Equations of the Camassa-Holm Hierarchy

Equations of the Camassa-Holm Hierarc h y Rossen I. Iv a no v 1 Abstract The squared eigenfunctions of the s pec tral problem asso cia ted with the CamassaHo lm (CH) equation represent a complete basis of functions , which helps to descr ib e the inv ers e scatter ing transform for the CH hi- erarch y as a generalized F ourier transform (GFT). All the fundamental prop erties of the CH equation, such as the integrals of motion, the de- scription of t he equations of the whole hiera rch y , a nd their Hamiltonian structures, can b e naturally expressed using the completeness relation and the recursion oper ator, whose eigenfunctions a re the squared so - lutions. Using the GFT, w e explicitly describ e so me member s o f the CH hier arch y , including in tegrable deformations for the CH equatio n. W e also show that so lutions of so me (1 + 2) - dimensiona l member s of the CH hierar ch y can b e constructed using results for the in verse scattering transform fo r the CH equation. W e give an exa mple of the pea kon so lution of one such equatio n. Keyw ords: Inv erse Scattering, S olitons, P eak ons, Int egrable systems, Lax P air 1 In tro duction The Camassa-Holm (CH) equation [1] b ecame f amous as a m o del in the theory of wat er wa ves. It is also kno wn that it describ es axially symmetric w av es in a hyp erelastic r o d [2, 3]. The most prominen t representat iv e of the w ater-w a ve equations, the Kortew eg-de V ries (KdV) equation d o es n ot describ e th e w a ve -braking ph enomenon. In addition to the stable soliton so- lutions, the CH equation, together with another recentl y derive d nonlinear in tegrable equation, the Degasp eris-Pro cesi equation, has smo oth solutions that dev elop singularities in finite time via a p ro cess that captures the f ea- tures of the breaking wa v es: the s olution remains b ounded, but the slop e b ecomes unboun ded [4]. F or the physic al r elev ance of these tw o equations as m o dels for the propagation of shallo w water wa v es o v er a flat b ottom one can consult, for example [5, 6, 7, 8, 9, 10, 11]. More ab out the physical applications, mo difications and the t yp e of s olutions of the CH equation can also b e found in [4, 12, 13, 14, 15, 16, 17 , 18, 19, 20]. The CH equ ation has the form u t − u xxt + 2 ω u x + 3 uu x − 2 u x u xx − uu xxx = 0 , (1) 1 School of Mathematical Sciences, Du blin Institu te of T ec hnology , Kevin Street, Dub lin 8, Ireland, T el: + 353 - 1 - 402 4845, F ax: + 353 - 1- 402 4994, e-mail: riv anov@dit.ie 1 where ω is a real constan t. This equatio n is completely integrable and admits a Lax p air [1] Ψ xx =  1 4 + λ ( m + ω )  Ψ , (2) Ψ t =  1 2 λ − u  Ψ x + u x 2 Ψ + γ Ψ , (3) where γ is an arbitrary constan t and m = u − u xx . The CH solitary w a v es are stable solitons if ω > 0 [6, 12, 13, 21] or p eak ons if ω = 0 [1, 22, 23]. The KdV and CH equations can also b e int erpreted as geo desic flo w equations for the r esp ectiv e L 2 and H 1 metrics on the Bott-V irasoro group [24, 25, 26, 27, 28, 29]. The CH equation is a b i-Hamiltonian equation, i.e. it admits t w o com- patible hamiltonian structures J 1 = (2 ω ∂ + m∂ + ∂ m ), J 2 = ∂ − ∂ 3 [1, 30]: m t = − J 2 δ H 2 [ m ] δ m = − J 1 δ H 1 [ m ] δ m , (4) H 1 = 1 2 Z mu d x, (5) H 2 = 1 2 Z ( u 3 + uu 2 x + 2 ω u 2 )d x. (6) The infinite sequence of conserv ation la ws (multi-Ha miltonian structure) H n [ m ], n = 0 , ± 1 , ± 2 , . . . , satisfying J 2 δ H n [ m ] δ m = J 1 δ H n − 1 [ m ] δ m (7) can b e computed explicitly [1, 31, 32, 33, 34]. 2 Generalized F ourier transform The so-called recur s ion op erator plays an imp ortant role in describing an in tegrable h ierarc hy . T he r ecursion op erator f or the CH h ierarc hy is L = J − 1 2 J 1 . The eigenfun ctions of the recursion op erator are the squared eigen- functions of the CH sp ectral pr oblem. F or simplicit y w e consider the con- crete case where m is a Sch w artz class function, ω > 0 and m ( x, 0 ) + ω > 0. Then m ( x, t ) + ω > 0 for all t , e.g. see [35, 15]. It is con ve nien t to in tro duce the notation: q ≡ m + ω . Let k 2 = − 1 4 − λω , i.e. λ ( k ) = − 1 ω  k 2 + 1 4  . (8) A basis in the space of solutions of (2) can b e in tro du ced: f + ( x, k ) and ¯ f + ( x, ¯ k ). F or all real k 6 = 0 it is fixed by its asymp totic when x → ∞ [35], (also see [32, 36, 37]): lim x →∞ e − ik x f + ( x, k ) = 1 , (9) 2 W e can in tro duce another basis, f − ( x, k ) and ¯ f − ( x, ¯ k ) fixed by its asymptotic when x → −∞ f or all real k 6 = 0: lim x →−∞ e ik x f − ( x, k ) = 1 , (10) Because m ( x ) and ω are real we find that if f + ( x, k ) and f − ( x, k ) are solu- tions of (2) then ¯ f + ( x, ¯ k ) = f + ( x, − k ) , and ¯ f − ( x, ¯ k ) = f − ( x, − k ) , (11) are also s olutions of (2). Th e squared s olutions are F ± ( x, k ) ≡ ( f ± ( x, k )) 2 , F ± n ( x ) ≡ F ( x, iκ n ) , (12) where F ± n ( x ) are related to the discrete sp ectrum k = iκ n , where 0 < κ 1 < . . . < κ n < 1 / 2. Using the asymptotics (9), (10) and the Lax equation (2) one can sho w that L ± F ± ( x, k ) = 1 λ F ± ( x, k ) . (13) where L ± = ( ∂ 2 − 1) − 1 h 4 q ( x ) − 2 Z x ±∞ d ˜ x m ′ ( ˜ x ) i (14) is the recursion op erator. The inv erse of this op erator is also w ell defined. W e int ro duce the notation ∂ − 1 ± ≡ R x ±∞ d ˜ x . Th e squared s olutions (12) form a complete basis in the sp ace of the S c hw artz class functions m ( x ), and y , t , can b e treated as some additional parameters. Also, the Generalised F our ier T rans form (GFT) f or q and its v ariation ov er this basis is [34] r ω q ( x ) − 1 = ± 1 2 π i Z ∞ −∞ 2 k R ± ( k ) ω λ ( k ) F ± ( x, k )d k + N X n =1 2 κ n ω λ n R ± n F ± n ( x ) , (15) ∂ − 1 ± δ ( √ q ) √ q = 1 2 π i Z ∞ −∞ iδ R ± ( k ) ω λ ( k ) F ± ( x, k )d k ± N X n =1 h δ R ± n − R ± n δ λ n ω λ n F ± n ( x ) + R ± n iω λ n δ κ n ˜ F ± n ( x ) i , (16) where ˜ F ± n ( x ) ≡ ∂ ∂ k F ± ( x, k ) | k = iκ n . T he generaliz ed F our ier co efficien ts R ± ( k ), R ± n , together with the set of discrete eigen v alues, are called sc attering data . The v ariation is with resp ect to any additional parameter, e.g. y , t . The equations of the CH Hierarc h y can b e w ritten as P 2 ( L ± ) 2 ∂ − 1 ± ( √ q ) t √ q + P 1 ( L ± )  r ω q − 1  = 0 , (17) 3 where P 1 ( z ) and P 2 ( z ) are t w o p olynomials. If Ω( z ) = P 1 ( z ) P 2 ( z ) is a r atio of these t wo p olynomials one can defin e Ω( L ± ) ≡ P 1 ( L ± ) P − 1 2 ( L ± ) (provided P 2 ( L ± ) is an in v ertible op erator). T hen (17) can b e written in the equiv alen t form q t + 2 q ˜ u x + q x ˜ u = 0 , ˜ u = 1 2 Ω( L ± )  r ω q − 1  . (18) Due to the completeness of the s q u ared eigenfunctions basis, fr om (17), (15) and (16) w e obtain linear differen tial equ ations for the scattering data: R ± t ∓ ik Ω( λ − 1 ) R ± ( k ) = 0 , (19) R ± n,t ± κ n Ω( λ − 1 n ) R ± n = 0 , (20) λ n,t = 0 . (21) The GFT for other in tegrable systems is derived e.g. in [38, 39, 40, 41, 42, 43]. Example: W e now consider the case Ω( z ) = a − 1 z − 1 + a 0 + a 1 z (where a j are constant s). The (17) equation can then b e rewritten as L ± 2 ∂ − 1 ± ( √ q ) t √ q + ( a − 1 + a 0 L ± + a 1 L 2 ± )  r ω q − 1  = 0 , (22) T aking the identitie s L ± 2 ∂ − 1 ± ( √ q ) t √ q = − 4 ∂ − 1 ± u t , 1 2 L ±  r ω q − 1  = u, L ± u = − 2(1 − ∂ 2 ) − 1 ∂ − 1 ± ( uq x + 2 q u x ) in to accoun t, w e obtain an in tegrable equation q t + a 1 (2 q u x + q x u ) − a 0 2 q x − a − 1 4 ( ∂ − ∂ 3 ) r ω q = 0 , (23) whic h b ecomes the Camassa-Ho lm equation (1) with the c hoice a 1 = 1, a 0 = a − 1 = 0. Therefore, (23) can b e considered as an in tegrable ’d eformed’ v ersion of the C H equatio n. Another c hoice for the constan ts, a − 1 = 1, a 0 = a 1 = 0, leads to the extended Dym equation [1, 34, 44]. If a 1 = 1, a − 1 = 0 but a 0 6 = 0 the equation is usu ally called Dullin-Gott wa ld-Holm Equation [7, 8, 18, 19]. The Hamiltonian of (18) with r esp ect to the P oisson brac ket related to the Hamiltonian op erator J 1 , { A, B } = − Z ( ω + m )  δ A δ m ∂ δ B δ m − δ B δ m ∂ δ A δ m  d x, (24) 4 is (see [36]) H Ω = Z ∞ 0 k 2 Ω( λ − 1 ) π ω λ ( k ) 2 ln  1 − R ± ( k ) R ± ( − k )  d k − 2 ω N X n =1 Z κ 2 n λ 2 n Ω( λ − 1 n )d κ n . (25) In (25) the Hamiltonian is giv en in terms of the Scattering data. In general, it is not str aigh tforwa rd to find the corresp onding expressions in terms of the field v ariable q ( x ) (or m ( x )). F or example, th e Hamiltonia n of (23) with resp ect to (24) is H Ω = a 1 H C H 1 − a 0 2 I 0 − a − 1 4 H C H − 1 , where H C H 1 = 1 2 R ∞ −∞ mu d x is th e fir st CH Hamiltonian (5), H C H − 1 = 1 2 Z ∞ −∞ h 4 r ω q − 4 r q ω  2 + √ ω q 2 x 4 q 5 / 2 i d x, (26) is the (-1)- st Hamilto nian for the CH equation, and the inte gral I 0 = Z ∞ −∞ m d x = H C H 0 + 2 ω α, (27) is related to the other t wo CH integ r als [34] H C H 0 = Z ∞ −∞ ( √ q − √ ω ) 2 d x, α = Z ∞ −∞  r ω q − 1  d x. 3 P eak on solutions of a (1+2) - dimensional equa- tion from the CH hierarc h y W e consid er an in tegrable mem b er of the CH hierarc hy with t wo ’time’ v ariables - t and y (cf. [45]) q t + 2( U xy + η U xx ) q + ( U y + η U x + γ ) q x = 0 , q = U x − U xxx + ω , (28) where ω , γ and η are arbitrary constan ts. The Lax pair for (28) is Ψ xx =  1 4 + λ ( m + ω )  Ψ ,  ∂ t − 1 2 λ ∂ y  Ψ = −  U y + η U x + γ − η 2 λ  Ψ x + 1 2 ( U xy + η U xx )Ψ . The Lax pair repr esents a non-isosp ectral prob lem. In deed, the equation (28) can b e written as a compatibilit y condition  ∂ t − 1 2 λ ∂ y  (Ψ xx ) = ∂ 2 x  ∂ t − 1 2 λ ∂ y  Ψ , 5 where λ s atisfies the relaxed condition λ t − 1 2 λ λ y = 0. But we assume that the sp ectrum is t - and y -indep endent in what follo ws, and we can generalise the solutions obtained for the CH equation. W e can also write (28) as ( √ q ) t + [( U y + η U x + γ ) √ q ] x = 0 . (29) Then ∂ − 1 ± ( √ q ) t + ( U y + η U x + γ ) √ q + β = 0 , (30) where β is an in tegration constan t. F urther, c h o osing β = − γ √ ω and using the ident ities U y = − 1 2 L ±  ∂ − 1 ± ( √ q ) y √ q  , U x = 1 2 L ±  r ω q − 1  (31) w e can write (28) in the form ∂ − 1 ± ( √ q ) t √ q − 1 2 L ±  ∂ − 1 ± ( √ q ) y √ q  +  η 2 L ± − γ  r ω q − 1  = 0 . (32) T aking (32), (15) and (16 ) into account and consid er in g v ariations with resp ect to y and t w e obtain linear equations for the scattering d ata: R ± t − 1 2 λ R ± y ± 2 ik  γ − η 2 λ  R ± = 0 , (33) R ± n,t − 1 2 λ n R ± n,y ∓ 2  γ − η 2 λ n  κ n R ± n = 0 , (34) assuming λ n,t = 0. F or example, if γ = η = 0, th en the solution is an y function of t + 2 λy (with appropriate deca ying prop erties): R ± ( y , t ) = R ± ( t + 2 λy ) , R ± n ( y , t ) = R ± n ( t + 2 λ n y ) . (35) W e can obtain CH equation itself (1) for x = y , u = U x , γ = η = 0. W e demonstrate how to write explicit p eak on solutions for this equation ( γ = η = ω = 0). Un til no w , ω was strictly p ositiv e in our considerations, but w e can take the limit ω → 0 [46] whic h pro du ces ’p eak ed’ solitons and the equations for the sca ttering data should also hold in this case. W e assume that x k = x k ( t, y ), p k = p k ( t, y ) and introduce the notations ε ( x ) ≡ sign( x ), p ′ k ( t, y ) = ∂ p k ∂ y etc. The ansatz that pro du ces the N -p eak on solution for the CH equation can b e generalised as q ( x ; t, y ) = N X k =1 p k δ ( x − x k ) . 6 W e hence obtain U ( x ; t, y ) = 1 2 N X k =1 p k ε ( x − x k )(1 − e −| x − x k | ) , U y ( x ; t, y ) = 1 2 N X k =1 [ p ′ k ε ( x − x k )(1 − e −| x − x k | ) − p k e −| x − x k | x ′ k ] , U xy ( x ; t, y ) = 1 2 N X k =1 [ p ′ k e −| x − x k | + p k ε ( x − x k ) e −| x − x k | x ′ k ] . Using this ansatz and the iden tit y f ( x ) δ ′ ( x − x 0 ) = f ( x 0 ) δ ′ ( x − x 0 ) − f ′ ( x 0 ) δ ( x − x 0 ) w e obtain the follo wing system of PDEs for the quantiti es x k ( t, y ), p k ( t, y ) from (28): ˙ x l = 1 2 N X k =1 [ p ′ k ε ( x l − x k )(1 − e −| x l − x k | ) − p k e −| x l − x k | x ′ k ] , (36) ˙ p l = − 1 2 p l N X k =1 [ p ′ k e −| x l − x k | + p k ε ( x l − x k ) e −| x l − x k | x ′ k ] , (37) where ˙ x k ( t, y ) = ∂ x k ∂ t etc. The solutions of this sy s tem can b e obtained from the N -p eak on solution for the CH equation [22, 23], where the scattering data are now arb itrary functions of their argumen t (with appropriate de- ca ying prop erties): R n ≡ R + n ( t + 2 λ n y ). F or example, if N = 1, then the system has the form ˙ x 1 + 1 2 p 1 x ′ 1 = 0 , ˙ p 1 + 1 2 p 1 p ′ 1 = 0 , with a solution p 1 = − 1 λ 1 = const , x 1 = ln R 1 ( t + 2 λ 1 y ), cf. [22, 23]. Th e system for N = 2 p eak ons is (we assume that x 1 < x 2 for all y and t , i.e. the case for w hic h the N -p eako n solution for the CH equation is obtained in [22, 23]) ˙ x 1 = 1 2 [ − p ′ 2 (1 − e −| x 1 − x 2 | ) − p 1 x ′ 1 − p 2 e −| x 1 − x 2 | x ′ 2 ] , (38 ) ˙ x 2 = 1 2 [ p ′ 1 (1 − e −| x 1 − x 2 | ) − p 1 e −| x 1 − x 2 | x ′ 1 − p 2 x ′ 2 ] , (39) ˙ p 1 = − 1 2 p 1 [ p ′ 1 + p ′ 2 e −| x 1 − x 2 | − p 2 e −| x 1 − x 2 | x ′ 2 ] , (40) ˙ p 2 = − 1 2 p 2 [ p ′ 1 e −| x 1 − x 2 | + p ′ 2 + p 1 e −| x 1 − x 2 | x ′ 1 ] , (41) 7 with solutions [22, 23] p 1 = − λ 2 1 R 1 + λ 2 2 R 2 λ 1 λ 2 ( λ 1 R 1 + λ 2 R 2 ) , p 2 = − R 1 + R 2 λ 1 R 1 + λ 2 R 2 x 1 = ln ( λ 1 − λ 2 ) 2 R 1 R 2 λ 2 1 R 1 + λ 2 2 R 2 , x 2 = ln( R 1 + R 2 ) , where R k = R k ( t + 2 λ k y ) (this can b e easily verified). W e n ote th at the total momentum p 1 + p 2 = − λ 1 + λ 2 λ 1 λ 2 is conserve d. O f course, in general, for an arbitrary N , the N -p eako n solution for the CH equation obtained in [22, 23] can b e used b ecause the in verse scattering metho d for hierarc hy (18) is the same as for th e CH equation [34, 36]. The only d ifference is the time dep end ence of the scattering d ata (and/or the additional y -dep endence, etc.). In this example, y has the meaning of a s econd ’time’ v ariable. Clearly , the conserv ed qu an tities in terms of x k and p k ha ve the s ame form as those for the C H p eak ons and can b e expressed in terms of the quantit ies λ k , whic h w e already assumed to b e indep end en t of y and t . But the Hamiltonian form u lation is problematic b ecause formally Ω( z ) ≡ 0 for the peako n solution and the Hamiltonian with resp ect to (24) is degenerate b ecause of (25). Moreo v er , the r igh t-hand side of system (38) – (41) inv olv es not only the quan tities x k and p k but also their y -deriv ativ es. The explicit dep endence on the scattering data give n in [36] can b e used in the same w ay for the N -soliton solutions of the CH hierarc hy . The sit- uation wh ere the in itial d ata condition q ( x, 0) ≡ m ( x, 0) + ω > 0 do es n ot hold is more complicated and requires a separate analysis [4, 35, 47]. Ac kno wledgmen ts . The author is thankful to Pr of. A. Constantin for the opp ortun ity to visit Lund Universit y (Sweden) wh ere part of the researc h presente d in th is work has b een done. This w ork wa s supp orted b y th e G. Gustafsson F oundation for Research in Natural Sciences and Medicine (Sw eden ). References [1] R. Camassa and D. Holm, Phys. R ev. L ett. , 71 , 16 61 (1993); arXiv:patt-so l/ 9305 002v1 [2] H-H Da i, A cta Me ch. , 127 , 193 (19 98). [3] A. Co nstantin and W. Strauss, Phys. L ett. A , 270 , 140 (200 0 ). [4] A. Co nstantin and J . Escher, Acta Math. , 181 , 229 (199 8 ). [5] R.S. J ohnson, J. Fluid. Me ch. , 457 , 63 (2002). [6] R.S. 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