nlin.SI 2015-05-13 0

Tzitzeica solitons vs. relativistic Calogero-Moser 3-body clusters

We establish a connection between the hyperbolic relativistic Calogero-Moser systems and a class of soliton solutions to the Tzitzeica equation (aka the Dodd-Bullough-Zhiber-Shabat-Mikhailov equation). In the 6N-dimensional phase space $\Omega$ of th…

Authors: ** M. G. G. K. M. S. (가명) – 실제 논문에 기재된 저자명은 원문을 참고하십시오. **

Tzitzeica solitons vs. relativistic Calogero-Moser 3-body clusters
Tzitzeica solitons vs. relativistic Calogero-Moser 3-b o dy clusters J. J. C. Nimmo ∗ S. N. M. Ruijsenaars † ‡ Abstract W e establish a connection b et w een the h yperb olic relativistic Calogero-Moser sys- tems and a class of soliton solutions to the Tzitzeica equation (ak a the Do dd- Bullough-Zhib er-Shabat-Mikhailo v equation). In the 6 N -dimensional phase space Ω of the relativistic systems with 2 N particles and N an tiparticles, there exists a 2 N - dimensional P oincar ´ e-in v ariant submanifold Ω P corresp onding to N free particles and N bound particle-antiparticle pairs in their ground state. The Tzitzeica N - soliton tau-functions under consideration are real-v alued, and obtained via the dual Lax matrix ev aluated in p oints of Ω P . This corresp ondence leads to a picture of the soliton as a cluster of tw o particles and one an tiparticle in their low est internal energy state. Con ten ts 1 In tro duction 2 2 The soliton reduction 2D T oda → Tzitzeica 8 3 Explicit form of the tau-functions 11 4 Tzitzeica solitons vs. relativistic Calogero-Moser dynamics 13 5 The N = 1 case 18 6 The Darb oux and Kaptso v-Shank o solitons 22 7 Concluding remarks 25 A F usion 28 B Pfaffian identities 29 C Solutions obtained b y binary Darb oux transformation 32 ∗ Departmen t of Mathematics, Univ ersit y of Glasgo w, Glasgo w, G12 8QW, UK † Departmen t of Applied Mathematics, Universit y of Leeds, Leeds LS2 9JT, UK ‡ Departmen t of Mathematical Sciences, Loughborough Universit y , Loughborough LE11 3TU, UK 1 References 33 1 In tro duction The equation Ψ uv = e Ψ − e − 2Ψ (1.1) has a curious history . It first arose a century ago in the work of the Rumanian mathe- matician Tzitzeica [1, 2]. He arriv ed at it from the viewp oint of the geometry of surfaces, obtaining an asso ciated linear represen tation and a B¨ ac klund transformation. F or man y decades after Tzitzeica’s work, the equation (1.1) w as not studied, t w o pap ers by Jonas [3, 4] b eing a notable exception. Thirty y ears ago, it was rein tro duced within the area of soliton theory , indep endently b y Dodd and Bullough [5] and Zhib er and Shabat [6], cf. also Mikhailo v’s pap er [7]. In this setting, (1.1) is viewed as an in tegrable relativistic theory for a field Ψ( t, y ) in tw o space-time dimensions, written in terms of ligh t cone (characteristic) co ordinates, t = u − v , y = u + v . (1.2) Accordingly , the PDE (1.1) is kno wn under v arious names, and has b een studied from sev eral p ersp ectiv es, including geometry [1–4], classical soliton theory [5–20], and quan tum soliton theory [21–26]. Moreo ver, it has sho wn up within the context of gas dynamics [27, 28]. The principal aim of this pap er is to tie in a class of soliton solutions to the Tzitzeica equation (1.1) with integrable particle dynamics of relativistic Calogero-Moser t yp e. (A surv ey co vering both relativistic and nonrelativistic Calogero-Moser systems can be found in [29].) The intimate relation of the latter in tegrable particle systems to soliton solutions of v arious evolution equations (including the sine-Gordon, T o da lattice, KdV and mo dified KdV equations) w as already revealed in the pap er in which they w ere in tro duced [30], and w as elab orated on in [31]. Later on, the list of equations whose soliton solutions are connected to the relativistic Calogero-Moser systems w as considerably enlarged [32–34]. In all of these cases, the N solitons corresp ond to N p oin t particles. The nov elty of the presen t soliton-particle corresp ondence is that the Tzitzeica N - soliton solutions at issue correspond to an in tegrable reduction of the 3 N -bo dy relativistic Calogero-Moser dynamics. Physically sp eaking, a Tzitzeica soliton ma y b e view ed as a lo w est energy b ound state of three Calogero-Moser ‘quarks’, one of which has negative c harge, whereas the other t wo hav e p ositiv e c harge. A crucial ingredien t for establishing the correspondence is the relation b etw een an extensiv e class of 2D T o da solitons and the relativistic Calogero-Moser systems, already studied in [33]. Indeed, the relation can b e com bined with the link b etw een the Tzitzeica equation and the 2D T o da equation. The latter link has b een known for quite a while, and w e pro ceed to sk etch it in a form that suits our later requiremen ts. Assume that φ n is a solution to the 2D T o da equation in the form [35] φ n,uv = exp( φ n − φ n − 1 ) − exp( φ n +1 − φ n ) , n ∈ Z , (1.3) whic h has the symmetry property φ − n = − φ n , (1.4) 2 and whic h is moreo ver 3-p erio dic, i.e., φ n +3 = φ n . (1.5) Then one has in particular φ 0 = 0 , φ 2 = − φ 1 , (1.6) so that Ψ = φ 1 (1.7) satisfies (1.1). Con v ersely , a solution Ψ to (1.1) yields a solution φ n to (1.3) satisfying (1.4) and (1.5) when one sets φ 3 k = 0 , φ 1+3 k = − φ 2+3 k = Ψ , k ∈ Z . (1.8) The p oint is now that there exist soliton solutions to (1.3) that can b e made to satisfy the extra requirements (1.4)–(1.5), hence yielding soliton solutions to (1.1). The relev an t 2D T o da solitons are those found b y the Kyoto sc ho ol [36, 37]. These solitons also formed the starting p oin t for [33]. They are most easily expressed in tau-function form, the relation of τ n to φ n b eing giv en by φ n = ln( τ n +1 /τ n ) , n ∈ Z . (1.9) In terms of τ n , the evolution equation b ecomes ∂ u ∂ v ln τ n = 1 − τ n − 1 τ n +1 /τ 2 n , (1.10) and the extra features (1.4) and (1.5) amoun t to τ − n +1 = τ n , (1.11) and τ n +3 = τ n . (1.12) As we sho w in Section 2, one can make sp ecial parameter c hoices in the 2D T o da 2 N -soliton solutions τ n ( u, v ) so that they satisfy (1.11)–(1.12). The function Ψ = ln( τ 2 /τ 1 ) (1.13) then satisfies (1.1), and can b e view ed as a Tzitzeica N -soliton solution. (In Lie algebraic terms, the successive requirements (1.11) and (1.12) amoun t to reductions A ∞ → B ∞ → A (2) 2 .) More specifically , the 2D T o da 2 N -solitons of Section 2 are of the form τ n ( u, v ) = det( 1 2 N + D ( n, u, v ) C ) , (1.14) where the dep endence of the 2 N × 2 N (Cauc h y type) matrix C and diagonal matrix D on the parameters a, b, ξ 0 ∈ C 2 N is suppressed. T o satisfy the B ∞ restriction (1.11) and to prepare for the 3-perio dicit y restriction (1.12), these 6 N parameters are expressed in terms of 2 N parameters φ, θ ∈ C N and a coupling parameter c . W e then sho w that the tau-functions hav e p erio d l for c equal to π /l , so that the Tzitzeica restrictions are satisfied for c = π / 3. 3 In Section 3 we make a further parameter change, trading φ 1 , . . . , φ N for ‘p ositions’ q 1 , . . . , q N . This reparametrization ensures in particular that the summand in v olving all exp onen tials has co efficient 1. Restricting attention to the parameter set P = { ( q , θ ) ∈ R 2 N | θ N < · · · < θ 1 } , (1.15) the tau-functions tak e their simplest and most natural form. In particular, for parameters in P the tau-functions are real-v alued. F urthermore, their space-time dep endence is such that the q j ’s and θ j ’s can be interpreted as relativistic p ositions and rapidities. Last but not least, it is in this form that the c = π / 3 Tzitzeica N -soliton tau-functions can b e most easily compared to the tau-functions arising in the framework of the 3 N -b o dy relativistic Calogero-Moser systems. Section 4 is devoted to this comparison. The c hoice of regime for the Calogero-Moser systems is the same as for almost all other soliton equations. Specifically , the regime is the h yp erb olic one, with the P oincar´ e group generators giv en b y H = M 0 2 ( S + + S − ) , P = M 0 2 ( S + − S − ) , B = N + X i =1 x + i + N − X j =1 x − j , (1.16) S ± = X 1 ≤ i ≤ N + exp( ± p + i ) V + i + X 1 ≤ j ≤ N − exp( ± p − j ) V − j , (1.17) ( V + i ) 2 = Y 1 ≤ k ≤ N + ,k 6 = i  1 + sin 2 c sinh 2 ( x + i − x + k ) / 2  Y 1 ≤ j ≤ N − 1 − sin 2 c cosh 2 ( x + i − x − j ) / 2 ! , (1.18) ( V − j ) 2 = Y 1 ≤ l ≤ N − ,l 6 = j 1 + sin 2 c sinh 2 ( x − j − x − l ) / 2 ! Y 1 ≤ i ≤ N + 1 − sin 2 c cosh 2 ( x − j − x + i ) / 2 ! . (1.19) Here, the generalized p ositions and momen ta v ary ov er the phase space Ω = { ( x + , x − , p + , p − ) ∈ R 2( N + + N − ) | x + N + < · · · < x + 1 , x − N − < · · · < x − 1 } , (1.20) equipp ed with the symplectic form ω = X 1 ≤ i ≤ N + dx + i ∧ dp + i + X 1 ≤ j ≤ N − dx − j ∧ dp − j . (1.21) (Only the Landau-Lifshitz solitons of [38] in v olv e the more general elliptic regime [32].) The coupling c that is needed, ho w ever, differs from the v alue π / 2 relev ant for most soliton equations (whic h correspond to A (1) 1 ). Indeed, as already men tioned, the connection of the space-time dynamics (1.16)–(1.19) to the Tzitzeica tau-functions arises for the ‘ A 2 ’-v alue c = π / 3 . (1.22) In order to clarify further restrictions and to describe our main result in some detail, a few more ingredien ts need to b e in tro duced. First, the Poisson comm uting ‘light-cone’ Hamiltonians S + and S − can b e expressed in terms of an ( N + + N − ) × ( N + + N − ) Lax matrix on Ω via S + = T r( L ) , S − = T r( L − 1 ) . (1.23) 4 This matrix has a pro duct structure D M D , with a diagonal matrix D and a matrix M that arises by suitable substitutions in Cauch y’s matrix C ( x, y ), i.e., a matrix with elemen ts ( x i − y j ) − 1 , whose determinant is giv en by Cauch y’s iden tit y |C ( x, y ) | = Q i 0 ho w ev er, L is not self-adjoint. Instead, L is J -self-adjoin t, that is, the adjoin t L ∗ equals J LJ , where J = diag( 1 N + , − 1 N − ) . (1.28) When one tak es the in terparticle distances to infinit y , it b ecomes clear that there is a subset of Ω on which L is diagonalizable with real eigen v alues. (Of course, for the one- c harge case this is true on all of Ω.) How ever, already for the simplest non-self-adjoint case N + = N − = 1, complex-conjugate eigenv alues are also presen t, reflecting the existence of b ound states of a particle and antiparticle. More precisely , c ho osing from now on c ∈ (0 , π / 2) , (1.29) these bound states are enco ded b y eigen v alues exp( ˆ θ + ) , exp( ˆ θ − ) , ˆ θ ± = θ ± iκ, κ ∈ (0 , c ] . (1.30) Moreo v er, the case of an eigenv alue pair on the b oundary of this sector corresp onds to phase space p oints x + = x − = x, p + = p − = p. (1.31) 5 F or p = 0 this yields a 1-parameter set with minimal energy H = 2 M 0 cos c , cf. (1.16)– (1.19), and, more generally , no oscillation tak es place for the 2-parameter set (1.31). The sp ecial case just discussed is the only one that can be explicitly understoo d in an elemen tary wa y . Already for N + + N − = 3 (the simplest case relev ant for this pap er) a complete account in volv es considerable analysis. The general case has b een elucidated in great detail in [31], and we need to make extensive use of Sections 2B–6B of that pap er, whic h p ertain to the c -range (1.29). The action-angle map constructed there mak es it possible to understand all of the Poisson comm uting dynamics at once, including their soliton type long-time asymptotics (conserv ation of momenta and factorized p osition shifts). It is b eyond our scope to recapitulate ev en the case N + + N − = 3, but we do men tion the k ey starting p oint for the analysis p erformed in [31]. This is b ecause it clarifies why the diagonal matrix A ( x ) = diag (exp( x + 1 ) , . . . , exp( x + N + ) , − exp( x − 1 ) , . . . , − exp( x − N − )) , (1.32) is of piv otal importance. This matrix is referred to as the dual Lax matrix, and its relation to L is enco ded in the commutation relation i cot( c )( LA − AL ) = LA + AL − 2 e ⊗ e. (1.33) (This relation is readily v erified; w e do not need the dyadic in the sequel.) One need only insp ect (1.33) to see that A and L pla y symmetric roles. In the one- c harge case it is p ossible to diagonalize the self-adjoint matrix L in such a w a y that A tak es essen tially the same form in terms of suitable v ariables (whic h are just the action- angle v ariables). This self-dualit y is no longer presen t for N + N − > 0, ho w ev er. Indeed, A is then still self-adjoin t, whereas L is not. Ev en so, (1.33) can again b e used to great adv an tage in the construction of the action-angle map, as detailed in [31]. W e are no w prepared to sp ecialize the ab o v e to the arena with which the presen t pap er is concerned. First of all, w e c ho ose N + = 2 N , N − = N . (1.34) This c hoice ensures that there exists a 2 N -dimensional P oincar ´ e-in v ariant submanifold Ω P ' P , (1.35) with P giv en b y (1.15), of the 6 N -dimensional phase space h Ω , ω i . This submanifold will b e described in detail in Section 4, and we shall presently add a brief qualitativ e description. Consider no w the solution exp( tH − y P ) Q, Q = ( x, p ) ∈ Ω , (1.36) to the join t Hamilton equations for H and P . Such a joint solution exists and is given b y (1.36), since the H -flo w and P -flow commute. Using (1.2) and (1.16), w e can rewrite (1.36) as exp( M 0 ( uS − − v S + )) Q = ( x ( u, v ) , p ( u, v )) . (1.37) Next, specializing to M 0 = √ 3 , c = π / 3 , (1.38) 6 w e introduce τ n ( u, v ) = det( 1 3 N + exp( iπ (1 − 2 n ) / 3) A ( x ( u, v ))) . (1.39) F or a giv en p oin t Q in the phase space Ω, this yields a function dep ending on n ∈ Z and ( u, v ) ∈ R 2 . W e are finally in the p osition to state the principal result of this pap er: When the tau- function (1.39) is restricted to the 2 N -dimensional P oincar ´ e-in v ariant subspace Ω P of Ω, it coincides with th e Tzitzeica tau-function (1.14) ev aluated on P (1.15). Our demonstration of this equality uses in particular sp ecial cases of the fusion identities obtained in [33]. (F or completeness w e add the pro of of these specializations in App endix A.) The reduction in the matrix size from 3 N × 3 N to 2 N × 2 N hinges on all p oints in Ω P yielding pairs of complex-conjugate eigen v alues exp( θ j ± iπ / 3) for the Lax matrix L on the boundary of the allo w ed angular sector. F or each of these pairs there is an additional eigenv alue exp( θ j ), so that the sp ectrum of L on Ω P in v olves only N degrees of freedom θ 1 , . . . , θ N ∈ R (and not the 3 N of the full phase space). These v ariables ma y b e view ed as action v ariables, and there are N canonically conjugate ‘angle’ v ariables q 1 , . . . , q N ∈ R . As a consequence, Ω P can be iden tified with P , cf. (1.35) and (1.15). A b etter understanding of ho w Ω P arises as a submanifold of the 6 N -dimensional phase space can only b e ac hieved by inv oking a great many details concerning the action-angle map constructed in [31], whic h are summarized in Section 4. A t this p oint we only add a few qualitativ e remarks, so as to render these details more accessible. First, the ab o v e co ordinates are not quite action-angle co ordinates. Rather, the precise definition of Ω P in v olves the harmonic oscillator transform. This transform is an extension of the action-angle transform, whic h takes into account that Ω contains an op en dense subset that is the union of submanifolds for whic h the angles v ary ov er R 3 N − l × T l , where T ' ( − π , π ] and l tak es all v alues in { 0 , 1 , . . . , N } . F rom a ph ysical point of view, these submanifolds can b e regarded as the subsets of Ω on whic h l particle-an tiparticle bound states are presen t. No w when the b ound state internal actions conv erge to their minima, the l -torus collapses to low er-dimensional tori in precisely the same w a y as for a harmonic oscillator Hamiltonian P l j =1 ( p 2 j + x 2 j ), whic h motiv ates the terminology . In these terms, Ω P amoun ts to the subset that arises from the submanifold with N b ound states b y taking the torus T N to a p oin t, so that each of the pairs is in its ground state; moreo v er, each of the remaining particles has action and angle v ariables that are paired with those of the b ound states, in such a w a y that the asymptotic space-time dep endence of Ψ is that of N 3-bo dy clusters mo ving apart, each of the clusters sta ying together as in the N = 1 case. In Section 5 w e presen t a close-up of the latter case. F or N = 1, v arious questions can b e rather easily answ ered, and this sp ecial case is also useful as an illustration of the general case. In particular, w e obtain some explicit information on the 2-dimensional space Ω P for c ∈ (0 , π / 2), and study the Tzitzeica 1-soliton solution corresp onding to c = π / 3. In Section 6 we compare the Tzitzeica solitons under consideration to the ones obtained b y Kaptso v and Shanko [16] and to the solitons arising by the ab o v e t w o-step reduction (1.11)–(1.12) from a class of 2D T o da solitons constructed via Darb oux transformations. A t face v alue, these t w o t yp es of Tzitzeica solitons seem different from the ones obtained from the Kyoto solitons in Section 2. As w e show, ho wev er, the latter are a sub class of the former. 7 W e ha v e added Section 6 primarily because it yields an affirmativ e answer to the t w o natural equality questions at issue, but as a b onus it yields a new insigh t on the Tzitzeica tau-function τ 0 (giv en b oth b y (1.39) and (1.14)): It is in fact p ositive, and equal to the square of a simpler tau-function o ccurring in the Kaptso v-Shank o work [16]. In Section 7 w e consider further asp ects of our results, in the form of several remarks. Sp ecifically , we show that the space-time translation generators restricted to Ω P giv e rise to a reduced integrable system, we introduce and comment on space-time tra jectories for the Tzitzeica solitons, isolate the difference with the setup of [33], and comment on an ev en tual quantum analog of the corresp ondence b etw een 3-b o dy clusters and solitons. A final remark concerns the solitons of the Demoulin system of equations [39, 40]. As it turns out, their tau-function form is obtained b y taking c = π / 6 in Sections 2 and 3. It is a challenging question whether they can also be tied in with the relativistic Calogero-Moser systems. W e ha v e relegated a pro of of the fusion identities used in Section 4 to App endix A. In App endix B w e collect some auxiliary results concerning pfaffians, which w e need in Section 6. Finally , App endix C contains a sk etc h of the Darb oux type construction of explicit tau-function solutions to the 2D T o da equation (1.10). 2 The soliton reduction 2D T o da → Tzitzeica Our starting p oin t is the 2D T o da M -soliton solutions in tro duced b y the Ky oto sc ho ol [36, 37] in the form τ n = X µ 1 ,...,µ M =0 , 1 Y 1 ≤ j ≤ M exp( µ j ξ j,n ) · Y 1 ≤ j 1 , t 1 , + = − v , t 1 , − = u, (2.4) to obtain a solution. W e proceed in several steps to sho w ho w a suitable sp ecialization of the 3 M parameters a, b and ξ 0 giv es rise to a tau-function with the ‘ B ∞ ’-symmetry (1.11). This also in volv es the c hoice M = 2 N , N ∈ N ∗ , (2.5) whic h will b e in force from no w on. Prop osition 2.1. Define a Cauchy typ e matrix C with elements C j k = a j − b j a j − b k , j, k = 1 , . . . , M , (2.6) 8 and a diagonal matrix D n = diag(exp( ξ 1 ,n ) , . . . , exp( ξ M ,n )) . (2.7) Then τ n may b e r ewritten as τ n = | 1 M + D n C | . (2.8) Pr o of. Using Cauch y’s iden tit y (1.24), the principal minor expansion of the rhs of (2.8) yields τ n = M X l =0 X I ⊂{ 1 ,...,M } , | I | = l Y j ∈ I exp( ξ j,n ) · Y j,k ∈ I ,j N , with j 6 = k ⇒ p j k = sinh 2 (( θ j − θ k ) / 2) + 3 / 4 sinh 2 (( θ j − θ k ) / 2) , (4.34) j ≤ N , k > N or j > N , k ≤ N ⇒ p j k = sinh 2 (( θ j − θ k ) / 2) + 1 sinh 2 (( θ j − θ k ) / 2) + 1 / 4 . (4.35) T aking p ositive square ro ots of the p j k , w e deduce from (4.33) that w e hav e lim p I ( ˆ θ ) = s J Y j ∈ J,k / ∈ J p 1 / 2 j k , (4.36) for a certain sign s J . W e shall determine this sign later on, cf. (4.45). Next, we fo cus on the principal minor expansion of the tau-function (1.39), restricted to the submanifold of Ω with N b ound states present, with in ternal actions δ 1 , . . . , δ N in ( − 2 π / 3 , 0). It reads τ n ( u, v ) = 3 N X l =0 exp( iπ l (1 − 2 n ) / 3) S l ( A ( x ( u, v ))) = 3 N X l =0 exp( iπ l (1 − 2 n ) / 3) X | I | = l exp( X i ∈ I ˆ q i ( u, v )) p I ( ˆ θ ) . (4.37) T aking the actions to their minima − 2 π / 3, we deduce from the ab o v e that the limiting tau-function is a sum of nonzero con tributions C J for all J of the form (4.31), with C J = exp( iπ ( s + 2 b )(1 − 2 n ) / 3) s Y σ =1 exp( q i σ ( u, v ) / 2) × b Y β =1 exp  q j β ( u, v ) / 2 + iπ 2  1 − ( − ) N   · s J Y j ∈ J,k / ∈ J p 1 / 2 j k . (4.38) W e are no w prepared to state and pro v e the principal result of this pap er. Theorem 4.1. L et c = π / 3 . Then the r elativistic Calo ger o-Moser tau-function (1.39) r estricte d to the subsp ac e Ω P ' P is e qual to the Tzitzeic a tau-function (2.9) with the factors given by (3.8)–(3.9) and (3.12)–(3.15) and with p ar ameters in P . Pr o of. W e need only show equality of the general contribution C J to the general term T S in (2.9), cf. (3.18). First, we compare f j k , given by (3.8)–(3.9) with c = π / 3, to p j k giv en b y (4.34)–(4.35). F rom this w e easily deduce F S ( θ ) = Y j ∈ J,k / ∈ J p 1 / 2 j k . (4.39) Next, w e note that the factors in v olving q j ( u, v ) are in agreemen t. Hence the asserted equalit y comes down to an equality of the remaining numerical factors. Sp ecifically , it remains to show ( − ) b exp  iπ b 2  1 − ( − ) N   s J = 1 . (4.40) 17 T o this end we no w calculate the sign s J in (4.36). W e b egin by noting that for ˆ θ ∈ C 3 N giv en b y (4.2)–(4), the pro duct p I ( ˆ θ ) (4.6) has a p ositive n umerator. (Indeed, eac h radicand is either p ositiv e or has a nonzero imaginary part; in the latter case it is matc hed b y a factor with the complex-conjugate radicand.) W e are therefore reduced to analyzing the phase of Π I = Y i ∈ I ,j / ∈ I s ij , s ij = sinh(( ˆ θ min( i,j ) − ˆ θ max( i,j ) ) / 2) , (4.41) for the sp ecial ˆ θ under consideration, namely , ˆ θ = ( θ 1 , . . . , θ N , θ 1 + iπ / 3 , . . . , θ N + iπ / 3 , θ 1 − iπ / 3 , . . . , θ N − iπ / 3) , θ N < · · · < θ 1 . (4.42) W e already kno w from (4.36) that this phase is just the sign s J . Indeed, w e ha ve sinh(( θ j − θ k − iψ ) / 2) sinh(( θ j − θ k + iψ ) / 2) > 0 , ψ = π / 3 , 2 π / 3 , (4.43) whic h confirms that Π I is either p ositive or negativ e. W e con tin ue to analyze the con tributions to s J from indices in I (4.30), taking (4.42) in to accoun t. First, we consider the contribution of i σ ∈ I . Due to the ordering of the θ j in (4.42), any j / ∈ I with j ≤ N yields s i σ j > 0. Also, an y B k that is not a subset of I yields a con tribution of the form (4.43) with ψ = π / 3. Hence all indices i 1 , . . . , i s in I yield positive signs. Next, we study the contribution of B j β ⊂ I . F or i ∈ I with i ≤ N it again follows from (4.43) that we get a p ositiv e sign. No w supp ose B k is not a subset of I and consider the pertinent pro duct s j β + N ,k + N s j β + N ,k + M s j β + M ,k + N s j β + M ,k + M . (4.44) Both for k < j β and for k > j β this pro duct is easily seen to b e negativ e. Therefore an y pair B j , B k with B j included in I and B k not included in I giv es rise to a min us sign in Π I . Now I con tains b breather index sets, so there are N − b breather sets not con tained in I . Th us we finally deduce s J = ( − ) b ( N − b ) . (4.45) With this explicit formula in hand, it is routine to verify (4.40). Consequen tly , our pro of of the asserted tau-function equalit y is no w complete. 5 The N = 1 case In this section we consider v arious asp ects of the sp ecial case N = 1, whic h yields an illuminating illustration of the ab ov e constructions and equalities. More sp ecifically , we fo cus on features of the Tzitzeica 1-soliton solution and the space Ω P for c ∈ (0 , π / 2). F or N = 1 it follows from (3.28)–(3.29) that we hav e τ 0 = τ 1 = 1 + 2 F + F 2 , τ 2 = 1 − 4 F + F 2 , F = exp( q ( u, v ) / 2) , (5.1) with q ( u, v ) = q − 2 √ 3( ue − θ + v e θ ) = q + 2 √ 3( t sinh( θ ) − y cosh( θ )) , (5.2) 18 cf. (1.2). Th us the tw o τ 2 -zeros for F = 2 ± √ 3 yield tw o parallel space-time lines y ± ( t ) = t tanh( θ ) + 1 2 √ 3 cosh( θ ) ( q − 2 ln(2 ± √ 3)) , (5.3) where Ψ diverges, cf. Fig. 1. In this figure and in Fig. 2, w e plot for clarity − Ψ rather than Ψ, and truncate − Ψ at a finite cutoff v alue. The regions betw een the singularities, where τ 2 is negative so that Ψ = ln( τ 2 /τ 0 ) takes complex v alues, are appro ximated by the ‘plateaux’ in these figures. Figure 1: One-soliton solution. − Ψ( t, y ) is shown, the y -axis p oints left to right and the t -axis p oints in to the page. The ‘plateau’ approximates the region in which Ψ( t, y ) is complex-v alued. Recalling (1.39), we see that the τ 2 -zeros corresp ond to x + 1 ( t, y ) or x + 2 ( t, y ) b eing zero. The space-time lines may therefore b e view ed as the tra jectories of the t w o particles in the 1-soliton cluster. F or N = 1 it might seem an easy matter to lo cate the 2-dimensional space Ω P within the 6-dimensional phase space Ω. In fact, ho w ever, it is not even easy to find the 1- dimensional submanifold of Ω P consisting of H -equilibrium points, i.e., Ω E = { ( q , θ ) ∈ Ω P ' R 2 | θ = 0 } , (5.4) in explicit form. (Note that for c = π / 3 this yields the stationary 1-soliton tau-functions.) Of course, it is immediate from (1.16)–(1.19) that we need p + 1 = p + 2 = p − 1 = 0 , (5.5) for H to hav e an equilibrium. Requiring this, one exp ects from ph ysical considerations to get an equilibrium point for x + 1 = − x + 2 = d > 0 , x − 1 = 0 , (5.6) and a suitable d . W e no w confirm this for c ∈ (0 , π / 2). Then we hav e from (1.16)–(1.19) H ( d, − d, 0 , 0 , 0 , 0) M 0 = 2( f + f − ) 1 / 2 + f − , f + = 1 + sin 2 c sinh 2 d , f − = 1 − sin 2 c cosh 2 ( d/ 2) . (5.7) 19 Th us w e ha v e H → 3 M 0 for d → ∞ and H → ∞ for d → 0. Since H is equal to M 0 P 3 j =1 cosh( ˆ θ j ), it has an absolute minim um M 0 p ( c ) for ˆ θ = (0 , ic, − ic ). Now we can still choose x s 1 , x 1 ∈ R (cf. (4.7)–(4.8)), so H has a 2-parameter family of stable equilibria. It is straightforw ard to c heck that for d equal to d e = cosh − 1 (1 + cos c ) , (5.8) w e hav e f + = 1 + 2 cos c (2 + cos c ) cos c , f − = (1 + 2 cos c ) cos c 2 + cos c . (5.9) Hence the rhs of (5.7) equals M 0 p ( c ), so we do get a stable equilibrium for d = d e . This implies that all of the p oin ts E ( σ ) = ( d e + σ, − d e + σ, σ, 0 , 0 , 0) , σ ∈ R , (5.10) are also equilibria. Since H equals M 0 p ( c ) on Ω E , one migh t exp ect that the 1-parameter family of equi- libria E ( σ ) yields Ω E . In fact, ho w ever, only E (0) b elongs to Ω E . T o b e sp ecific, we assert that it corresp onds to q = 0. T o sho w this, w e use (4.5) with N = 1 to find the symmetric functions of A on Ω E . W e determined the relev an t limits b elo w (4.28). In particular, (4.33) yields p ( c ) 2 for I equal to { 1 } or { 2 , 3 } , since θ 1 = 0 on Ω E . Using also (4.25) and (4.28), w e readily obtain S 1 ( A ) = p ( c ) exp( q /p ( c )) , (5.11) S 2 ( A ) = − p ( c ) exp(2 cos( c ) q /p ( c )) , (5.12) S 3 ( A ) = − exp( q ) . (5.13) F or q = 0 this implies that the sp ectrum of A is given by σ ( A ) = { 1 + cos( c ) ± (cos 2 ( c ) + 2 cos( c )) 1 / 2 , − 1 } . (5.14) The first t w o eigen v alues can b e written as exp( ± d e ), cf. (5.8). Thus the origin of Ω E ' R corresp onds to generalized p ositions x + 1 = d e , x + 2 = − d e , x − 1 = 0. Since Ω E consists of H -equilibria, we also ha v e p + 1 = p + 2 = p − 1 = 0. Hence w e deduce E (0) ∈ Ω E and q = 0, as asserted. On the other hand, the equilibrium p oints in Ω P with q 6 = 0 are harder to find explicitly . As already announced, they do not include the equilibria E ( σ ) for σ 6 = 0. (Indeed, E (0) do es belong to Ω E , as just sho wn. No w the σ -shift of the generalized p ositions corresponds to a shift of x s 1 and x 1 b y σ , so that the Ω P -condition x 1 = cos( c ) x s 1 no longer holds true for σ 6 = 0.) Rather, they are obtained b y acting with the comm uting flo w exp( y P / M 0 ) , y ∈ R , on the equilibrium E (0), yielding translated equilibria T ( y ) = ( x + 1 ( y ) , x + 2 ( y ) , x − 1 ( y ) , 0 , 0 , 0) , y ∈ R . (5.15) In view of (5.13) and (4.22) this yields the sum rule x + 1 ( y ) + x + 2 ( y ) + x − 1 ( y ) = p ( c ) y . (5.16) 20 Also, from Hamilton’s equations for P / M 0 w e hav e x + j ( y ) 0 > cos( c ) , j = 1 , 2 , x − 1 ( y ) 0 ≥ cos 2 ( c ) , (5.17) so that x + 1 ( y ) , x + 2 ( y ) and x − 1 ( y ) are strictly increasing functions of y . Next, w e deduce from (5.11)–(5.13) that w e hav e the reflection symmetry x + 1 ( − y ) = − x + 2 ( y ) , x − 1 ( − y ) = − x − 1 ( y ) , (5.18) so it remains to determine the functions for y > 0. This is presumably p ossible, but w e ha ve not pursued this. How ever, the large- y asymptotics can b e established from the ab o v e. Sp ecifically , setting  = exp(cos c − 1) y , (5.19) w e obtain for  → 0 x + 1 ( y ) = y + ln p ( c ) + O (  2 ) , (5.20) x + 2 ( y ) = cos( c ) y − 1 2 ln p ( c ) − 1 2 p p ( c )(1 − p ( c ) − 2 )  + O (  2 ) , (5.21) x − 1 ( y ) = cos( c ) y − 1 2 ln p ( c ) + 1 2 p p ( c )(1 − p ( c ) − 2 )  + O (  2 ) . (5.22) A final observ ation of interest concerns the kinetic and p oten tial energy densit y of the stationary 1-soliton solution, i.e., the functions E K ( y ) = ( ∂ y Ψ) 2 / 2 , E P ( y ) = exp(Ψ) + exp( − 2Ψ) / 2 − 3 / 2 , Ψ = ln( τ 2 /τ 0 ) , (5.23) obtained from (5.1) for θ = 0. It is far from obvious, but true that these functions are equal. This equalit y can b e v erified directly by a straightforw ard, but quite tedious calculation. A more conceptual deriv ation of this virial type identit y will no w b e giv en. First, w e note that any t -indep endent solution to (1.1) satisfies the ODE f y y = e f − e − 2 f . (5.24) No w from (5.1) w e obtain a solution to (5.24) of the form f ( y ) = g (exp(( q − 2 √ 3 y ) / 2)) , g ( z ) = ln  1 − 4 z + z 2 1 + 2 z + z 2  . (5.25) Defining z ± = 2 ± √ 3 , (5.26) it satisfies e g ( z ) > 0 , z > z + , z < z − , (5.27) e g ( z ) < 0 , z ∈ ( z − , z + ) , (5.28) and g ( z ) , g 0 ( z ) → 0 , z → ±∞ , (5.29) exp( g (1)) = − 1 / 2 , g 0 (1) = 0 . (5.30) 21 Next, consider the Hamiltonians H ± ( x, p ) = p 2 / 2 − V ± ( x ) , V ± ( x ) = ± e x + e − 2 x / 2 − 3 / 2 . (5.31) The potential − V + ( x ) has a maxim um 0 at x = 0 and yields a Newton equation ¨ x = e x − e − 2 x . (5.32) Comparing this to (5.24), (5.27) and (5.29), w e see that g ( z ) for z > z + corresp onds to the E = 0 orbit with x ( t ) < 0 coming from 0 for t → −∞ , and g ( z ) for z < z − to the E = 0 orbit with x ( t ) < 0 going to 0 for t → ∞ . By energy conserv ation, w e hav e ˙ x 2 / 2 = V + ( x ), whic h implies E K ( y ) = E P ( y ) for the y -interv als where e f ( y ) > 0. It remains to show the iden tit y for the y -interv al where e f ( y ) < 0. Then we can compare (5.24) to the Newton equation ¨ x = − e x − e − 2 x , (5.33) corresp onding to H − . The E = 0 orbit stays to the left of the origin and has its turning p oin t at x = ln(1 / 2), cf. (5.31). Comparing this to (5.30), w e see that it corresponds to the real part of g ( z ) for z ∈ ( z − , z + ). Hence E K ( y ) = E P ( y ) no w follo ws from ˙ x 2 / 2 = V − ( x ). 6 The Darb oux and Kaptso v-Shank o solitons In this section we b egin by showing that the solitons obtained by the B ∞ -reduction from the Kyoto 2D T o da solitons are equal to those obtained b y a similar reduction from a seemingly different class of 2D T o da solitons. The latter are constructed via rep eated Darboux transformations. As will transpire, this pro cedure yields a larger class of solutions, in volving an arbitrary constant an tisymmetric matric C . In order to obtain equalit y to the B ∞ solitons of Section 2, this matrix m ust be suitably specialized. Our demonstration of equalit y leads to a new represen tation of the latter solitons. This represen tation can b e exploited to sho w that τ 0 equals the square of a simpler tau- function. This is b ecause it readily leads to τ 0 b eing the determinan t of an an tisymmetric M × M matrix A . Hence it follows that w e hav e τ 0 = τ 2 , τ = Pf ( A ) . (6.1) The pfaffian can b e explicitly ev aluated, and when the Tzitzeica substitutions of Sec- tions 2–3 are made in A , then the resulting τ is the one obtained b y Kaptsov and Shanko in their study of the Tzitzeica equation [16]. W e pro ceed with the details. W e start from the tau-function τ n (2.19), with J , D and A giv en by (2.17), (2.20) and (2.21), and with the quantities χ j in D arbitrary at this stage. W e claim that τ n is equal to the determinan t ˜ τ n of the matrix M n = C R + B ˜ A n B , (6.2) where C R =  0 R N −R N 0  , (6.3) B = diag ( β 1 , . . . , β M ) , (6.4) 22 ˜ A n,j k = ( − a j ) n a − n +1 k a j + a k , j, k = 1 , . . . , M , (6.5) pro vided that β 1 , . . . , β M are c hosen suc h that β j β M − j +1 = χ 2 j , j = 1 , . . . , N . (6.6) (Recall R N denotes the N × N reversal p ermutation matrix.) In order to pro v e this, w e denote the columns of M n b y c 1 , . . . , c M , and use ˜ τ n = |M n | = | Col( c 1 , . . . , c M ) | = | Col( c M , . . . , c N +1 , − c N , . . . , − c 1 ) | . (6.7) Th us, ˜ τ n equals the determinant of the matrix 1 M + N n J , N n,j k = β j β M − k +1 ( − a j ) n ( a M − k +1 ) − n +1 a j + a M − k +1 . (6.8) T ransforming this matrix with the similarit y matrix S = diag ( γ 1 , . . . , γ M ) , γ j = ( β M − j +1 /β j ) 1 / 2 , j = 1 , . . . , M , (6.9) w e obtain ˜ τ n = | 1 M + ˜ D A n ˜ D J | , (6.10) with ˜ D = diag (( β 1 β M ) 1 / 2 , ( β 2 β M − 1 ) 1 / 2 , . . . , ( β M β 1 ) 1 / 2 ) . (6.11) Hence the β -constrain t (6.6) en tails equalit y of ˜ τ n and τ n , as advertised. W e can no w compare the new representation τ n = |C R + D ˜ A n D | , (6.12) obtained b y c ho osing β j = β M − j +1 = χ j , j = 1 , . . . , N , (6.13) (so that (6.6) is ob eyed, and B and ˜ D both equal D ), with the solitons obtained by a B ∞ symmetry reduction from the Darb oux t ype 2D T o da solitons. (See App endix C for a sk etc h of their construction.) The reduced solitons are of the form τ D n = |C + ˜ B ˜ A n ˜ B | , (6.14) where C is an arbitrary antisymmetric M × M matrix and ˜ B is a diagonal matrix ˜ B = diag ( ˜ β 1 , . . . , ˜ β M ) , (6.15) with diagonal elements of the form ˜ β j = α j exp( − iv a j + iua − 1 j ) , j = 1 , . . . , M . (6.16) Recalling the definition (2.13)–(2.15) of χ j , we see that (6.6) is ob eyed when we choose α 1 , . . . , α M suc h that α j α M − j +1 = exp( ϕ j ) , j = 1 , . . . , N . (6.17) 23 Pro vided w e also sp ecialise the arbitrary an tisymmetric matrix C to C R (cf. (6.3)), we therefore conclude equalit y of the reduced Darb oux t yp e solitons to the reduced Kyoto solitons. W e con tin ue b y using the new representation (6.12) to sho w the square prop ert y of τ 0 , cf. (6.1). As it stands, the matrix C R + D ˜ A 0 D is not an tisymmetric. But w e ha v e ˜ A 0 ,j k = a k a j + a k = 1 2  1 − a j − a k a j + a k  , (6.18) so that we can write ˜ A 0 = 1 2 ζ ⊗ ζ − 1 2 A , ζ = (1 , . . . , 1) , (6.19) where A is the an tisymmetric matrix with elements A j k = a j − a k a j + a k . (6.20) As sho wn in Appendix B, the determinan t τ 0 = |C R + 1 2 ( D ζ ⊗ D ζ − D AD ) | (6.21) equals the square of τ = Pf ( C R − D AD / 2) , (6.22) and the pfaffian can be explicitly ev aluated. The resulting formula is τ = N X l =0 X 1 ≤ j 1 < ··· 0 and exp( s j ) > 0, so that d j > 0 and c ij > 0. The function τ is therefore p ositiv e, implying τ 0 is p ositive. Note that τ can b e rewritten in the form (2.1), with M , ξ j,n and f j k replaced b y N , d j and c j k , respectively . Last but not least, the tau-function (6.23) with the substitutions (6.31)–(6.32) co- incides with the tau-function obtained in Section 2 of [16]. Moreo v er, com bining the relations exp(Ψ) = τ 2 /τ 0 , τ 0 = τ 1 , τ 0 = τ 2 , (6.33) (cf. (1.13), (1.11) and (6.1)), and the 2D T o da equation of motion (1.10) with n = 1, we deduce 2 ∂ u ∂ v ln τ = 1 − exp(Ψ) . (6.34) Therefore, the function exp(Ψ) coincides with the function v employ ed in [16]. 7 Concluding remarks (i) ( Inte gr ability on Ω P ) It is not hard to see that the flo ws generated b y the Hamiltonians H r = T r( L r ) , r ∈ R ∗ , (7.1) lea v e Ω P in v ariant, provided cos( r c ) = cos( c ) . (7.2) Indeed, this readily follo ws from (4.11) and (4.17), noting that (7.1) corresp onds to (4.10) with h ( z ) = e rz . Thus the restricted phase space h Ω P , P N j =1 dq j ∧ dθ j i and the Hamilto- nians H r , ± r = 1 + 2 π k /c, k ∈ Z , (7.3) giv e rise to an in tegrable system on Ω P . F or the Tzitzeica case c = π / 3, one can c ho ose as the N independent Hamiltonians the pow er traces T r( L 1+6 k ) , k = 0 , 1 , . . . , N − 1 . (7.4) (ii) ( Sp ac e-time tr aje ctories ) As we hav e sho wn in Theorem 4.1, on Ω P the tau-function (1.39) equals the Tzitzeica tau-function, and hence is real, cf. Proposition 3.1. This realit y prop ert y is far from ob vious for n = 0 , 1, but for n = 2 realit y on all of Ω is in fact clear from real-v aluedness of A , cf. (1.32). Sp ecializing to Ω P , w e ha v e τ 2 ( u, v ) = 2 N Y i =1 [1 − exp( x + i ( u, v ))] · N Y j =1 [1 + exp( x − j ( u, v ))] . (7.5) No w it follows from Section 6 that τ 0 ( u, v ) is positive. Therefore, the Tzitzeica solution Ψ( u, v ) = ln( τ 2 ( u, v ) /τ 0 ( u, v )) , (7.6) 25 has logarithmic singularities at the zeros of τ 2 , as w e ha ve already seen for N = 1 in the previous section. W e pro ceed to analyze these in terms of the space-time co ordinates t and y , cf. (1.2). Fixing t , the function τ 2 is p ositiv e for y → ±∞ , and has 2 N sign changes for finite y . The lo cations of these zeros on the y -axis are distinct for all t (since x + 2 N < · · · < x + 1 ). Th us one obtains 2 N space-time tra jectories, which may b e viewed as the lo cations of the particles in the N -soliton solution. F or t → ±∞ these tra jectories exhibit soliton scattering, with a factorized phase shift in terms of the function ln( c ij ), cf. (6.32). A more systematic analysis would b e feasible b y following the path laid out in Chapter 7 of [31], but this is b eyond our presen t scope. See, how ever, Fig. 2 for a plot of the 2-soliton collision. Figure 2: Tw o-soliton interaction. − Ψ( t, y ) is shown, the y -axis p oints left to right and the t -axis points in to the page. The ‘plateau’ appro ximates the region in whic h Ψ( t, y ) is complex-v alued. (iii) ( Comp arison with [33]) The 2D T o da soliton tau-functions studied in [33] do not include the ab ov e real-v alued Tzitzeica tau-functions. This is b ecause in [33] the condition τ − n = τ n is imp osed (cf. (2.10) in [33]), whereas for the Tzitzeica case we ha v e τ − 1 = τ 2 6 = τ 1 = τ 1 . T o accommo date this differen t starting p oint, the fusion pro cedure in Section 2 of [33] starts from tau-functions that in terms of the relativistic Calogero-Moser systems amoun t to τ n = | 1 M + exp( − 2 inc ) A ( x ( u, v )) | , A ( x ) = diag (exp( x + 1 ) , . . . , exp( x + M )) . (7.7) Ev en so, we could sp ecialize the fusion iden tities of [33], since they only p ertain to the action v ariables, and the dependence on the latter is gov erned b y the same function ((2.15) in [33]) for particles and antiparticles. (iv) ( Quantum analo gs ) W e ha v e c hosen N + = 2 N and N − = N throughout the pap er (cf. (1.34)), but w e could just as well hav e started from N particles and 2 N an tiparticles. Indeed, this amounts to w orking with the ‘c harge conjugate’ dual Lax matrix A C ( x ) = diag (exp( x + 1 ) , . . . , exp( x + N ) , − exp( x − 1 ) , . . . , − exp( x − 2 N )) , (7.8) 26 and tau-function τ C n ( u, v ) = det( 1 3 N − exp( iπ (1 − 2 n ) / 3) A C ( x ( u, v ))) , (7.9) instead of (1.32) and (1.39). Clearly , one can then proceed in the same wa y as before. On the other hand, one cannot enlarge the 3-b o dy–soliton corresp ondence without v en turing in to complexified phase spaces, losing con trol of the action-angle map in the pro cess. More precisely , there exist real-v alued Tzitzeica soliton tau-functions that corre- sp ond to tw o differen t c harges, hence giving rise to breather-lik e b ound states. (Indeed, this is not hard to see from the explicit form of the solitons at the end of Section 3; for example, one need only perform a suitable analytic con tin uation in the 2-soliton solution to obtain the 1-breather solution.) Ho w ev er, it can b e sho wn that these do not correspond to subspaces of the real Calogero-Moser phase spaces with arbitrary N + and N − . It ma y b e exp ected that this picture p ersists on the quantum lev el. T o b e specific, a suitable reduction of the unitary joint eigenfunction transform for the comm uting quan tum Hamiltonians with c = π / 3 and 3 N v ariables (which, to b e sure, is not ev en known to exist to date) should give rise to a unitary transform with N v ariables, whic h can b e in terpreted as a transform for N quan tum solitons with the same charge. Ho w ev er, no suc h reductions are likely to exist when different charges are inv olved. In particular, if quan tum breathers for the Tzitzeica quan tum field theory do exist (a feature that is tak en for gran ted in most of the work within the form factor program), then they are not lik ely to hav e analogs in the Calogero-Moser particle picture. (By con trast, for the sine-Gordon case a complete corresp ondence is expected [41].) (v) ( Demoulin solitons ) F rom Prop ositions 2.2 and 2.3 it is clear that for c = π / 6 (7.10) the tau-function satisfies τ − n +1 = τ n , τ n +6 = τ n . (7.11) As a consequence, w e ha ve τ 1 = τ 0 , τ 2 = τ 5 , τ 3 = τ 4 . (7.12) Setting h n = τ n +1 τ n − 1 /τ 2 n = exp( φ n − φ n − 1 ) , (7.13) (where w e used (1.9)), we deduce h 1 = h 0 , h 2 = h 5 , h 3 = h 4 , (7.14) and h 1 h 2 h 3 = 1 . (7.15) No w from (1.3) w e hav e (ln h n ) uv = 2 h n − h n +1 − h n − 1 . (7.16) Hence, setting h = h 1 = τ 2 /τ 1 , k = h 3 = τ 2 /τ 3 , (7.17) 27 and using (7.15), w e obtain (ln h ) uv = h − 1 hk , (ln k ) uv = k − 1 k h . (7.18) This system of relativistic wa ve equations is the Demoulin system, cf. [40], p. 343, Eq. (9.58). Therefore the c = π / 6 tau-functions with parameters in P yield real-v alued Demoulin solitons in tau-function form. In particular, from (3.15)–(3.27) w e see that the one-soliton case is giv en b y (7.12) and τ 1 = 1 + 2 G + G 2 , τ 2 = 1 + G 2 , τ 3 = 1 − 2 G + G 2 , G = exp( q / 2 − v e θ − ue − θ ) . (7.19) App endices A F usion In this app endix w e detail ho w the formula (4.6) for p I ( ˆ θ ) 2 leads to (4.33) in the collapsing torus limit. W e b egin b y recalling that in this limit p I ( ˆ θ ) v anishes, unless the subset I of { 1 , . . . , 3 N } has the form I = [ j ∈ J I j (A.1) where J = I ∩ { 1 , . . . , 2 N } , I j = { j } , I N + j = { N + j, 2 N + j } , j = 1 , . . . , N , (A.2) cf. the paragraph preceding (4.30). F or a subset I of this form, (4.6) en tails p I ( ˆ θ ) 2 = Y j ∈ J,k 6∈ J   Y m ∈ I j ,n ∈ I k sinh 2 (( ˆ θ m − ˆ θ n ) / 2) + sin 2 c sinh 2 (( ˆ θ m − ˆ θ n ) / 2)   . (A.3) Next, w e substitute (4.26) and (4.27) in this form ula and cancel terms in the in terior pro duct using fusion identities. A general fusion identit y , in whic h the sets I k are of arbitrary cardinality , can b e found in [33], but here we only detail the sp ecial cases w e need. Using the trigonometric/hyperb olic identit y sinh 2 x + sin 2 y = sinh( x + iy ) sinh( x − iy ) , (A.4) the general pair factor in (A.3) can be written as sinh(( ˆ θ m − ˆ θ n ) / 2 + ic ) sinh(( ˆ θ m − ˆ θ n ) / 2 − ic ) sinh 2 (( ˆ θ m − ˆ θ n ) / 2) . (A.5) F or a soliton-soliton interaction included in J , there is only one pair factor in the product. Sp ecifically , we hav e I j = { j } , I k = { k } and (from (4.26) and (4.32)) ˆ θ j = η j , ˆ θ k = η k , so the con tribution to the pro duct for a soliton-soliton pair is sinh 2 (( η j − η k ) / 2) + sin 2 c sinh 2 (( η j − η k ) / 2) = sinh(( η j − η k ) / 2 + ic ) sinh(( η j − η k ) / 2 − ic ) sinh 2 (( η j − η k ) / 2) = s (2) s ( − 2) s 2 (0) , (A.6) 28 where w e hav e set s ( l ) = sinh(( η j − η k + lic ) / 2). F or a soliton-breather in teraction w e ha v e I j = { j } , I N + k = { N + k , 2 N + k } and ˆ θ j = η j , ˆ θ N + k = η k + ic , ˆ θ 2 N + k = η k − ic , yielding t wo pair factors s (1) s ( − 3) s 2 ( − 1) s (3) s ( − 1) s 2 (1) = s (3) s ( − 3) s (1) s ( − 1) = sinh 2 (( η j − η k ) / 2) + sin 2 (3 c/ 2) sinh 2 (( η j − η k ) / 2 + sin 2 ( c/ 2) . (A.7) Finally , for a breather-breather in teraction in J , we get I N + j = { N + j, 2 N + j } , I N + k = { N + k , 2 N + k } , and hence four pairs whose contribution to the product is s (2) s ( − 2) s 2 (0) s (4) s (0) s 2 (2) s (2) s ( − 2) s 2 (0) s (0) s ( − 4) s 2 ( − 2) = s (4) s ( − 4) s 2 (0) = sinh 2 (( η j − η k ) / 2) + sin 2 (2 c ) sinh 2 (( η j − η k ) / 2) . (A.8) Using η l + N = η l and c l = c , c N + l = 2 c for l = 1 , . . . , N (cf. (4.32)), these three cases may b e enco ded in the single form ula sinh 2 (( η j − η k ) / 2) + sin 2 (( c j + c k ) / 2) sinh 2 (( η j − η k ) / 2 + sin 2 (( c j − c k ) / 2) , j, k = 1 , . . . , 2 N . (A.9) Hence (4.33) results. B Pfaffian iden tities In this app endix we sho w that the determinant on the rhs of (6.21) equals the square of the pfaffian on the rhs of (6.22), and that the latter has the explicit form (6.23)–(6.25). W e pro ceed in a sligh tly more general w a y , since this eases the notation and adds insigh t. First w e note that the determinant is of the form | v ⊗ v + A | , where A is a 2 N × 2 N an tisymmetric matrix. No w we inv oke the follo wing lemma. Lemma B.1. Assume A is an antisymmetric 2 N × 2 N matrix and v ∈ C 2 N . Then one has | v ⊗ v + A | = | A | . (B.1) Pr o of. By con tin uit y it suffices to prov e this for an in v ertible A . Then we hav e | v ⊗ v + A | = | A || 1 2 N + A − 1 v ⊗ v | = | A | (1 + ( v , A − 1 v )) . (B.2) No w since ( A − 1 ) t = ( A t ) − 1 = − A − 1 , it follows that A − 1 is also antisymmetric. Hence the inner product ( v , A − 1 v ) = v t A − 1 v v anishes, and (B.1) results. As a consequence, w e ha ve τ 0 = |C R − D AD / 2 | = τ 2 , (B.3) with τ given by (6.22). W e con tin ue to show (6.23). Again, we first study a slightly more general situation. W e begin b y recalling the definition Pf ( A ) = X ( − ) sgn( σ ) A i 1 j 1 · · · A i N j N , A ∈ M 2 N ( C ) , A t = − A, (B.4) 29 where the sum is o ver all index c hoices with 1 = i 1 < · · · < i N ≤ 2 N − 1 , i 1 < j 1 , . . . , i N < j N , (B.5) and σ is the permutation σ : ( i 1 , j 1 , . . . , i N , j N ) 7→ (1 , 2 , . . . , 2 N ) . (B.6) Denoting Pf ( A ) b y [1 , 2 , . . . , 2 N ], this implies an expansion formula [1 , 2 , . . . , 2 N ] = 2 N X n =2 ( − ) n [1 , n ][2 , . . . , ˆ n, . . . , 2 N ] , (B.7) where the notation on the rhs will b e clear from context. Consider no w the pfaffian of the an tisymmetric matrix B =  0 λ R N − λ R N 0  + A, λ ∈ C . (B.8) The relev ant (upp er triangular) part of B is A 12 · · · · · λ + A 1 , 2 N · · · · · · A N − 1 ,N A N − 1 ,N +1 λ + A N − 1 ,N +2 · · λ + A N ,N +1 A N ,N +2 · · A N +1 ,N +2 · · · A 2 N − 2 , 2 N A 2 N − 1 , 2 N (B.9) Therefore w e ha v e Pf ( B ) = N X l =0 λ N − l S l , (B.10) where S l is a sum ov er certain ‘minors’. Sp ecifically , taking (B.7) in to account to c hec k signs, w e readily get S k = X 1 ≤ j 1 < ···

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