g k p 1 − y 2 k 2 , whi h up on inserting (8.8 ) giv es (9.1 ) at one. The ev olution of b k omes from equation (4.23 ) . Example 9.2. The t w o-p eak on solution is x 1 = 1 2 ln Z 2 W 1 = 1 2 ln ( λ 1 − λ 2 ) 4 ( λ 1 + λ 2 ) 2 λ 1 λ 2 b 2 1 b 2 2 λ 1 b 2 1 + λ 2 b 2 2 + 4 λ 1 λ 2 λ 1 + λ 2 b 1 b 2 , x 2 = 1 2 ln Z 1 W 0 = 1 2 ln  b 2 1 λ 1 + b 2 2 λ 2 + 4 λ 1 + λ 2 b 1 b 2  , m 1 = p Z 2 W 1 U 2 U 1 =  ( λ 1 − λ 2 ) 4 b 2 1 b 2 2 ( λ 1 + λ 2 ) 2 λ 1 λ 2  λ 1 b 2 1 + λ 2 b 2 2 + 4 λ 1 λ 2 λ 1 + λ 2 b 1 b 2  1 / 2 ( λ 1 − λ 2 ) 2 b 1 b 2 λ 1 + λ 2 ( b 1 + b 2 ) =  λ 1 b 2 1 + λ 2 b 2 2 + 4 λ 1 λ 2 λ 1 + λ 2 b 1 b 2  1 / 2 p λ 1 λ 2 ( b 1 + b 2 ) , m 2 = p Z 1 W 0 U 1 U 0 =  b 2 1 λ 1 + b 2 2 λ 2 + 4 λ 1 + λ 2 b 1 b 2  1 / 2 b 1 + b 2 , (9.3) where the simpler of the t w o expressions for m 1 is obtained under the assumption that all sp etral data are p ositiv e, and therefore only an b e trusted in the pure p eak on ase. This w a y of writing the solution is simpler and more expliit than that found in [19 ℄. In order to translate (9.3) to the notation used there, write ( q k , p k ) instead of ( x k , m k ) , c k instead of 1 /λ k , and t 0 instead of ( λ − 1 1 − λ − 1 2 ) − 1 ln b 2 (0) b 2 (0) ; then tanh T = ( b 1 − b 2 ) / ( b 1 + b 2 ) and cosh − 2 T = 4 b 1 b 2 / ( b 1 + b 2 ) 2 , where T = 1 2 ( c 1 − c 2 )( t − t 0 ) . 21 Example 9.3. The three-p eak on solution is x 1 = 1 2 ln Z 3 W 2 , x 2 = 1 2 ln Z 2 W 1 , x 3 = 1 2 ln Z 1 W 0 , m 1 = p Z 3 W 2 U 3 U 2 , m 2 = p Z 2 W 1 U 2 U 1 , m 3 = p Z 1 W 0 U 1 U 0 , (9.4) where U 0 = W 0 = 1 , U 1 = b 1 + b 2 + b 3 , U 2 = Ψ 12 b 1 b 2 + Ψ 13 b 1 b 3 + Ψ 23 b 2 b 3 , U 3 = Ψ 123 b 1 b 2 b 3 , (9.5) W 1 = λ 1 b 2 1 + λ 2 b 2 2 + λ 3 b 2 3 + 4 λ 1 λ 2 λ 1 + λ 2 b 1 b 2 + 4 λ 1 λ 3 λ 1 + λ 3 b 1 b 3 + 4 λ 2 λ 3 λ 2 + λ 3 b 2 b 3 , W 2 = Ψ 2 12 λ 1 λ 2 b 2 1 b 2 2 + Ψ 2 13 λ 1 λ 3 b 2 1 b 2 3 + Ψ 2 23 λ 2 λ 3 b 2 2 b 2 3 + 4 Ψ 13 Ψ 23 λ 1 λ 2 λ 3 λ 1 + λ 2 b 1 b 2 b 2 3 + 4 Ψ 12 Ψ 23 λ 1 λ 2 λ 3 λ 1 + λ 3 b 1 b 2 2 b 3 + 4 Ψ 12 Ψ 13 λ 1 λ 2 λ 3 λ 2 + λ 3 b 2 1 b 2 b 3 , (9.6) Z 1 = b 2 1 λ 1 + b 2 2 λ 2 + b 2 3 λ 3 + 4 λ 1 + λ 2 b 1 b 2 + 4 λ 1 + λ 3 b 1 b 3 + 4 λ 2 + λ 3 b 2 b 3 , Z 2 = Ψ 2 12 λ 1 λ 2 b 2 1 b 2 2 + Ψ 2 13 λ 1 λ 3 b 2 1 b 2 3 + Ψ 2 23 λ 2 λ 3 b 2 2 b 2 3 + 4 Ψ 13 Ψ 23 ( λ 1 + λ 2 ) λ 3 b 1 b 2 b 2 3 + 4 Ψ 12 Ψ 23 ( λ 1 + λ 3 ) λ 2 b 1 b 2 2 b 3 + 4 Ψ 12 Ψ 13 ( λ 2 + λ 3 ) λ 1 b 2 1 b 2 b 3 , Z 3 = Ψ 2 123 λ 1 λ 2 λ 3 b 2 1 b 2 2 b 2 3 , (9.7) and Ψ 12 = ( λ 1 − λ 2 ) 2 λ 1 + λ 2 , Ψ 13 = ( λ 1 − λ 3 ) 2 λ 1 + λ 3 , Ψ 23 = ( λ 2 − λ 3 ) 2 λ 2 + λ 3 , Ψ 123 = ( λ 1 − λ 2 ) 2 ( λ 1 − λ 3 ) 2 ( λ 2 − λ 3 ) 2 ( λ 1 + λ 2 )( λ 1 + λ 3 )( λ 2 + λ 3 ) . (9.8) Theorem 9.4 (Asymptotis) . L et the eigenvalues b e numb er e d so that 0 < λ 1 < · · · < λ n . Then x k ( t ) ∼ t λ k + lo g b k (0) − 1 2 ln λ k + n X i = k +1 ln ( λ i − λ k ) 2 ( λ i + λ k ) λ i , as t → −∞ , x k ′ ( t ) ∼ t λ k + lo g b k (0) − 1 2 ln λ k + k − 1 X i =1 ln ( λ i − λ k ) 2 ( λ i + λ k ) λ i , as t → + ∞ , (9.9) 22 wher e k ′ = n + 1 − k . Mor e over, lim t →−∞ m k ( t ) = 1 √ λ k = lim t → + ∞ m k ′ ( t ) . (9.10) In wor ds: asymptoti al ly as t → ±∞ , the k th fastest p e akon has velo ity 1 /λ k and amplitude 1 / √ λ k . Pr o of. This is just a matter of iden tifying the dominan t terms; b 1 ( t ) = b 1 (0) e t/λ 1 gro ws m u h faster as t → + ∞ than b 2 ( t ) , whi h in turn gro ws m u h faster than b 3 ( t ) , et., and as t → − ∞ it is the other w a y around. Th us, for example, U k ∼ Ψ 12 ...k b 1 b 2 . . . b k as t → + ∞ . A similar analysis of W k and Z k leads qui kly to the stated form ulas. The only dierene ompared to the x k asymptotis for Degasp erisPro esi p eak ons [28 , Theorem 2.25℄ is that (9.9 ) on tains an additional term − 1 2 ln λ k . Sine this term anels in the subtration, the phase shifts for No vik o v p eak ons are exatly the same as for Degasp erisPro esi p eak ons [28 , Theorem 2.26℄: lim t →∞  x k ′ ( t ) − t λ k  − lim t →−∞  x k ( t ) − t λ k  = = k − 1 X i =1 log ( λ i − λ k ) 2 ( λ i + λ k ) λ i − n X i = k +1 log ( λ i − λ k ) 2 ( λ i + λ k ) λ i . (9.11) A Com binatorial results This app endix on tains some material related to the om binatorial struture of the onstan ts of motion H 1 , . . . , H n of the No vik o v p eak on ODEs; see Setion 4, and in partiular Theorem 4.2 . Reall that A ( λ ) = 1 − λH 1 + · · · + ( − λ ) n H n = det( I − λT P E P ) , where I is the n × n iden tit y matrix, and T , E , P are n × n matries dened b y T j k = 1 + sgn( j − k ) , E j k = e −| x j − x k | , and P = diag( m 1 , . . . , m n ) . The rst thing to pro v e is that the matrix T P E P is osillatory if all m k > 0 , whi h sho ws that the zeros of A ( λ ) are p ositiv e and simple. Then w e sho w ho w to easily ompute the minors of P E P , and nally w e pro v e the Canada Da y Theorem (Theorem 4.1 ) whi h implies that H k equals the sum of all k × k minors of P E P . A.1 Preliminaries In this setion w e ha v e olleted some fats ab out total p ositivit y [21 , 15 , 13 ℄ that will b e used b elo w. Denition A.1. If X is a matrix and I and J are index sets, the submatrix ( X ij ) i ∈ I ,j ∈ J will b e denoted b y X I J (or sometimes X I ,J ). The set of k -elemen t subsets of the in teger in terv al [1 , n ] = { 1 , 2 , . . . , n } will b e denoted  [1 ,n ] k  , and 23 elemen ts of su h a subset I will alw a ys b e assumed to b e n um b ered in asending order i 1 < · · · < i k . Denition A.2. A square matrix is said to b e total ly p ositive if all its minors of all orders are p ositiv e. It is alled total ly nonne gative if all its minors are nonnegativ e. A matrix is osil latory if it is totally nonnegativ e and some p o w er of it is totally p ositiv e. Theorem A.3. A l l eigenvalues of a total ly p ositive matrix ar e p ositive and of algebr ai multipliity one, and likewise for osil latory matri es. A l l eigenval- ues of a total ly nonne gative matrix ar e nonne gative, but in gener al of arbitr ary multipliity. Theorem A.4. The pr o dut of an osil latory matrix and a nonsingular total ly nonne gative matrix is osil latory. Denition A.5. A planar network (Γ , ω ) of order n is an ayli planar direted graph Γ with arro ws going from left to righ t, with n soures (v erties with outgoing arro ws only) on the left side, and with n sinks (v erties with inoming arro ws only) on the righ t side. The soures and sinks are n um b ered 1 to n , from b ottom to top, sa y . All other v erties ha v e at least one arro w oming in and at least one arro w going out. Ea h edge e of the graph Γ is assigned a salar w eigh t ω ( e ) . The weight of a direted path in Γ is the pro dut of all the w eigh ts of the edges of that path. The weighte d p ath matrix Ω(Γ , ω ) is the n × n matrix whose ( i, j ) en try Ω ij is the sum of the w eigh ts of the p ossible paths from soure i to sink j . The follo wing theorem w as diso v ered b y Lindström [ 25 ℄ and made famous b y Gessel and Viennot [ 16 ℄. A similar theorem also app eared earlier in the w ork of Karlin and MGregor on birth and death pro esses [22 ℄. Theorem A.6 (Lindström's Lemma) . L et I and J b e subsets of { 1 , . . . , n } with the same  ar dinality. The minor det Ω I J of the weighte d p ath matrix Ω(Γ , ω ) of a planar network is e qual to the sum of the weights of al l p ossible families of noninterse ting p aths (i.e., p aths having no verti es in  ommon)  onne ting the sour  es lab el le d by I to the sinks lab el le d by J . (The weight of a family of p aths is dene d as the pr o dut of the weights of the individual p aths.) Corollary A.7. If al l weights ω ( e ) ar e nonne gative, then the weighte d p ath matrix is total ly nonne gative. Remark A.8. Bew are that ha ving p ositive w eigh ts do es not in general imply total p ositivit y of the path matrix Ω , sine some minors det Ω I J ma y b e zero due to absene of nonin terseting path families from I to J , in whi h ase Ω is only totally nonnegativ e. A.2 Pro of that T P E P is osillatory The matrix T is the path matrix of the planar net w ork whose struture is illustrated b elo w for the ase n = 4 (with all edges, and therefore all paths and families of paths, ha ving unit w eigh t): 24 1 2 3 4 1 2 3 4 Indeed, there is learly one path from soure i to sink j if i = j , t w o paths if i > j , and none if i < j , and this agrees with T ij = 1 + sgn( i − j ) =      1 , i = j, 2 , i > j, 0 , i < j. Similarly one an  he k that the matrix P E P is the w eigh ted path matrix of the planar net w ork illustrated b elo w for the ase n = 5 (w e are assuming that x 1 < · · · < x n , so that E 12 E 23 = e x 1 − x 2 e x 2 − x 3 = E 13 , et.): 1 2 3 4 5 1 2 3 4 5 m 1 m 2 m 3 m 4 m 5 m 1 m 2 m 3 m 4 m 5 E 12 E 12 1 − E 2 12 E 23 E 23 1 − E 2 23 E 34 E 34 1 − E 2 34 E 45 E 45 1 − E 2 45 By Corollary A.7, b oth T and P E P are totally nonnegativ e (if all m k > 0 ). F urthermore, ( P E P ) N is the w eigh ted path matrix of the planar net w ork ob- tain b y onneting N opies of the net w ork for P E P in series, and if N is large enough, there is learly enough wiggle ro om in this net w ork to nd a nonin- terseting path family from an y soure set I to an y sink set J with | I | = | J | . Th us ( P E P ) N is totally p ositiv e for suien tly large N ; in other w ords, P E P is osillatory . (Another w a y to see this is to use a riterion [ 15 , Chapter I I, Theorem 10℄ whi h sa ys that a totally nonnegativ e matrix X is osillatory if and only if it is nonsingular and X ij > 0 for | i − j | = 1 .) Sine T is nonsingular, Theorem A.4 implies that T P E P is osillatory , whi h w as the rst thing w e w an ted to pro v e. 25 A.3 Minors of P E P Ha ving a planar net w ork for P E P mak es it easy to ompute its minors using Lindström's Lemma. Example A.9. Consider the onstan t of motion H 3 in the ase n = 6 . F or soures I = { 1 , 2 , 3 } and sinks J = { 1 , 2 , 3 } there is only one family of nonin terseting paths, namely the paths going straigh t aross. The w eigh ts of these paths are m 1 m 1 , m 2 (1 − E 2 12 ) m 2 and m 3 (1 − E 2 23 ) m 3 , and the total w eigh t of that family is therefore (1 − E 2 12 )(1 − E 2 23 ) m 2 1 m 2 2 m 2 3 , whi h will b e the rst term in H 3 . A similar term results whenev er I = J . F or instane, when I = J = { 1 , 2 , 4 } the paths starting at soures 1 and 2 m ust go straigh t aross, while the path from soure 4 to to sink 4 an go straigh t aross, or do wn to line 3 and up again. The on tributions from these t w o p ossible nonin terseting path families add up to m 1 m 1 · m 2 (1 − E 2 12 ) m 2 ·  m 4 (1 − E 2 34 ) m 4 + m 4 E 34 (1 − E 2 23 ) E 34 m 4  = (1 − E 2 12 )(1 − E 2 24 ) m 2 1 m 2 2 m 2 4 . F rom I = { 1 , 2 , 3 } to J = { 1 , 2 , 4 } there is one nonin terseting path family , and there is another one with the same w eigh t from I = { 1 , 2 , 4 } to J = { 1 , 2 , 3 } ; the t w o add up to the term 2(1 − E 2 12 )(1 − E 2 23 ) E 24 m 2 1 m 2 2 m 3 m 4 . Con tin uing lik e this, one nds that the t yp es of terms that app ear in H 3 are H 3 = (1 − E 2 12 )(1 − E 2 23 ) m 2 1 m 2 2 m 2 3 + . . . + 2 (1 − E 2 12 )(1 − E 2 23 ) E 34 m 2 1 m 2 2 m 3 m 4 + . . . + 4 (1 − E 2 12 )(1 − E 2 34 ) E 23 E 45 m 2 1 m 2 m 3 m 4 m 5 + . . . + 8 (1 − E 2 23 )(1 − E 2 45 ) E 12 E 34 E 56 m 1 m 2 m 3 m 4 m 5 m 6 . (A.1) The last term omes from the 8 p ossible nonin terseting path families from I = { i 1 , i 2 , i 3 } to J = { j 1 , j 2 , j 3 } where ( i 1 , j 1 ) = (1 , 2) or (2 , 1) , ( i 2 , j 2 ) = (3 , 4) or (4 , 3) , and ( i 3 , j 3 ) = (5 , 6) or (6 , 5) . Remark A.10. Alternativ ely , the m k an b e fatored out from an y minor of P E P , lea ving the orresp onding minor of E , whi h an b e omputed using a result from Gan tma her and Krein [ 15 , Setion I I.3.5℄, sine the matrix E is what they all a single-p air matrix . This means a symmetri n × n matrix X with en tries X ij = ( ψ i χ j , i ≤ j, ψ j χ i , i ≥ j. (A.2) The k × k minors of a single-pair matrix are giv en b y the follo wing rule: det X I J = 0 , unless I , J ∈  [1 ,n ] k  satisfy the ondition ( i 1 , j 1 ) < ( i 2 , j 2 ) < · · · < ( i k , j k ) , (A.3) 26 where the notation means that b oth n um b ers in one pair m ust b e less than b oth n um b ers in the follo wing pair; in this ase, det X I J = ψ α 1     χ β 1 χ α 2 ψ β 1 ψ α 2         χ β 2 χ α 3 ψ β 2 ψ α 3     . . .     χ β k − 1 χ α k ψ β k − 1 ψ α k     χ β k , (A.4) where ( α m , β m ) =  min( i m , j m ) , max( i m , j m )  . (A.5) In the ase of E w e ha v e ψ i = e x i and χ i = e − x i (assuming as usual that x 1 < · · · < x n ), and (A.4) b eomes det E I J = (1 − E 2 β 1 α 2 )(1 − E 2 β 2 α 3 ) . . . (1 − E 2 β k − 1 α k ) E α 1 β 1 E α 2 β 2 . . . E α k β k . (A.6) A.4 Pro of of the Canada Da y Theorem The result to b e pro v ed (Theorem 4.1 ) is that for an y symmetri n × n matrix X , the o eien t of s k in the p olynomial det( I + s T X ) equals the sum of all k × k minors of X : det( I + s T X ) = 1 + n X k =1    X I ∈ ( [1 ,n ] k ) X J ∈ ( [1 ,n ] k ) det X I J    s k . (A.7) W e start from the elemen tary fat that for an y matrix Y , the o eien ts in its  harateristi p olynomial are giv en b y the sums of the prinip al minors, det( I + s Y ) = 1 + n X k =1    X J ∈ ( [1 ,n ] k ) det Y J J    s k . Applying this to Y = T X and omputing the minors of T X using the Cau h y Binet form ula [14 , Ch. I,  2℄ det( T X ) AB = X I ∈ ( [1 ,n ] k ) det T AI det X I B , for A, B ∈  [1 ,n ] k  , (A.8) w e nd that det( I + s T X ) = 1 + n X k =1    X I ∈ ( [1 ,n ] k ) X J ∈ ( [1 ,n ] k ) det T J I det X I J    s k . Comparing this to (A.7), it is lear that what w e need to sho w is that, for an y k , X I ∈ ( [1 ,n ] k ) X J ∈ ( [1 ,n ] k ) det T J I det X I J = X I ∈ ( [1 ,n ] k ) X J ∈ ( [1 ,n ] k ) det X I J . (A.9) The rst thing to do is alulate the minors det T J I . 27 Denition A.11. Giv en I , J ∈  [1 ,n ] k  , the set I is said to interla e with the set J , denoted I ≤ J , if i 1 ≤ j 1 ≤ i 2 ≤ j 2 ≤ . . . ≤ i k ≤ j k . (A.10) If all the inequalities are strit, then I is said to stritly interla e with J , in whi h ase w e write I < J . If I ≤ J , then I ′ and J ′ denote the stritly in terlaing subsets (p ossibly empt y) I ′ = I \ ( I ∩ J ) , J ′ = J \ ( I ∩ J ) , (A.11) whose ardinalit y (p ossibly zero) will b e denoted b y p ( I , J ) = | I ′ | = | J ′ | . (A.12) Lemma A.12. F or I , J ∈  [1 ,n ] k  , the  orr esp onding k × k minor of T is det T J I = ( 2 p ( I ,J ) , if I ≤ J , 0 , otherwise . (A.13) Pr o of. W e will use Lindström's Lemma (Theorem A.6 ) on the planar net w ork for T giv en in Setion A.2 ab o v e; the minor det T J I equals the total n um b er of families of nonin terseting paths onneting the soure no des (on the left) indexed b y J to the sink no des (on the righ t) indexed b y I . The pro of pro eeds b y indution on the size n of T . The laim is trivially true for n = 1 . Consider an arbitrary n > 1 , and supp ose the laim is true for size n − 1 . If neither I nor J on tain n , the laim follo ws immediately from the indution h yp othesis, and lik ewise if I and J b oth on tain n , b eause there is only one path onneting soure n to sink n . If I on tains n but J do es not, then det T J I = 0 b eause there are no paths going up w ard; this agrees with the laim, sine in this ase I do es not in terlae with J . The only remaining ase is therefore J = J 1 ∪ { n } , I = I 1 ∪ { i k } , with i k < n . But then det T J I = det T J 1 I 1 ×      2 , if j k − 1 < i k , 1 , if j k − 1 = i k , 0 , if j k − 1 > i k , dep ending on whether the path onneting soure n with sink i k has to ross the j k − 1 lev el; if it do es not, there are t w o a v ailable paths, if it do es, there is only one a v ailable path pro vided j k − 1 = i n , otherwise the path in tersets the path oming from soure j k − 1 . In the last instane, I do es not in terlae with J , while in the other t w o I ≤ J if and only if I 1 ≤ J 1 , th us pro ving the laim. A ording to this lemma, the struture of (A.9 ) (whi h is what w e w an t to pro v e) is X I ,J ∈ ( [1 ,n ] k ) I ≤ J 2 p ( I ,J ) det X I J = X A,B ∈ ( [1 ,n ] k ) det X AB , (A.14) 28 and w e m ust sho w that those terms det X I J that o ur more than one on the left-hand side exatly omp ensate for those that are absen t. This will follo w from another te hnial lemma: Lemma A.13 (Relations b et w een k × k minors of a symmetri matrix) . Supp ose I , J ∈  [1 ,n ] k  and I ≤ J . Then, for any symmetri n × n matrix X , X A,B ∈ ( I ∪ J k ) A ∩ B = I ∩ J det X AB = 2 p ( I ,J ) det X I J . (A.15) Before pro ving Lemma A.13 , w e will use it to nish the pro of of the main theorem. The t w o lemmas ab o v e sho w that the sum on the left-hand side of (A.14 ) equals X I ,J ∈ ( [1 ,n ] k ) I ≤ J 2 p ( I ,J ) det X I J = X I ,J ∈ ( [1 ,n ] k ) I ≤ J      X A,B ∈ ( I ∪ J k ) A ∩ B = I ∩ J det X AB      , (A.16) whi h in turn equals the sum on the righ t-hand side of (A.14), X A,B ∈ ( [1 ,n ] k ) det X AB . (A.17) Th us (A.14) holds, and the theorem is pro v ed. The nal step from (A.16 ) to (A.17 ) is justied b y the observ ation that an y giv en pair ( A, B ) of the t yp e summed o v er in (A.17) app ears exatly one in the righ t-hand side of (A.16), namely for the sets I and J dened as follo ws. Let M = A ∩ B , A ′ = A \ M , B ′ = B \ M , and let p ≥ 0 b e the ardinalit y of the disjoin t sets A ′ and B ′ (they are empt y sets if p = 0 ). Then dene I ′ and J ′ b y en umerating the 2 p elemen ts of A ′ ∪ B ′ in the stritly in terlaing order I ′ < J ′ , and let I = M ∪ I ′ and J = M ∪ J ′ . Con v ersely , no other terms than these app ear in the righ t hand side of (A.16 ) , and it is therefore indeed equal to ( A.17). Pr o of of L emma A.13 . The sets I ≤ J and I ′ < J ′ (as in Denition A.11 ), with | I | = | J | = k , | I ′ | = | J ′ | = p ( I , J ) = p , will b e xed throughout the pro of, and for on v eniene w e also in tro due M = I ∩ J and U = I ∪ J , with | M | = k − p and | U | = k + p . W e an assume that p > 0 , sine the ase p = 0 is trivial; it o urs when I = J , and then b oth sides of (A.15 ) simply equal det X I I . The set U \ M onsists of the 2 p n um b ers whi h b elong alternatingly to I ′ and to J ′ . The sum (A.15) runs o v er all pairs of sets ( A, B ) obtained b y splitting these 2 p n um b ers in to t w o disjoin t p -sets A ′ and B ′ in an arbitrary w a y and letting A = M ∪ A ′ and B = M ∪ B ′ . W rite Q for this set; that is, Q denotes 29 the set of pairs ( A, B ) ∈  [1 ,n ] k  ×  [1 ,n ] k  su h that A ∪ B = U and A ∩ B = M . After expanding det X AB , w e an then write the left-hand side of ( A.15) as X (( A,B ) ,σ ) ∈Q×S k ( − 1) σ X a 1 b σ (1) X a 2 b σ (2) . . . X a k b σ ( k ) , (A.18) where S k is the group of p erm utations of { 1 , 2 , . . . , k } , and ( − 1) σ denotes the sign of the p erm utation σ . F or ea h (( A, B ) , σ ) ∈ Q × S k , w e let A ′ = A \ M and B ′ = B \ M , and set up a ( σ -dep enden t) bijetion b et w een A ′ and B ′ as follo ws: a ′ ∈ A ′ is paired up with b ′ ∈ B ′ if and only if the pro dut X a 1 b σ (1) X a 2 b σ (2) . . . X a k b σ ( k ) on tains either the fator X a ′ b ′ or a sequene of fators X a ′ r , X r s , . . . , X tb ′ where r , s, . . . , t ∈ M . Let us sa y that a ′ and b ′ are linke d if they are paired up in this manner. A link ed pair ( a ′ , b ′ ) ∈ A ′ × B ′ will b e alled hostile if ( a ′ , b ′ ) b elongs to I ′ × I ′ or J ′ × J ′ , and friend ly if ( a ′ , b ′ ) b elongs to I ′ × J ′ or J ′ × I ′ . T o ea h term in the sum (A.18) there will th us orresp ond p su h link ed pairs, and what w e will sho w is that the terms on taining at least one hostile pair will anel out, and that the remaining terms (with all friendly pairs) will add up to the righ t-hand side of (A.15). Next w e dene what w e mean b y ipping a link ed pair ( a ′ , b ′ ) . This means that, in the pro dut X a 1 b σ (1) X a 2 b σ (2) . . . X a k b σ ( k ) , those fators X a ′ r X r s . . . X tb ′ that link a ′ to b ′ are replaed b y X b ′ t . . . X sr X r a ′ , with all the indies in rev ersed order. (When the linking in v olv es just a single fator X a ′ b ′ , ipping means replaing it b y X b ′ a ′ .) Sine the matrix X is symmetri, this do es not  hange the v alue of the pro dut, but it  hanges the w a y it is indexed. The n um b er a ′ whi h used to b e in the rst slot (in X a ′ r ) is no w in the seond slot (in X r a ′ ), and vie v ersa for b ′ . The onneting indies r , s, . . . , t ∈ M do not on tribute to an y  hange in the indexing sets, sine, for example, the r in X a ′ r is mo v ed from the seond slot to the rst, while the other r in X r s is mo v ed from the rst to the seond. The new pro dut (the result of the ipping) is therefore indexed b y the sets  A \ { a ′ }  ∪ { b ′ } =: e A = { e a 1 < · · · < e a k } and  B \ { b ′ }  ∪ { a ′ } =: e B = { e b 1 < · · · < e b k } resp etiv ely , and after reordering the fators so that the rst indies ome in asending order, it an b e written X e a 1 e b e σ ( 1) X e a 2 e b e σ ( 2) . . . X e a k e b e σ ( k ) for some uniquely determined p erm utation e σ ∈ S k . Flipping a giv en pair th us tak es (( A, B ) , σ ) to (( e A, e B ) , e σ ) . This op eration is in v ertible, with in v erse giv en b y simply ipping the same pair again, no w view ed as a pair ( b ′ , a ′ ) ∈ (( e A ) ′ , ( e B ) ′ ) link ed via the indies t, . . . , s, r . Beause of the symmetry of the matrix X , the term in (A.18) orresp onding to (( e A, e B ) , e σ ) is equal to the term orresp onding 30 to (( A, B ) , σ ) , exept p ossibly for a dierene in sign, dep ending on whether the signs of σ and e σ ome out equal or not: ( − 1) e σ X e a 1 e b e σ (1) X e a 2 e b e σ ( 2) . . . X e a k e b e σ ( k ) = ± ( − 1) σ X a 1 b σ (1) X a 2 b σ (2) . . . X a k b σ ( k ) . W e will sho w b elo w that the p ermutation e σ has the same sign as σ when a friend ly p air is ipp e d, and the opp osite sign when a hostile p air is ipp e d . T aking this for gran ted for the momen t, divide the set Q × S k in to the t w o sets ( Q × S k ) hostile , onsisting of those (( A, B ) , σ ) for whi h at least one link ed pair is hostile, and ( Q × S k ) friendl y , onsisting of those (( A, B ) , σ ) for whi h all p link ed pairs are friendly . The mapping ip that out of all hostile pairs ( a ′ , b ′ ) for whi h min( a ′ , b ′ ) is smallest is an in v olution on ( Q × S k ) hostile that pairs up ea h term with a partner term that is equal exept for ha ving the opp osite sign (sine it is a hostile pair that is ipp ed). Consequen tly these terms anel out, and the on tribution from ( Q × S k ) hostile to (A.18) is zero. The sum therefore redues to X (( A,B ) ,σ ) ∈ ( Q×S k ) friendly ( − 1) σ X a 1 b σ (1) X a 2 b σ (2) . . . X a k b σ ( k ) . (A.19) No w equip the set ( Q × S k ) friendl y with an equiv alene relation; (( e A, e B ) , e σ ) and (( A, B ) , σ ) are equiv alen t if one an go from one to another b y ipping friendly pairs. Ea h equiv alene lass on tains 2 p elemen ts, sine ea h of the p friendly pairs an b elong to either I ′ × J ′ or J ′ × I ′ . Moreo v er, the terms orresp onding to the elemen ts in one equiv alene lass are all equal (inluding the sign, sine only friendly pairs are ipp ed), and ea h lass has a anonial represen tativ e with all link ed pairs b elonging to I ′ × J ′ , ( − 1) σ X i 1 j σ (1) X i 2 j σ (2) . . . X i k j σ ( k ) , where the p erm utation σ is uniquely determined b y the equiv alene lass (and vie v ersa). Th us (A.19 ) b eomes 2 p X σ ∈S k ( − 1) σ X i 1 j σ (1) X i 2 j σ (2) . . . X i k j σ ( k ) = 2 p det X I J , (A.20) whi h is what w e w an ted to pro v e. T o nish the pro of, it no w remains to demonstrate the rule that e σ has the same (opp osite) sign as σ when a friendly (hostile) pair is ipp ed. T o this end, w e will represen t (( A, B ) , σ ) with a bipartite graph, with the n um b ers in U = A ∪ B (in inreasing order) as no des b oth on the left and on the righ t, and the left no des a i ∈ A onneted b y edges to the orresp onding righ t no des b σ ( i ) ∈ B . The sign of σ will then b e equal to ( − 1) c , where c is the rossing n um b er of the graph. As an aid in explaining the ideas w e will use the follo wing example with U = [1 , 8] , where the no des in M = A ∩ B are mark ed with diamonds, and the no des in A ′ and B ′ are mark ed with irles: 31 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 A = { 2 , 3 , 4 , 5 , 6 , 8 } = { 2 , 4 , 5 , 8 } ∪ { 3 , 6 } = M ∪ A ′ B = { 1 , 2 , 4 , 5 , 7 , 8 } = { 2 , 4 , 5 , 8 } ∪ { 1 , 7 } = M ∪ B ′ Clearly , A ′ ∪ B ′ = { 3 , 6 } ∪ { 1 , 7 } = { 1 , 3 , 6 , 7 } = { i ′ 1 < j ′ 1 < i ′ 2 < j ′ 2 } , so that I ′ = { i ′ 1 , i ′ 2 } = { 1 , 6 } and J ′ = { j ′ 1 , j ′ 2 } = { 3 , 7 } . Consequen tly , I = M ∪ I ′ = { 1 , 2 , 4 , 5 , 6 , 8 } and J = M ∪ J ′ = { 2 , 3 , 4 , 5 , 6 , 7 } . The  hosen p erm utation is σ (123456) = 632415 , where the notation means that σ (1) = 6 , σ (2) = 3 , et.; for example, the latter equalit y omes from the seond smallest n um b er a 2 in A b eing onneted to the third smallest n um b er b 3 in B . There are 9 rossings, so σ is an o dd p erm utation, and this graph therefore represen ts the term − X 28 X 34 X 42 X 55 X 61 X 87 , app earing with a min us sign in the sum (A.18). The link ed pairs ( a ′ , b ′ ) ∈ A ′ × B ′ are (6 , 1) (diretly link ed) and (3 , 7) (link ed via 4 , 2 , 8 ∈ M ). Both pairs are hostile, sine (6 , 1) ∈ I ′ × I ′ and (3 , 7) ∈ J ′ × J ′ . W e will illustrate in detail what happ ens when the pair (3 , 7) is ipp ed. The ip is eeted b y replaing the fators X 34 X 42 X 28 X 87 b y X 78 X 82 X 24 X 43 and sorting the resulting pro dut so that the rst indies ome in asend- ing order; this giv es X 24 X 43 X 55 X 61 X 78 X 82 . Th us e A = { 2 , 4 , 5 , 6 , 7 , 8 } , e B = { 1 , 2 , 3 , 4 , 5 , 8 } , and e σ (123456 ) = 4 35162 (an ev en p erm utation). In terms of the graph, the no des that are in v olv ed in the ip are, on b oth sides, { 2 , 3 , 4 , 7 , 8 } (the t w o no des in the pair b eing ipp ed, plus the no des linking them), and the edges in v olv ed are { 34 , 42 , 2 8 , 8 7 } , whi h get  hanged in to { 43 , 24 , 8 2 , 7 8 } . In other w ords, the ip orresp onds to this ative sub gr aph b eing mirror reeted aross the en tral v ertial line. T o understand ho w the pro ess of reetion aets the rossing n um b er, it an b e brok en do wn in to t w o steps, as follo ws. On the left, no de 7 is uno upied to b egin with, so w e an  hange the edge 87 to 77 . This frees no de 8 on the left, so that w e an  hange the edge 28 to 88 , whi h frees no de 2 on the left. (Think of this edge as a rubb er band onneted at one end to no de 8 on the righ t; w e're disonneting its other end from no de 32 2 on the left and sliding it past all the other no des do wn to no de 8 on the left. Ob viously the rossing n um b er inreases or dereases b y one ev ery time w e slide past a no de that has an edge atta hed to it.) Con tin uing lik e this, w e get the result illustrated in Step 1 b elo w; the edges  hanged are 87 → 77 , 28 → 88 , 42 → 22 , 34 → 44 . 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 In termediate stage (after Step 1) 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 Result of the ip (after Step 2) In Step 2, w e w ork similarly on the righ t-hand side: no de 3 is uno upied to b egin with, so w e an  hange edge 44 to 43 , and so on. The list of edge mo v es is 44 → 43 , 22 → 24 , 88 → 82 , 77 → 78 . (In the graph on the righ t w e see that the rossing n um b er after the ip is 8 , v erifying the laim that e σ is an ev en p erm utation.) W e need to k eep tra k of the  hanges in the rossing n um b er aused b y sliding ativ e edges past no des that ha v e edges atta hed to them. This is most easily done b y follo wing the dotted lines in the gures, and oun ting whether the no des that are mark ed (with irles and diamonds) are passed an ev en or an o dd n um b er of times. Ho w ev er, sine the ativ e subgraph simply gets reeted, the rossings among its edges will b e the same b efore and after the ip, so w e need in fat only oun t ho w man y times w e pass a p assive mark ed no de. (The passiv e no des in the example are { 1 , 5 , 6 } .) If a passiv e no de b elonging to M is passed in Step 1, then it is passed the same n um b er of times in Step 2 as w ell, sine the no des in M are mark ed b oth on the left and on the righ t. Therefore they do not aet the parit y of the rossing n um b er either, and w e an ignore the no des mark ed with diamonds, and only lo ok at the p assive ir le d no des (all the no des in A ′ and B ′ exept for the t w o ativ e no des that are b eing ipp ed). P assiv e no des b elonging to A ′ are oun ted only in Step 1 and passiv e no des in B ′ only in Step 2; they get oun ted an o dd n um b er of times if they lie b etwe en the t w o ipp ed no des (lik e no de 6 in the example, oun ted one), and an even 33 n um b er of times otherwise (lik e no de 1 , nev er oun ted). Consequen tly , what determines whether the parit y of the rossing n um b er  hanges is the n um b er of no des b etwe en the ipp ed ones that b elong to A ′ ∪ B ′ = I ′ ∪ J ′ . And for a friendly pair, this n um b er is ev en, while for a hostile pair, it is o dd. This sho ws that the rossing n um b er k eeps its parit y (so that ( − 1) σ = ( − 1) e σ ) when a friendly pair is ipp ed, and the opp osite when a hostile pair is ipp ed. The pro of is nally omplete. B V eriation of the Lax pair for p eak ons The purp ose of this app endix is to arefully v erify that the Lax pair form ulation (4.1 )(4.2 ) of the No vik o v equation really is v alid for the lass of distributional solutions that w e are onsidering. This is not at all ob vious, as should b e lear from the omputations b elo w. B.1 Preliminaries W e will need to b e more preise regarding the notation here than in the main text. A w ord of w arning righ t a w a y: our notation for deriv ativ es here will dier from that used in the rest of the pap er (where subsripts should b e in terpreted as distributional deriv ativ es). T o b egin with, giv en n smo oth funtions x = x k ( t ) su h that x 1 ( t ) < · · · < x n ( t ) , let x 0 ( t ) = −∞ and x n +1 ( t ) = + ∞ , and let Ω k (for k = 0 , . . . , n ) denote the region x k ( t ) < x < x k +1 ( t ) in the ( x, t ) plane. Our omputations will deal with a lass that w e denote P C ∞ , onsisting of pieewise smo oth funtions f ( x, t ) su h that the restrition of f to ea h region Ω k is (the restrition to Ω k of ) a smo oth funtion f ( k ) ( x, t ) dened on an op en neigh b ourho o d of Ω k (so that f ( k ) and its partial deriv ativ es mak e sense on the urv es x = x k ( t ) ). F or ea h xed t , the funtion f ( · , t ) denes a regular distribution T f in the lass D ′ ( R ) , dep ending parametrially on t (and written T f ( t ) where needed). After ha ving made lear exatly what is mean t, w e will mostly b e less strit, and write f instead of T f for simpliit y . The v alues of f on the urv es x = x k ( t ) need not b e dened; the funtion de- nes the same distribution T f no matter what v alues are assigned to f ( x k ( t ) , t ) . But our assumptions imply that the left and righ t limits of f exist, and (sup- pressing the time dep endene) they will b e denoted b y f ( x − k ) := f ( k − 1) ( x k ) and f ( x + k ) := f ( k ) ( x k ) , resp etiv ely . The jump and the a v erage of f at x k will b e denoted b y  f ( x k )  := f ( x + k ) − f ( x − k ) and  f ( x k )  := f ( x + k ) + f ( x − k ) 2 , (B.1) resp etiv ely . They satisfy the pro dut rules  f g  =  f  g  +  f  g  ,  f g  =  f  g  + 1 4  f  g  . (B.2) 34 W e will use subsripts to denote partial deriv ativ es in the lassial sense, so that (for example) f x denotes the pieewise smo oth funtion whose restrition to Ω k is giv en b y ∂ f ( k ) /∂ x (and whose v alues at x = x k ( t ) are in general undened). On the other hand, D x will denote the distributional deriv ativ e, whi h in addition pi ks up Dira delta on tributions from jump dison tin uities of f at the urv es x = x k ( t ) . That is, D x T f = T f x + P n k =1  f ( x k )  δ x k , or, in less strit notation, D x f = f x + n X k =1  f ( x k )  δ x k . (B.3) The time deriv ativ e D t is dened as a limit in D ′ ( R ) , D t T f ( t ) = lim h → 0 T f ( t + h ) − T f ( t ) h , (B.4) and it omm utes with D x b y the on tin uit y of D x on D ′ ( R ) . F or our lass P C ∞ of pieewise smo oth funtions, w e ha v e D t T f = T f t − P n k =1 ˙ x k  f ( x k )  δ x k , or simply D t f = f t − n X k =1 ˙ x k  f ( x k )  δ x k , (B.5) where ˙ x k = dx k /dt . W e also note that d dt f ( x ± k ( t ) , t ) = f x ( x ± k ( t ) , t ) ˙ x k ( t ) + f t ( x ± k ( t ) , t ) , whi h giv es d dt  f ( x k )  =  f x ( x k )  ˙ x k +  f t ( x k )  , d dt  f ( x k )  =  f x ( x k )  ˙ x k +  f t ( x k )  . (B.6) B.2 The problem of m ultipliation If the funtion f is on tin uous at x = x k , then the Dira delta at x k an b e m ultiplied b y the orresp onding distribution T f aording to the w ell-kno wn form ula T f δ x = f ( x k ) δ x k . (B.7) But b elo w w e will ha v e to onsider this pro dut for funtions in the lass P C ∞ desrib ed ab o v e, where the v alue f ( x k ) is not dened. It will turn out that in the presen t on text, the righ t thing to do is to use the a v erage v alue of f at the jump, and th us dene T f δ x :=  f ( x k )  δ x k . Ho w ev er, sine w e w an t this to b e a onsequene of the analysis, rather than an a priori assumption, w e will, to b egin with, just assign a h yp othetial v alue f ( x k ) and use that v alue in (B.7 ) . This assignmen t is justied in the presen t on text, as w e will see b elo w. Ho w ev er, w e are not laiming that this addresses an y of the deep er issues; for example, this assignmen t do es not resp et the pro dut struture of pieewise on tin uous funtions. See [32 , Ch. 5℄ for more information ab out the strutural problems asso iated with an y attempt to dene a pro dut of distributions in D ′ ( R ) . 35 B.3 Distributional Lax pair P eak on solutions u ( x, t ) = n X k =1 m k ( t ) e −| x − x k ( t ) | (B.8) b elong to the pieewise smo oth lass P C ∞ . They are on tin uous and satisfy D x u = u x = n X k =1 m k sgn( x k − x ) e −| x − x k | , D 2 x u = D x ( u x ) = u xx + n X k =1  u x ( x k )  δ x k = u + n X k =1 ( − 2 m k ) δ x k , whi h implies m := u − D 2 x u = 2 n X k =1 m k δ x k . (B.9) The Lax pair (4.1 )(4.2 ) will in v olv e the funtions u and D x u , as w ell as the purely singular distribution m . W e will tak e ψ 1 , ψ 2 , ψ 3 to b e funtions in P C ∞ , and separate the regular (funtion) part from the singular (Dira delta) part. The form ulation obtained in this w a y reads D x Ψ = b L Ψ , D t Ψ = b A Ψ , (B.10) where Ψ = ( ψ 1 , ψ 2 , ψ 3 ) t , b L = L + 2 z n X k =1 m k δ x k ! N , L =   0 0 1 0 0 0 1 0 0   , N =   0 1 0 0 0 1 0 0 0   , (B.11) and b A = A − 2 z n X k =1 m k u ( x k ) 2 δ x k ! N , A =   − uu x u x /z u 2 x u/z − 1 /z 2 − u x /z − u 2 u/z uu x   . (B.12) Note that (B.10 ) in v olv es m ultiplying N Ψ = ( ψ 2 , ψ 3 , 0) b y δ x k , and some v alue ψ 2 ( x k ) m ust b e assigned in order for this to b e w ell-dened (w e will so on see that ψ 3 m ust b e on tin uous and therefore it is only ψ 2 that presen ts an y problems). Theorem B.1. Pr ovide d that the pr o dut mψ 2 is dene d using the aver age value ψ 2 ( x k ) :=  ψ 2 ( x k )  at the jumps, mψ 2 := 2 n X k =1 m k  ψ 2 ( x k )  δ x k , (B.13) the fol lowing statement holds. With u and m given by (B.8)  (B.9) , and with Ψ ∈ P C ∞ , the L ax p air (B.10 )  (B.12) satises the  omp atibility  ondition D t D x Ψ = D x D t Ψ if and only if the p e akon ODEs (3.4) ar e satise d: ˙ x k = u ( x k ) 2 and ˙ m k = − m k u ( x k )  u x ( x k )  . 36 Pr o of. F or simpliit y , w e will write just P instead of P n k =1 . Iden tifying o- eien ts of δ x k in the t w o Lax equations (B.10 ) immediately giv es  Ψ( x k )  = 2 z m k N Ψ ( x k ) and − ˙ x k  Ψ( x k )  = − 2 z m k u ( x k ) 2 N Ψ ( x k ) , resp etiv ely . Th us, [ ψ 3 ( x k )] = 0 (in other w ords, ψ 3 is on tin uous) and ˙ x k = u ( x k ) 2 . Next w e ompute the deriv ativ es of (B.10 ): D t ( D x Ψ) = D t ( L Ψ + 2 z  X m k δ x k  N Ψ ) = L ( b A Ψ) + 2 z N X d dt  m k Ψ( x k )  δ x k − 2 z N X m k Ψ( x k ) ˙ x k δ ′ x k , D x ( D t Ψ) = D x ( A Ψ − 2 z  X m k u ( x k ) 2 δ x k  N Ψ ) = ( A Ψ) x + X  A Ψ( x k )  δ x k − 2 z N X m k Ψ( x k ) u ( x k ) 2 δ ′ x k . The regular part of (B.10 ) giv es Ψ x = L Ψ , so that ( A Ψ) x = A x Ψ + AL Ψ , and it is easily v eried that LA = A x + AL holds iden tially (sine u xx = u ). This implies that the regular parts of the t w o expressions ab o v e are equal, and the terms in v olving δ ′ x k are also equal sine ˙ x k = u ( x k ) 2 . Therefore the ompatibilit y ondition D t ( D x Ψ) = D x ( D t Ψ) redues to an equalit y b et w een the o eien ts of δ x k , − 2 z m k u ( x k ) 2 LN Ψ( x k ) + 2 z N d dt  m k Ψ( x k )  =  A Ψ( x k )  . (B.14) Using the pro dut rule (B.2), the expression for  Ψ( x k )  ab o v e, and  u x ( x k )  = − 2 m k , w e nd that the righ t-hand side of (B.14 ) equals  A ( x k )  2 z m k N Ψ ( x k ) +  A ( x k )  Ψ( x k )  = 2 z m k 0 − u  u x   u x  /z 0 u/z − 1 /z 2 0 − u 2 u/z ! x k Ψ( x k ) + 2 m k u − 1 /z − 2  u x  0 0 1 /z 0 0 − u ! x k  Ψ( x k )  . (B.15) The (3,2) en try − u 2 in the matrix in the rst term will anel against the whole rst term on the left-hand side of (B.14 ) , sine the only nonzero en try of LN is ( LN ) 32 = 1 . Th us (B.14 ) is equiv alen t to ˙ m k N Ψ ( x k ) + m k N d dt Ψ( x k ) = m k 0 − u  u x   u x  /z 0 u/z − 1 /z 2 0 0 u/z ! x k Ψ( x k ) + m k u/z − 1 /z 2 − 2  u x  /z 0 0 1 /z 2 0 0 − u/z ! x k  Ψ( x k )  . (B.16) T o mak e it lear ho w the assumption ( B.13 ) en ters the pro of, w e w an t to a v oid assigning a v alue to ψ 2 ( x k ) for as long as p ossible. Therefore w e an't ompute d dt Ψ( x k ) quite y et. But  Ψ( x k )  is w ell-dened, and its time deriv ativ e an b e 37 omputed using Ψ x = L Ψ and Ψ t = A Ψ in (B.6 ) : N d dt  Ψ( x k )  = N  L Ψ( x k )  ˙ x k + N  A Ψ( x k )  = N  Lu ( x k ) 2 +  A ( x k )    Ψ( x k )  + N 1 4  A ( x k )  Ψ( x k )  = u/z − 1 /z 2 −  u x  /z 0 u/z u  u x  0 0 0 ! x k  Ψ( x k )  + 1 4 N  A ( x k )  N | {z } =0 2 z m k Ψ( x k ) . A bit of manipulation using this result, as w ell as  ψ 3  ( x k ) = ψ 3 ( x k ) , sho ws that the ompatibilit y ondition (B.16 ) an b e written as m k N d dt  Ψ( x k ) −  Ψ( x k )   +  ˙ m k + m k u ( x k )  u x ( x k )   N Ψ ( x ) = m k  0 0 0 0 u/z 0 0 0 0  x k  Ψ( x k ) −  Ψ( x k )   (B.17) The third ro w is zero, and the rst t w o ro ws sa y that  ˙ m k + m k u ( x k )  u x ( x k )   ψ 2 ( x k ) = − m k d dt  ψ 2 ( x k ) −  ψ 2 ( x k )   ,  ˙ m k + m k u ( x k )  u x ( x k )   ψ 3 ( x k ) = 1 z m k u ( x k )  ψ 2 ( x k ) −  ψ 2 ( x k )   . A t this p oin t w e  ho ose to assign ψ 2 ( x k ) :=  ψ 2  ( x k ) , and then it is lear that (B.17 ) is satised if and only if ˙ m k = − m k u ( x k )  u x ( x k )  . A  kno wledgemen ts HL is supp orted b y the Sw edish Resear h Counil (V etensk apsrådet), and JS b y the National Sienes and Engineering Resear h Counil of Canada (NSER C). HL and JS w ould lik e to a kno wledge the hospitalit y of the Mathemati- al Resear h and Conferene Cen ter, Bdlew o, P oland, and the Departmen t of Mathematis and Statistis, Univ ersit y of Sask at hew an. Referenes [1℄ Ri hard Beals, Da vid H. Sattinger, and Jaek Szmigielski. Multi-p eak ons and a theorem of Stieltjes. Inverse Pr oblems , 15(1):L1L4, 1999. [2℄ Ri hard Beals, Da vid H. Sattinger, and Jaek Szmigielski. Multip eak ons and the lassial momen t problem. A dvan es in Mathematis , 154:229257, 2000. 38 [3℄ Maro Bertola, Mi hael Gekh tman, and Jaek Szmigielski. P eak ons and Cau h y biorthogonal p olynomials. [nlin.SI℄ , 2007. [4℄ Harish S. Bhat and Razv an C. F eteau. A Hamiltonian Regularization of the Burgers Equation. Journal of Nonline ar Sien e , 16(6):615638, Deem b er 2006. [5℄ Rob erto Camassa and Darryl D. Holm. An in tegrable shallo w w ater equa- tion with p eak ed solitons. Phys. R ev. L ett. , 71(11):16611664, 1993. [6℄ Giusepp e M. Co lite and Kenneth H. Karlsen. On the w ell-p osedness of the Degasp erisPro esi equation. J. F unt. A nal. , 233(1):6091, 2006. [7℄ Giusepp e M. Co lite and Kenneth H. Karlsen. On the uniqueness of dis- on tin uous solutions to the Degasp erisPro esi equation. J. Dier ential Equations , 234(1):142160, 2007. [8℄ A drian Constan tin and Joa him Es her. W a v e breaking for nonlinear non- lo al shallo w w ater equations. A ta Math. , 181(2):229243, 1998. [9℄ A drian Constan tin and Henry P . MKean. A shallo w w ater equation on the irle. Comm. Pur e Appl. Math. , 52(8):949982, 1999. [10℄ An tonio Degasp eris, Darryl D. Holm, and Andrew N. W. Hone. A new in tegrable equation with p eak on solutions. The or eti al and Mathemati al Physis , 133:14631474, 2002. [11℄ An tonio Degasp eris and Mi hela Pro esi. Asymptoti in tegrabilit y . In A. Degasp eris and G. Gaeta, editors, Symmetry and p erturb ation the ory (R ome, 1998) , pages 2337. W orld Sien ti Publishing, Riv er Edge, NJ, 1999. [12℄ Harry Dym and Henry P . MKean. Gaussian pr o  esses, funtion the- ory, and the inverse sp e tr al pr oblem . A ademi Press [Harourt Brae Jo v ano vi h Publishers℄, New Y ork, 1976. Probabilit y and Mathematial Statistis, V ol. 31. [13℄ Sergey F omin and Andrei Zelevinsky . T otal p ositivit y: tests and parametrizations. Math. Intel ligen er , 22(1):2333, 2000. [14℄ F elix R. Gan tma her. The the ory of matri es. Vol. 1 . AMS Chelsea Pub- lishing, Pro videne, RI, 1998. T ranslated from the Russian b y K. A. Hirs h, Reprin t of the 1959 translation. [15℄ F elix R. Gan tma her and Mark G. Krein. Osil lation matri es and ker- nels and smal l vibr ations of me hani al systems . AMS Chelsea Publishing, Pro videne, RI, revised edition, 2002. T ranslation based on the 1941 Rus- sian original, edited and with a prefae b y Alex Eremenk o. [16℄ Ira Gessel and Gérard Viennot. Binomial determinan ts, paths, and ho ok length form ulae. A dv. in Math. , 58(3):300321, 1985. 39 [17℄ Darryl D. Holm and Andrew N. W. Hone. A lass of equations with p eak on and pulson solutions. Journal of Nonline ar Mathemati al Physis , 12(2):46 62, 2005. [18℄ Andrew N. W. Hone and Jing Ping W ang. Prolongation algebras and Hamiltonian op erators for p eak on equations. Inverse Pr oblems , 19(1):129 145, 2003. [19℄ Andrew N. W. Hone and Jing Ping W ang. In tegrable p eak on equations with ubi nonlinearit y . Journal of Physis A: Mathemati al and The or eti al , 41(37):372002, 2008. [20℄ Rossen Iv ano v. On the in tegrabilit y of a lass of nonlinear disp ersiv e w a v e equations. J. Nonline ar Math. Phys. , 12(4):462468, 2005. [21℄ Sam uel Karlin. T otal p ositivity. Vol. I . Stanford Univ ersit y Press, Stanford, Calif, 1968. [22℄ Sam uel Karlin and James MGregor. Coinidene probabilities. Pai J. Math. , 9:11411164, 1959. [23℄ Jennifer K ohlen b erg, Hans Lundmark, and Jaek Szmigielski. The in v erse sp etral problem for the disrete ubi string. Inverse Pr oblems , 23:99121, 2007. [24℄ Christian Kratten thaler. A dv aned determinan t alulus. Sém. L othar. Combin. , 42:Art. B42q, 67 pp. (eletroni), 1999. The Andrews F ests hrift (Maratea, 1998). [25℄ Bern t Lindström. On the v etor represen tations of indued matroids. Bul l. L ondon Math. So . , 5:8590, 1973. [26℄ Hans Lundmark. F ormation and dynamis of sho  k w a v es in the Degasp erisPro esi equation. J. Nonline ar Si. , 17(3):169198, 2007. [27℄ Hans Lundmark and Jaek Szmigielski. Multi-p eak on solutions of the Degasp erisPro esi equation. Inverse Pr oblems , 19:12411245, Deem b er 2003. [28℄ Hans Lundmark and Jaek Szmigielski. Degasp erisPro esi p eak ons and the disrete ubi string. IMRP Int. Math. R es. Pap. , 2005(2):53116, 2005. [29℄ Ian G. Madonald. Symmetri funtions and Hal l p olynomials . Oxford Univ ersit y Press, New Y ork, seond edition, 1995. [30℄ Alexander V. Mikhailo v and Vladimir S. No vik o v. P erturbativ e symmetry approa h. J. Phys. A , 35(22):47754790, 2002. [31℄ Jürgen Moser. Three in tegrable systems onneted with isosp etral defor- mations. A dvan es in Mathematis , 16:197220, 1975. 40 [32℄ Lauren t S h w artz. Thé orie des distributions. Tome I . A tualités Si. Ind., no. 1091 = Publ. Inst. Math. Univ. Strasb ourg 9. Hermann & Cie., P aris, 1950. 41