Generalised Fourier Transform and Perturbations to Soliton Equations

Generalise d F ourier T ransform and P erturbatio ns to Soliton Equations Georgi G. Graho vski † 1 2 and Rossen I. Iv ano v ‡ 3 † Scho ol of Ele ctr oni c Eng i n e ering, Dublin City University, Glasne v i n, Dublin 9, Ir ela n d ‡ Scho ol of Mathematic al Scienc es, Dublin Institute of T e chnolo gy, Kevin Str e et, Dublin 8, Ir ela nd Abstract A brief surve y of the theory of soliton p ertu rbations is presented. The fo cus is on the usefulness of the so -called Generalised F ourier T ransform (GFT). This is a metho d that in v olv es expansions ov er the co mplete basis of “squared solutions” of the sp ectral pr ob lem, asso ciated to the soliton equation. The In ve rse Scattering T rans form f or the corresp onding hierarch y of soliton equ ations can b e view ed as a GFT wh ere the expansions of the solutions hav e generalised F ourier co efficien ts giv en b y the scattering data. The GFT pro vides a natural setting f or the analysis of small p erturbations to an in tegrable equation: starting fr om a purely solito n solution one can ’mo dify’ the soliton parameters suc h as to incorp orate the c hanges caused by the p ertur bation. As illustrativ e exa mples the p ertur b ed equations of the Kd V hierarc hy , in partic- ular the Ostro vsky equation, follo wed b y the p ertur bation theory for the Camassa- Holm hierarc hy are presen ted. AMS sub ject classification n umbers Pr imary: 37K15, 37K40, 37K55; Secondary: 35P10, 35P25, 35P30 Key W ords : In ve rse Scattering Metho d, Soliton P erturbations, KdV equation, Camassa-Holm equation, Ostro vs ky equation 1 In t ro du ction In tegrable equations are widely used as mo del equations in v arious pro blems. The in te- grabilit y concept o r ig inates from the fact that these equations are in some sense exactly solv able, e.g. b y the inv erse scattering metho d (ISM), and exhibit global r egular solu- tions. This feature is v ery import an t for applications, where in general analytical resu lts (first integrals, pa rticular solutions) are preferable to n umerical computations, whic h are not only long and costly , but also intrins ically sub ject to n umerical error. In a h ydro dy- namic con text, ev en though w a ter w av es are expected to b e unstable in general, they do exhibit certain stability prop erties in ph ysical regimes where in tegrable mo del equations are accurate approx imations for t he ev o lutio n o f the free surface w ater w av e cf. [1, 2 1]. There a r e situations ho w eve r where the mo del equation is not in tegrable, but is some- ho w close to an in tegrable equation, i.e. can b e considered as a p erturbatio n of an in te- grable e quation. In suc h case it is still p ossible to obtain approximate analytical solutions. 1 E-mail: grah@i nrne.b as.bg 2 On leav e from Institute for Nuclear Research and Nuclear Energy , Bulgarian Academy of Sciences, 72 Tsarigr asko c haussee, 1784 Sofia, Bulgaria 3 E-mail: rivano v@dit. ie 1 There are sev eral approaches treating t he perturbations of in tegrable equations. One p os- sibilit y is to consider expanding the solutions of the p erturb ed nonlinear equation around the corresp onding unp erturb ed solution and to determine the corrections due to p ertur- bations. In other w ords, o ne represen ts the solutions ˜ u ( x, t ) in the form: ˜ u ( x, t ) = u ( x, t ) + ∆ u ( x, t ) , where u ( x, t ) is the solution of the corr esp onding unperturb ed nonlinear evolutionary equation and ∆ u ( x, t ) is a p erturbation. The strength of the p erturbation is measured b y a parameter ǫ , ∆ u ( x, t ) = O ( ǫ ). By smal l (we ak) p erturb ation o ne means 0 < ǫ ≪ 1. Suc h p erturbat io ns can b e studied dir e ctly in the configuration (co o rdinate) space , while the effect of t he p erturbations o n the corresp onding scattering data can b e studied in the sp e ctr al sp ac e (usually the complex plane of the spectral pa r ameter) of the asso ciated sp ectral problem. F or a di r e ct study of soliton p erturba t ions, one can use the m ulti-scale expansion metho d [29, 30], introducing multiple scales, i.e. transforming t he indep enden t time v ariable t in to sev eral v ariables t n , ( n = 0 , 1 , 2 , . . . ) b y t n = ǫ n t, n = 0 , 1 , 2 , . . . , where each t n is an order of ǫ smaller than the previous time t n − 1 . Then, the time-deriv ativ e are replaced by the expansion (the so- called “ deriv ativ e expansion”) with respect to the m ultiple scales : ∂ t = ∞ X n =0 ǫ n ∂ t n . The dependen t v ariable is expanded in an asymptotic series u ( x, t ) = ∞ X n =0 ǫ n u n ( x, t ) . These expre ssions are substituted bac k in to the equation, g iving a sequence of equations for u n ( x, t ), correspo nding to eac h order of ǫ (each time scale t n = ǫt ). Solving the system of equations for u n ( x, t ), one has to ensure tha t there a r e no singularities in the solutions (i.e. that the solutions do not blow up in time, etc.). This may lead to some additional conditions on the functions u n ( x, t ) (or on the parameters in them), kno wn as se cular c onditions . Sev eral author s had used v a rious ve rsions of the direct approac h in the study of soliton p erturbations: D . J. Kaup [66] had used a similar approac h for t he p erturb ed sine-Gordon equation. Keener and McLaughlin [69] had prop osed a direct approach by obtaining the appropriate G reen functions for the nonlinear Schrodinger and sine-Gordon equations. F or a comprehensiv e review of the direct p erturbation theory see e.g. [29, 44] and the references therein. In the sp ectral space, the study of the soliton p erturbations is based on the p erturba- tions o f the scattering data, associated to the spectral problem. Such metho ds are used b y a n um b er of authors, for studying p erturbations of v arious nonlinear ev olutionary equa- tions, like the sin-G ordon equation [65], the nonlinear Schr¨ odinger equation [70, 38 , 37 ] and of course, the KdV equation whic h is discussed in details in the f ollo wing sections. The metho d is based on expanding the ’p otential’ (i.e. t he dep enden t v aria ble) u ( x, t ) of 2 the associat ed sp ectral problem ov er the complete set of “squared solutions”, whic h are eigenfunctions of the correspo nding recursion op erator. The squared eigenfunctions of the sp ectral problem a sso ciat ed to an in tegrable equa- tion repre sen t a complete basis of functions, whic h helps to de scrib e the In v erse Scattering T ransform f or the corresp onding hierarc hy as a Generalised F ourier transform (GF T). The F ourier mo des for the GF T are the Scattering data. Th us all the fundamental prop erties of an in tegrable equation suc h as the in tegra ls of motion, the description of the equations of the whole hierarc h y and their Hamiltonian structures can b e naturally expressed mak- ing use of t he completeness relation for the squared eigenfunctions and the prop erties of the recursion op erator. The GFT also provides a natural setting for t he analysis of small p erturbations to an in tegrable equation. The leading idea is that starting from a purely soliton solution of a certain in t egr a ble equation one can ’mo dify’ the soliton parameters s uc h as to inc orp o r ate the changes caused b y the p erturbation. There is a con tributio n to t he equations for the scattering data that comes from the GFT-expansion of the p erturbation. In this review article w e illustrate these ideas with sev eral examples. F irstly w e con- sider the equations o f the KdV hierarch y a nd the KdV p erturb ed v ersion – the Ostrov sky equation. Then w e presen t the perturbatio n theory for the Camass a-Holm hierarc h y . 2 Basic facts for th e in v erse scattering metho d for the KdV hierarc hy 2.1 Direct scattering transform and scattering data The spectral problem for the equations of the KdV hierarc h y is [82, 48] − Ψ xx + u ( x )Ψ = k 2 Ψ , (1) in which the real-v alued p oten tial u ( x ) is ta k en for simplicit y to b e a function of Sc hw artz- t yp e: u ( x ) ∈ S ( R ), k ∈ C is sp ectral parameter. The con tin uous sp ectrum under these conditions corresp onds to real k . The discrete sp ectrum consists of finitely man y p oints k n = iκ n , n = 1 , . . . , N where κ n is real. The Jost solutions for (1) are as follo ws: f + ( x, k ) a nd ¯ f + ( x, ¯ k ) are fixed b y their asymptotic when x → ∞ for all real k 6 = 0 [82]: lim x →∞ e − ik x f + ( x, k ) = 1 , (2) f − ( x, k ) and ¯ f − ( x, ¯ k ) fixed by their asym ptotic when x → − ∞ for all real k 6 = 0: lim x →−∞ e ik x f − ( x, k ) = 1 , (3) Since u ( x ) is real then ¯ f + ( x, ¯ k ) = f + ( x, − k ) , and ¯ f − ( x, ¯ k ) = f − ( x, − k ) . (4) In par t icular, fo r real k 6 = 0 w e hav e: ¯ f ± ( x, k ) = f ± ( x, − k ) , (5) 3 and the v ectors of the t wo bases are related 4 : f − ( x, k ) = a ( k ) f + ( x, − k ) + b ( k ) f + ( x, k ) , Im k = 0 . (6) The co efficien t a ( k ) allows analytic extension in t he upp er half of the complex k -plane [82] and ¯ a ( ¯ k ) = a ( − k ) , ¯ b ( ¯ k ) = b ( − k ) , | a ( k ) | 2 − | b ( k ) | 2 = 1 . (7) The quan tities R ± ( k ) = b ( ± k ) /a ( k ) are kno wn a s reflection co efficien t s (to the righ t with sup erscript (+) and to the left with sup erscript ( − ) resp ectiv ely). It is sufficien t to know R ± ( k ) only on the half line k > 0, since from (7): R ± ( − k ) = ¯ R ± ( k ) and also (7) | a ( k ) | 2 = (1 − |R ± ( k ) | 2 ) − 1 , (8) F urthermore R ± ( k ) unique ly determines a ( k ) [82]. A t the p oin ts κ n of the discre te sp ec- trum, a ( k ) has simple zero es i.e.: a ( k ) = ( k − iκ n ) ˙ a n + 1 2 ( k − iκ n ) 2 ¨ a n + · · · , (9) The dot stands for a deriv a tiv e with resp ect to k and ˙ a n ≡ ˙ a ( iκ n ), ¨ a n ≡ ¨ a ( iκ n ), etc. The follo wing dispersion relation holds: a ( k ) = exp  − 1 2 π i Z ∞ −∞ ln(1 − |R ± ( k ′ ) | 2 ) k ′ − k d k ′  N Y j = 1 k − iκ j k + iκ j . (10) A t the p oints of the discrete spectrum f − and f + are linearly dep enden t: f − ( x, iκ n ) = b n f + ( x, iκ n ) . (11) In o t her w ords, the discrete spectrum is simple, there is only one (real) linearly indep en- den t eigenfunction, corresp onding to eac h eigen v a lue iκ n , sa y f − n ( x ) ≡ f − ( x, iκ n ) (12) F rom (12 ) a nd (2), (3) it follows that f − n ( x ) falls off exp onen tially for x → ±∞ , whic h allo ws one to sho w t ha t f n ( x ) is square integrable. Moreo v er, for compactly supp orted p oten tials u ( x ) (cf. (11) and (6)) b n = b ( iκ n ) , b ( − iκ n ) = − 1 b n . (13) One can argue [82], that the results from this case can b e extended to Sc h w arz-class p oten tials b y a n appropriate limiting pro cedure. The asym ptotic of f − n , according to (5), (2), (11) is f − n ( x ) = e κ n x + o ( e κ n x ) , x → −∞ ; (14) f − n ( x ) = b n e − κ n x + o ( e − κ n x ) , x → ∞ . (15) 4 According to the notations used in [82] f + ( x, k ) ≡ ¯ ψ ( x, ¯ k ), f − ( x, k ) ≡ ϕ ( x, k ). 4 The sign of b n ob viously dep ends on the n um b er of the zero es of f − n . Supp ose that 0 < κ 1 < κ 2 < . . . < κ N . Then f r o m the oscillation t heorem for the Sturm-Liouville problem [3], f − n has exactly n − 1 zeroes. Therefore b n = ( − 1) n − 1 | b n | . (16) The se ts S ± ≡ {R ± ( k ) ( k > 0) , κ n , R ± n ≡ b ± 1 n i ˙ a n , n = 1 , . . . N } (17) are called scattering data. Clearly , due to (10) eac h set – S + or S − of scattering data uniquely determines the other one and also the p otential u ( x ) [82, 48, 88]. 2.2 Generalised F ourier T ransform The recurs ion op erator for the KdV hiererc h y is L ± = − 1 4 ∂ 2 + u ( x ) − 1 2 Z x ±∞ d ˜ xu ′ ( ˜ x ) · . (18) The eigenfunctions of the recursion operato r are the squared eigenfunctions of the spectral problem: F ± ( x, k ) ≡ ( f ± ( x, k )) 2 , F ± n ( x ) ≡ F ( x, iκ n ) , (19) where F ± n ( x ) are r elat ed to the discrete spectrum k n = iκ n . Using (1) and the asy mptotics (2), (3 ) one can sho w that L ± F ± ( x, k ) = k 2 F ± ( x, k ) L ± F ± n ( x ) = k 2 n F ± n ( x ) . (20) The Generalised F ourier expansion can b e form ulated a s follo ws: Theorem 2.1 Supp ose that f + and f − ar e not line arly dep endent at x = 0 . F or e ach function g ( x ) ∈ S ( R ) the fol low ing exp ansion formulas hold: g ( x ) = ± 1 2 π i Z ∞ −∞ ˜ g ± ( k ) F ± x ( x, k )d k ∓ N X j = 1  g ± 1 ,j ˙ F ± j,x ( x ) + g ± 2 ,j F ± j,x ( x )  , wher e ˙ F ± j ( x ) ≡ [ ∂ ∂ k F ± ( x, k )] k = k j and the F ourier c o efficien ts ar e ˜ g ± ( k ) = 1 k a 2 ( k )  g , F ∓  , where ( g , F ) ≡ Z ∞ −∞ g ( x ) F ( x )d x, g ± 1 ,j = 1 k j ˙ a 2 j  g , F ∓ j  , g ± 2 ,j = 1 k j ˙ a 2 j   g , ˙ F ∓ j  −  1 k j + ¨ a j ˙ a j   g , F ∓ j   . 5 Pro of: The details of the deriv ation can b e found e.g. in [33, 48].  In part icular one can expand the p oten tial u ( x ), the co efficien ts ar e giv en through the scattering data [33, 48]: u ( x ) = ± 2 π i Z ∞ −∞ k R ± ( k ) F ± ( x, k )d k + N X j = 1 4 ik j R ± j F ± j ( x ) . (21) The v ariation δ u ( x ) under the a ssumption tha t t he n um b er of the discrete eigenv alues is conserv ed is δ u ( x ) = − 1 π Z ∞ −∞ δ R ± ( k ) F ± x ( x, k )d k ± 2 N X j = 1 h R ± j δ k j ˙ F ± j,x ( x ) + δ R ± j F ± j,x i . (22) An imp ort a n t sub class of v ariations are due to the time-ev olution of u , i.e. effectiv ely w e consider a one-parametric family of sp ectral problems, allowing a dep endence of an additional parameter t (time). Then δ u ( x, t ) = u t δ t + Q (( δ t ) 2 ), etc. The equations of the KdV hierar c hy can b e written as u t + ∂ x Ω( L ± ) u ( x, t ) = 0 , (23) where Ω( k 2 ) is a ra tional function sp ecifying the disp ersion law of the equation. The substitution of (22) and (21) in (23 ) due to (20) giv es a system of trivial linear or dinary differen tia l equations fo r the scattering data: R ± t ± 2 ik Ω( k 2 ) R ± = 0 , (24) R ± j,t ± 2 ik j Ω( k 2 j ) R ± j = 0 , (25) k j,t = 0 . (26) The KdV equation u t − 6 u u x + u xxx = 0 (27) can b e obtained for Ω( k 2 ) = − 4 k 2 . Once the scattering dat a are determined from (24 ) – (26) one can reco ver the solution from (22 ). Th us the Inv erse Scattering T ransform can b e view ed as a GFT. 3 P ert urbations of the equations of the KdV hierar- c hy Let us now consider a p erturb ed equation from the KdV hierarc h y , i.e. an equation of the fo rm u t + ∂ x Ω( L ± ) u ( x, t ) = P [ u ] , (28) where P [ u ] is a small p erturba tion, whic h is also assumed in the Sc h w artz-class. T he p erturb ed equation is, in general, non-integrable. One can expand P [ u ] according to 6 Theorem 2.1 and u t and u according to (22) and (21). Their substitution in (28 ) no w apparen tly leads to a mo dificatio n of (24) – (2 6) as follo ws: R ± t ± 2 ik Ω( k 2 ) R ± = ± ( P , F ∓ ) 2 k a 2 ( k ) , (29) R ± j,t ± 2 ik j Ω( k 2 j ) R ± j = − 1 2 k j ˙ a 2 j  ( P , ˙ F ∓ j ) −  1 k j + ¨ a j ˙ a j   P , F ∓ j   , (30) k j,t = − ( P , F ∓ j ) 2 k j ˙ a 2 j R ± j . (31) Note that due to (3 1) as a result of the p erturbation the discrete eigen v alues are time- dep enden t. Another feature is the contribution from the con tin uous sp ectrum: ev en if one start s with a pure solito n solution of the unperturb ed equation ( R ± ( k , 0) = 0) then, in g eneral R ± ( k , t ) 6 = 0 due to (29). F or practical applications it is easier to work with an equation for b n instead of (30). Suc h an equation can b e obta ined as f ollo ws. W e notice that R + n,t =  b n i ˙ a n  t = 1 i ˙ a n b n,t + b n  1 i ˙ a n  t , R − n,t =  1 ib n ˙ a n  t = − 1 i ˙ a n b 2 n b n,t + 1 b n  1 i ˙ a n  t , th us b n,t = i ˙ a n 2 ( R + n,t − b 2 n R − n,t ). Then using (30 ) and the fact t ha t F − n = b 2 n F + n , cf. (11) w e ha v e b n,t + 2 ik n Ω( k 2 n ) b n = i 4 k n ˙ a n  P , b 2 n ˙ F + n − ˙ F − n  . (32) As an example let us consider the adiabatic p erturbatio n of the one-soliton solution of the KdV equation. The o ne- solito n solution is u s ( x, t ) = − 2 κ 2 1 sec h 2 z , z = κ 1 ( x − ξ ) , ξ = 4 κ 2 1 t + ξ 0 . (33) The eigenfunc tions are f ± ( x, k ) = e ± ik x ( k ± iκ 1 tanh z ) k + iκ 1 , a ( k ) = k − iκ 1 k + iκ 1 , b 1 = e 2 κ 1 ξ . (34) The perturb ed solution is u ( x, t ) = − 2 κ 2 1 sec h 2 z + u r ( x, t ) , z = κ 1 ( t )[ x − ξ ( t )] . (3 5) Here u r ( x, t ) is the con tribution from the con tinuous sp ectrum (radiation). F rom (31) w e ha v e κ 1 ,t = − 1 4 κ 1 Z ∞ −∞ P [ u s ( z )]sec h 2 z d z . (36) 7 W riting b 1 = e 2 κ 1 ( t ) ξ ( t ) and using (36) and (32) we obtain ξ t = 4 κ 2 1 − 1 4 κ 3 1 Z ∞ −∞ P [ u s ( z )]  z + 1 2 sinh 2 z  sec h 2 z d z . (37) F or the reflection co efficien t (29 ) giv es R + t − 8 ik 3 R + = ie − 2 ik ξ 2 k κ 1 Z ∞ −∞ P [ u s ( z )] e − 2 ik z /κ 1 ( k − iκ 1 tanh z ) 2 d z . (38) then according to [59] using approximations in Gelfa nd- Levitan-Marc henk o equation one can o btain u r ( x, t ) = κ 1 π d d z Z ∞ −∞ R + ( k ) e 2 ikξ +2 ikz /κ 1  k + iκ 1 tanh z k + iκ 1  2 d k . (39) Alternativ ely , from (21) it follo ws that u r ( x, t ) = 2 π i Z ∞ −∞ k R + ( k ) F + ( x, k )d k = 2 π i Z ∞ −∞ k R + ( k ) e 2 ikξ +2 ikz /κ 1  k + iκ 1 tanh z k + iκ 1  2 d k . (40) Both form ulae giv e an approxim ation o f the radiation compo nen t since the z - deriv ativ e of tanh z can b e neglected [76]. The p erturbation results f o r the Zakharov-Shabat (ZS) t yp e sp ectral problems ha v e b een obta ined firstly in [63] and for KdV in [5 9 ]. As it has b een explained, the p erturbation theory is based on the completeness relatio ns for the squared eigenfunctions. F or t he Sturm-Liouville sp ectral problem suc h relations apparently hav e b een studied as early as in 19 4 6 [4] and then b y other authors, e.g. [71, 48]. The completeness relation for the eigenfunctions of the ZS sp ectral problem is deriv ed in [64] a nd generalisations are studied further in [36, 38, 39, 41, 87], see also [40]. Example: Ostr ovsky e quation. This equation has the form [83]: u t + u xxx − 6 uu x = γ ∂ − 1 u, (41) where ∂ − 1 is an op erator suc h that ( ∂ − 1 u ) x ≡ u , in general not uniquely determined. The Ostrovs ky eq uation can b e view ed as a time-dependen t nonlo cal perturbatio n of the KdV equation (27 ) . Here γ is a constant parameter. The equation is often called the Rotation-Mo dified Ko r t ew eg-de V ries equation. It describes gra vit y wa v es propagat ing do wn a c hannel under the influence of Coriolis force. In essence , u in the equation can b e regarded a s the fluid v elo city in the x − direction. The ph ysical parameter γ measures t he effect of the Earth’s rotation. More details ab out the Ostro vsky equation can b e found e.g. in [83, 7, 79, 85]. In the p erturbation theory γ ≪ 1 play s the role of a small parameter. In o rder to ensure tha t t he p erturbation is deca ying fast enough at x = ±∞ w e tak e the one-soliton KdV solutio n in the f orm u s = 2 κ 2 1 / sinh 2 z (42) 8 whic h can b e obtained formally from (33) for κ 1 ξ 0 = π i/ 2. It is not contin uous at z = 0 but decays fast enough at x = ±∞ . Using the fact that d d z coth z = − 1 sinh 2 z + 2 δ ( z ) = − 1 sinh 2 z + d d z [ θ ( z ) − θ ( − z )] (43) [34] we obtain P [ u s ] = γ ∂ − 1 u s = − 2 γ κ 1 [coth z − θ ( z ) + θ ( − z )] , (44) whic h is an o dd function of z and then (36) giv es κ 1 ,t = 0. Th us the amplitude of the soliton do es not c hange under this p erturbation. The computatio n o f (37 ) giv es a correction t o the v elo city of the soliton: ξ t = 4 κ 2 1 + π 2 γ 8 κ 2 1 . 4 Conser v ation la ws and p ertu rb e d s oliton equation s It is w ell kno wn [88, 82] that the KdV equation is a completely in tegrable Hamiltonia n system and p ossesses infinitely-many integrals of motion. Thes e inte grals can b e con- structed fro m the scattering co efficien ts a ( k ) of the asso ciated sp ectral problem (1) and are polynomials of the dep endent v ariable u ( x, t ) and its x -deriv ative s: I n = Z ∞ −∞ P n ( u, u x , u xx , . . . ) d x, (45) where P n is a p olynomial with resp ect to u and its deriv ative s. Since a ( k ) is time- indep enden t, it can b e view ed as generating functional of integrals of motion a k [82]: ln a ( k ) = ∞ X s =0 I s +1 (2i k ) s . (46) Skipping the details (see e.g. [82]), w e pro vide here o nly the list of the first few integrals of motio n: I 1 = − 1 2 Z ∞ −∞ u ( x ) d x ; (47) I 2 = − 1 2 Z ∞ −∞ u ( x ) 2 d x ; (48) I 3 = − 1 2 Z ∞ −∞  u 2 x ( x ) + 2 u 3 ( x )  d x ; (49) I 4 = − 1 2 Z ∞ −∞  u 2 xx − 5 u 2 u xx + 5 u 4  d x ; (50) The KdV equation (27) can b e written as a Hamiltonian syste m u t = ∂ ∂ x δ H δ u ( x ) , (51) 9 where the sym b ol δ /δ u denotes v ariatio nal deriv ativ e. Moreov er, (51) can b e further represen ted in its Hamiltonia n form with a Hamiltonian H : u t = { u, H } . (52) where H = I 3 (49). The P oisson brac k et is defined as { F , G } ≡ Z δ F δ u ( x ) ∂ ∂ x δ G δ u ( x ) d x. (53) The first three inte grals of motion (47)–(49) hav e the same interpretation for all mem b ers of KdV hirarch y: The first one, I 1 is related to the algebraic structure o f the P oisson brac k et (53): it follow s from t he presence o f the op erator ∂ / ∂ x in the P oisson brac k ets. The inte gral (48) has a me aning of a momen tum. It is related to the t r anslation inv ariance of the Hamiltonian. Since H [ u ( x + ε )] − H [ u ( x )] ≡ 0, the expans ion of R ( H [ u ( x + ε )] − H [ u ( x )])d x in ε ab o ut ε = 0 gives (note that u ( x ) = δ P /δ u ) 0 = Z δ H δ u ( x ) ∂ u ∂ x d x = Z δ H δ u ( x ) ∂ ∂ x δ P δ u ( x ) d x ≡ { P , H } = P t , Consider no w the p erturb ed KdV eq uation (28). W e will seek the integrals of motion for the p erturb ed equation ˜ I k in the form ˜ I k = I k + ∆ I k , k = 1 , 2 , . . . . Here ∆ I k can b e view ed as a correction to the inte grals of mo t io n of the unp erturb ed equation (27) coming fro m the p erturbations P [ u ]. After a direct in tegration of (28), one gets: ∆ I 1 = Z ∞ −∞ P [ u ] d x. (54) Then, m ultiplying (27) by u ( x, t ), and in tegrating leads to: ∆ I 2 = 2 Z ∞ −∞ uP [ u ] d x, (55) and so on. As an illustrativ e example w e will ta ke again the Ostro vsky equation (41). Due to the concrete c hoice of the p erturbation in the rig h t hand side of (41), the in tegrals in (54) and (55) v anish, so the p erturbatio ns do not con tribute to these integrals: the first t w o in tegrals of motio n are the same as for t he KdV equation. The non tr ivial con tributions of p erturba tions to the in tegrals of motion in the Hamiltonian I 3 are: ∆ I 3 = γ 2 Z ∞ −∞ ( ∂ − 1 u ) 2 d x. (56) Note also, that there is no second Hamiltonian formulation for the Ostrov sky equation, compatible with the one given ab o v e, i.e. the equation is no t bi-Hamiltonian – indeed (41) is not completely in tegrable for γ 6 = 0, [7]. 10 5 P ert urbations to the equations of the Camassa- Holm hierarc hy Closely related to the KdV hierarc h y is the hie rarch y of the Camassa-Holm (CH) eq uation [6]. This equation has the form u t − u xxt + 2 ω u x + 3 uu x − 2 u x u xx − u u xxx = 0 , (57) where ω is a real constan t. It is inte grable with a Lax pair [6] Ψ xx =  1 4 + λ ( m + ω )  Ψ (58) Ψ t =  1 2 λ − u  Ψ x + u x 2 Ψ + γ Ψ (59) where m ≡ u − u xx and γ is an a rbitrary constant. Both CH and KdV equations a pp eared initially as mo dels of the propagatio n o f t w o- dimensional shallo w w ater w a v es ov er a flat b ottom. Mor e ab o ut the phys ical relev ance of the CH equation can b e found e.g. in [6, 54 , 55, 31, 32, 51, 21, 49, 50]. The pap er [75] suggests tha t KdV and CH migh t b e relev an t to the mo delling of tsunami w av es (see also the discussion in [18]). While all smo o t h data yield solutions of the KdV equation existing for all times, certain smo oth initial data for CH lead to global solutions and others to breaking w a v es: the solution remains b o unded but its slop e b ecomes un b ounded in finite time (see [1 3, 8, 5]). The solitary wa v es of KdV are smo oth solitons, while the solitary w a v es of CH, whic h are also solitons, are smo oth if ω > 0 [6, 55] and p eak ed (called “p eakons” and represen ting w eak solutio ns) if ω = 0 [6 , 14, 2, 23, 77]. Both solita r y w av e forms for CH are stable [26, 24 , 2 7]. It could b e p ointed out that the p eakons app ear also as tra v elling wa v e solutions of greatest height ( f or the go v erning equations for w ater w av es), cf. [11, 12, 86]. In g eometric con text, the CH equation arises as a geo desic equation on the diffeo- morphism gro up (if ω = 0) [8, 19, 20 , 74] and on the Bott-Virasoro group (if ω > 0) [81]. CH equation a lso allow s for solutions with compactly suppor ted m ( x, t ), [10], how ev er u ( x, t ) lo oses instan tly its compact supp ort, whether ω 6 = 0 [42] or ω = 0 [78]. The problem o f p erturba tion of the CH equation arises when one deals with mo del equations that are in general non-in tegrable but close to the CH equation. A p erturbation could app ear for example when one takes in to accoun t the viscosit y effect [84]. Another p ossible sce nario comes from the so-called ’ b -equation’ [28, 47] m t + bω u x + bmu x + m x u = 0 . The b -equation generalizes the CH equation and is integrable only for b = 2 (when it coincides with the CH) and b = 3 (then kno wn as Degasp eris-Pro cesi equation) [80, 47, 52]. Qualitativ ely , the DP equation exhibits most of the features of the CH equation, e.g. the infinite propa gation sp eed for DP w as established in [43]. In [75] it is suggested that DP (as w ell as CH) might b e relev an t to the mo delling o f tsunami w a v es (see also the discussion in [18]). 11 The hydrodynamic relev ance o f the b -equation is discussed e.g. in [56, 51]. There- fore, the solutions o f the b - equation for v alues of b close to b = 2 can b e a nalyzed in the framework of the CH-p erturbatio n theory . W e can represen t the equation as a CH p erturbation m t + 2 ω u x + 2 mu x + m x u = (2 − b )( ω u x + mu x ) ≡ P [ u ] , for a small parameter ǫ = b − 2. 5.1 In v erse scattering and generalised F ourier transform for the CH sp ectral problem The CH sp ectral problem (58) can b e handled in a wa y , similar to the one, already outlined. F or simplicit y we consider the case where m is a Sc hw artz class function, ω > 0 and m ( x, 0) + ω > 0. Let is in tro duce the no t a tion q ( x, t ) = m ( x, t ) + ω . Then one can sho w that q ( x, t ) > 0 for all t [9]. Let k 2 = − 1 4 − λω , i.e. λ ( k ) = − 1 ω  k 2 + 1 4  . (60) The sp ectrum of the problem (58) under these conditions is describ ed in [9]. The con- tin uous sp ectrum in terms o f k corresp onds to k – real. The discrete sp ectrum consists of finitely man y p oin ts k n = iκ n , n = 1 , . . . , N where κ n is real and 0 < κ n < 1 / 2. The con tin uous spectrum v anishes for ω = 0 , [22]. All results (2) – ( 1 7) remain formally the same with the exception of (10) whic h no w has the form [17 , 1 5, 16] a ( k ) = exp  − iαk − 1 2 π i Z ∞ −∞ ln(1 − |R ± ( k ′ ) | 2 ) k ′ − k d k ′  N Y j = 1 k − iκ j k + iκ j . (61) where α = Z ∞ −∞  r q ( x ) ω − 1  d x = N X n =1 ln  1 + 2 κ n 1 − 2 κ n  2 + 4 π Z ∞ 0 ln(1 − |R + ( e k ) | 2 ) 4 e k 2 + 1 d e k is one of the CH integrals of motion (Casimir). With the asymptotics of the Jost solutions and (5 8) one can show that L ± F ± ( x, k ) = 1 λ F ± ( x, k ) L ± F ± n ( x ) = 1 λ n F ± n ( x ) , (62) where λ n = λ ( iκ n ); F ± are again the sq uares of the Jost solutio ns as in (19) and L ± = ( ∂ 2 − 1) − 1 h 4 q ( x ) − 2 Z x ±∞ d y m ′ ( y ) i · (63) is the recursion operat o r. The in v erse of this op erat o r is also w ell defined. 12 The complete ness relation for the eigenfunctions of the recurs ion op erator is [16] ω p q ( x ) q ( y ) θ ( x − y ) = − 1 2 π i Z ∞ −∞ F − ( x, k ) F + ( y , k ) k a 2 ( k ) d k + N X n =1 1 iκ n ˙ a 2 n h ˙ F − n ( x ) F + n ( y ) + F − n ( x ) ˙ F + n ( y ) −  1 iκ n + ¨ a n ˙ a n  F − n ( x ) F + n ( y ) i . (64) Therefore F ± , F ± n and ˙ F ± n can b e considered as ’generalised’ exp onen ts. Lik e in the KdV case it is p ossible to expand m ( x ) and it s v ariatio n ov er the aforemen tioned basis, or rather the quan tities that are determined b y m ( x ) and δ m ( x ), [16]: ω  r ω q ( x ) − 1  = ± 1 2 π i Z ∞ −∞ 2 k R ± ( k ) λ ( k ) F ± ( x, k )d k + N X n =1 2 κ n λ n R ± n F ± n ( x ); (65) ω p q ( x ) Z x ±∞ δ p q ( y ) d y = 1 2 π i Z ∞ −∞ i λ ( k ) δ R ± ( k ) F ± ( x, k )d k ± N X n =1 h 1 λ n ( δ R ± n − R ± n δ λ n ) F ± n ( x ) + R ± n iλ n δ κ n ˙ F ± n ( x ) i (66) The expansion co efficien ts as exp ected are giv en b y t he scattering data and their v aria- tions. This mak es eviden t the interpretation o f the ISM as a generalized F ourier transform. No w it is straightforw ard to describ e the hierarc h y of Camassa-Holm equations. T o eve ry c ho ice of the function Ω( z ), kno wn also a s the disp ersion law we can put in to corre- sp ondence the nonlinear ev olution equation (NLEE) that b elongs to the Camassa-Holm hierarc h y: 2 √ q Z x ±∞ ( √ q ) t d y + Ω( L ± )  r ω q − 1  = 0 . (67) An equiv alen t form of the equation is q t + 2 q ˜ u x + q x ˜ u = 0 , ˜ u = 1 2 Ω( L ± )  r ω q − 1  . (68) The c hoice Ω( z ) = z leads to ˜ u = u and th us to the CH equation (57). Other c hoices of the disp ersion la w and the corresp onding equations of the Camassa-Holm hierarc h y are discusse d in [16, 53]. By virtue of the expansions (65) and (66) the NLEE (67) is equiv alen t to t he follo wing linear ev olution equations for the scattering data: R ± t ( k ) ∓ ik Ω( λ − 1 ) R ± ( k ) = 0 , (69) R ± n,t ± κ n Ω( λ − 1 n ) R ± n = 0 , (70) κ n,t = 0 . (71) The time-ev olution of the scattering data f or the CH e quation (57) can b e compute d fr o m the a b o v e form ula e for Ω( z ) = z , see also [17, 15]. 13 5.2 P erturbation theory for the CH hierarc hy Let us start with a p erturb ed equation of the CH hierarc hy of the form q t + 2 q ˜ u x + q x ˜ u = P [ u ] , ˜ u = 1 2 Ω( L ± )  r ω q − 1  , (72) where again, P [ u ] is a small p erturbation, b y assumption in the Sc hw artz-class. It is useful to write (7 2) in the fo r m 2 √ q Z x ±∞ ( √ q ) t d y + Ω( L ± )  r ω q − 1  = 1 √ q Z x ±∞ P ( y ) p q ( y ) d y . (73) With the completeness relation (64) one can deduce the gereralised F ourier expansion for expressions , lik e the one on the right-hand side o f (72) Theorem 5.1 Assuming that f + and f − ar e not line arly dep endent at x = 0 an d g ( x ) ∈ S ( R ) , the fol lowing exp a nsion formulas hold: ω √ q Z x ±∞ g ( y ) p q ( y ) d y = ± 1 2 π i Z ∞ −∞ ˜ g ± ( k ) F ± x ( x, k )d k ∓ N X j = 1  g ± 1 ,j ˙ F ± j,x ( x ) + g ± 2 ,j F ± j,x ( x )  , (74) wher e ˙ F ± j ( x ) ≡ [ ∂ ∂ k F ± ( x, k )] k = k j and the F ourier c o efficien ts ar e ˜ g ± ( k ) = 1 k a 2 ( k )  g , F ∓  , g ± 1 ,j = 1 k j ˙ a 2 j  g , F ∓ j  , g ± 2 ,j = 1 k j ˙ a 2 j   g , ˙ F ∓ j  −  1 k j + ¨ a j ˙ a j   g , F ∓ j   . The substitution o f the expansions (74 ) for P [ u ], (65) and (6 6) in to the p erturb ed equation (7 3) give s the f o llo wing ex pressions for the mo dified scattering data : R ± t ∓ ik Ω(1 /λ ) R ± = ∓ iλ ( P , F ∓ ) 2 k a 2 ( k ) , (75) k j,t = λ j ( P , F ∓ j ) 2 k j ˙ a 2 j R ± j (76) R ± j,t − R ± j λ j,t ± κ j Ω(1 /λ j ) R ± j = − λ j 2 k j ˙ a 2 j  ( P , ˙ F ∓ j ) −  1 k j + ¨ a j ˙ a j   P , F ∓ j   , (77) F rom (77) w e obtain the follo wing for the co efficien t b j : b j,t + κ j Ω(1 /λ j ) b j = − λ j 4 κ j ˙ a j  P , b 2 j ˙ F + j − ˙ F − j  . 14 The ’p erturb ed’ solution for the hierarc h y in the adiabatic approximation can b e reco ve red from the following expansion for ˜ u ( x ) with the ’mo dified’ scattering data k eeping the unp erturb ed ’generalised’ exp onents: ˜ u ( x ) = ± 1 2 π i Z ∞ −∞ k Ω(1 /λ ( k )) ω λ ( k ) R ± ( k ) F ± ( x, k )d k + N X n =1 κ n Ω(1 /λ n ) ω λ n R ± n F ± n ( x ) . This f orm ula follo ws from the second part of (68) and (65). Note that for the CH equation (57) ˜ u ≡ u . 6 Discuss ions W e ha v e presen ted a review of some asp ects of t he p erturba t io n theory for in tegrable equations using as main examples the KdV and CH hierarc hies. In our deriv ations w e used completeness relations that are v alid only given the as- sumption that the Jo st solutions f + and f − are linearly indep endent at x = 0. The case when this condition is not satisfied is quite exceptional, how ev er this is exactly the case when one has purely solito n solution [33, 48]. Then one ha s to tak e in to accoun t a non trivial con tribution f r om the scattering data at k = 0 [4 6 ] and some of the presen t ed results r equire mo dification. E.g. (37) should b e [58] ξ t = 4 κ 2 1 − 1 4 κ 3 1 Z ∞ −∞ P [ u s ( z )]( z sec h 2 z + tanh z + tanh 2 z )d z . (78) In the presen ted example with the Ostro vsky equation Z ∞ −∞ P [ u s ( z )] tanh 2 z d z = 0 (79) since P ( z ) is an o dd function and the additional term do es not contribute. The meaning of the condition (79) is that no shelf is fo rmed behind the solito n [58 , 46]. The presence of shelf for KdV equation is observ ed e.g. under the p erturbatio n P [ u ] = ǫu [5 8 , 76]. The corrections to the conserv ation la ws due to p erturbations hav e b een used in studies of the effects of the disturbance on the initial soliton [58, 65], or as a correctness c hec k of results o btained ot herwise [67]. The ev aluat io n of the p erturbation terms for the CH hierarc h y could b e tec hnically difficult due to the complicated for m of the CH multisoliton solutions [9 , 15]. Ho w ev er the limit ω → 0 leads to the relatively simple p eakon solutions. Therefore, using the presen ted general form ula e one should b e able to access the p erturbations of the peakon parameters. 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