Two component integrable systems modelling shallow water waves: the constant vorticity case

Tw o comp onen t in tegrable systems mo delling shallo w w ater w a v es: the constan t v orticit y case Rossen I. Iv ano v 1 Scho ol of Mathematic al Scienc es, Dublin Inst itute of T e chnolo gy, Kevin Str e et, Dublin 8, Ir eland Abstract In this contribution w e describe the role of several t wo -comp onent in- tegrable systems in the classical problem of shallow w ater wa ves. The starting point in our deriv ation is the Euler equ ation for an incompress- ible fluid, the equation of mass conserv ation, the simplest b ottom and surface conditions and the constant vorticit y condition. The approximate mod el equ ations are generated by in tro duction of suitable scalings and by truncating asymptotic expansions of t h e quantities to appropriate order. The so obtained equations can b e related to three differen t integra ble sy s- tems: a tw o component generalization of the Camassa-Holm equation, the Zakharov-Ito sy stem and th e Kaup-Boussinesq system. The sig nificance of the results is the inclusion of vorticit y , an imp ortant feature of water wa ve s that has b een given increasing attention during the last decade. The presented inve stigation shows how – up to a certain order – the mo del equations relate to th e shear flow up on whic h the w av e resides. In particular, it shows exactly how th e constant vorticit y affects the equations. Key W ords: w ater wa ve, vorticit y , Camas sa-Holm equation, Za kharov-Ito system, Kaup-Bo ussinesq sy stem, Lax pair, soliton, p ea kon P A CS: 02.30 .Ik, 04.30.Nk, 47.3 5.Bb, 47.35.Fg 1 In tro duc tion The integrable nonlinear equations a re used extensively as approximate mo dels in hydrodyna mics. They descr ib e in a relatively s imple wa y the co mp e tition b e- t ween nonlinea r and disp ersive effects. The b est known example in this rega rd is the K orteweg-de V ries (KdV) equation. The use o f ter m inte gr able corres po nds to the idea that such equations a r e in some s ense exa ctly so lv able and exhibit global reg ular solutions. This featur e is very imp o rtant fo r applications wher e in gener a l a nalytical res ults a re preferable to numerical computations . The Ca massa- Holm (CH) and Degasp eris-Pr o cesi (DP) eq uations [1 , 2, 3] ar e another tw o integrable equations with a pplication in the theory o f water wa ves [1, 4, 5, 6, 7, 8, 9]. The excitemen t that greeted the CH and DP eq uations is due to their non-standar d pr op erties that set them apar t from the clas sical soliton equations such as KdV. The first mo st re ma rk able of these prop erties is 1 E-mail: riv anov@dit.ie 1 the pr esence of multi-soliton solutions consisting of a train of p eaked solitary wa ves (or ’p ea kons’) [1, 10]. Another remar k a ble prop er ty of the CH and DP equations is the o ccur rence of brea king wa ves [1, 11, 12, 13, 14] (i.e. a solution that rema ins bounded while its slop e b ecomes unbounded in finite time [15]) as well as that of smo o th s olutions defined for all times [10, 16, 17]. In many recent publica tions the problem of water wav es with nonzero vor- ticit y and esp ecially with constant vorticity is under inv estigation - e.g. see the publications [6, 1 8, 19, 20, 21, 22, 2 3, 2 4, 25, 26, 27] and the refer ences therein. The no nzero v orticity case arises for example in situations with underly ing shear flow [6]. Our a im is to des crib e the deriv ation of s hallow water mo del equations for the constant vorticit y cas e a nd to demo nstrate how these equatio ns can b e related to some other integrable sys tems: a t wo co mp onent gener alization of the Ca massa- Holm equatio n [2 8], Zakha rov-Ito s ystem [29, 30] and Kaup-Bo ussinesq system [31]. A starting p oint in o ur deriv ation are the equatio ns that ex pr ess the constant vorticity and the mass conserv ation. Another approa ch, based on the Green-Naghdi approximation is used in an alterna tive deriv a tion of the t wo comp onent Cama ssa-Holm equation in [3 2], where the o ccurrence of so lutio ns in the form of br eaking wa ves is als o established. The Ka up-Boussinesq sys tem is used as a mo del in hydrody na mics under slightly differen t assumptions in [15, 31, 3 3]. 2 Go v ern ing equations for the in viscid fluid mo- tion The motion of in viscid fluid with a constant density ρ is describ ed by the Euler’s equations: ∂ v ∂ t + ( v · ∇ ) v = − 1 ρ ∇ P + g , ∇ · v = 0 , where v ( x, y , z , t ) is the veloc ity of the fluid at the p oint ( x, y , z ) at the time t , P ( x, y, z , t ) is the pr essure in the fluid, g = (0 , 0 , − g ) is the constant E arth’s gravit y accelera tion. Consider now a motio n o f a shallow water ov er a flat b o ttom, which is lo cated at z = 0 (Fig. 1). W e a ssume that the motion is in the x -direction, and that the physical v ar iables do not dep end on y . Figure 1: W ater wa ves: general notations 2 Let h b e the mean lev el of the water and let η ( x, t ) describ es the shap e of the water surfa c e, i.e. the deviation from the av erage level. The pres sure is P ( x, z , t ) = P A + ρg ( h − z ) + p ( x, z , t ), wher e P A is the constant atmospheric pressure, and p is a press ure v aria ble, meas uring the deviation fr o m the hydro- static pressur e distribution. On the surface z = h + η , P = P A and therefor e p = η ρg . T aking v ≡ ( u, 0 , w ) we can write the kinematic condition on the surface as w = η t + uη x on z = h + η [34]. Finally , ther e is no vertical velo city at the bo ttom, thus w = 0 on z = 0. All these e quations can b e written as a system u t + uu x + w u z = − 1 ρ p x , w t + uw x + w w z = − 1 ρ p z , u x + w z = 0 , w = η t + uη x , p = η ρg on z = h + η , w = 0 on z = 0 . Let us introduce now dimensio nless parameters ε = a / h and δ = h/ λ , where a is the typical amplitude o f the wav e and λ is the typical wav elength of the wav e. Now we ca n introduce dimensionless quantities, ac cording to the mag nitude of the physical quantities, see [5, 34, 3 5] for deta ils: x → λx , z → z h , t → λ √ gh t , η → aη , u → ε √ g hu , w → εδ √ g hw , p → ερg h . This s caling is due to the observ ation that bo th w and p ar e pro po rtional to ε i.e. the wa ve amplitude, since at undisturb ed sur face ( ε = 0) bo th w = 0 a nd p = 0. The sy s tem in the new, dimensionles s v a r iables is u t + ε ( uu x + w u z ) = − p x , δ 2 ( w t + ε ( uw x + w w z )) = − p z , u x + w z = 0 , w = η t + εu η x , p = η on z = 1 + εη , w = 0 on z = 0 . 3 W a ve s in the presence of shear So far no a ssumptions hav e b een made on the presence of shea r. Now let us notice that there is an exac t solution of the gov erning equations of the form u = ˜ U ( z ), 0 ≤ z ≤ h , w ≡ 0 , p ≡ 0 , η ≡ 0. This solution represents an arbitrar y underlying ’shear ’ flow [6]. Let us consider wa ves in the presence of a s hear flow. In such case the horizontal velo city o f the flow will b e ˜ U ( z ) + u . The s c aling for such solution is clear ly u → √ g h  ˜ U ( z ) + εu  , where u o n the left-hand side is the horizontal velo city b efor e the initial s caling, and the s caling for the other v ar iables is as b efore. Th us, in this case we hav e (the prime denotes deriv ative with resp ect to z ): u t + ˜ U u x + w ˜ U ′ + ε ( uu x + w u z ) = − p x , (1) δ 2 ( w t + ˜ U w x + ε ( uw x + w w z )) = − p z , ( 2) 3 u x + w z = 0 , (3) w = η t + ( ˜ U + εu ) η x , p = η on z = 1 + εη (4) w = 0 on z = 0 . (5) W e consider only the simplest nontrivial case: a linear shear , ˜ U ( z ) = Az , where A is a cons ta nt ( 0 ≤ z ≤ 1). W e cho ose A > 0 , so that the underlying flow is pr opaga ting in the pos itive direction of the x -co o rdinate. Burns condition [3 6] gives the following express ion for c , the s p e e d of the tr avelling wa ves in linea r approximation : c = 1 2  A ± p 4 + A 2  . (6) This expressio n will b e derived again in our further considerations , e.g. see (1 5). Note that if there is no shear ( A = 0), then c = ± 1. Before the scaling the v or ticity is ω = ( U + u ) z − w x or in terms of the rescaled v ariables ( ω → p h/g ω ), ω = A + ε ( u z − δ 2 w x ) . W e are lo oking for a solutio n with co nstant vorticit y ω = A , and therefor e we require that u z − δ 2 w x = 0 . (7) This assumption amounts to considering approximate wav e-solutions that are in teractio ns o f an underlying shear flow and an irro tational disturba nce thereof. F rom (7), (3) and (5) we o btain u = u 0 − δ 2 z 2 2 u 0 xx + O ( ε 2 , δ 4 , εδ 2 ) , (8) w = − z u 0 x + δ 2 z 3 6 u 0 xxx + O ( ε 2 , δ 4 , εδ 2 ) , (9) where u 0 ( x, t ) is the leading order approximation for u . Note that u 0 do es no t depe nd on z since from (7) it follows that u z = 0 when δ → 0. F rom (4) with (8) a nd (9) we obtain η t + Aη x + h (1 + εη ) u 0 + ε A 2 η 2 i x − δ 2 1 6 u 0 xxx = 0 , (10) ignoring terms of order O ( ε 2 , δ 4 , εδ 2 ). F rom (2), (4), (8) and (9) we ha ve (again, ignoring terms o f order O ( ε 2 , δ 4 , εδ 2 )) p = η − δ 2 h 1 − z 2 2 u 0 xt + 1 − z 3 3 Au 0 xx i . Then (1) gives (note that there is no z -de p endence !)  u 0 − δ 2 1 2 u 0 xx  t + εu 0 u 0 x + η x − δ 2 A 3 u 0 xxx = 0 . (11) Letting b oth the par ameters ε and δ to 0, we obtain fr om (10), (11) the system of linear equa tio ns u 0 t + η x = 0 , (12) η t + Aη x + u 0 x = 0 , (13) 4 giving η tt + Aη tx − η xx = 0 . (14) The linear equa tion (14) has a travelling w ave s olution η = η ( x − ct ) with a velocity c sa tisfying c 2 − Ac − 1 = 0 . (15) This gives the same solutio n for c that follows fro m the Bur ns condition (6). There is o ne positive and one negative solution, representing left and r ight running wav es. W e assume that w e ha ve only one of these w av es, then (e.g. from (12)) η = cu 0 + O ( ε, δ 2 ) . (16) Let us introduce a new v ariable ρ = 1 + ε αη + ε 2 β η 2 + εδ 2 γ u 0 xx , (17) for s ome co nstants α , β and γ . These co nstants will b e determined in our further considera tio ns. The v a riable ρ will b e used instead of η as a tool for mathematical s implification of o ur eq ua tions. The expans ion of ρ 2 in the same order of ε and δ 2 is ρ 2 = 1 + ε (2 α ) η + ε 2 ( α 2 + 2 β ) η 2 + εδ 2 (2 γ ) u 0 xx . (18) With this definitio n o ne can expr ess η in ter ms o f ρ and write equa tion (10) in the for m (keeping only ter ms of or der O ( ε, δ 2 )): ρ t + Aρ x αε + δ 2  γ α ( c − A ) − 1 6  u 0 xxx + h (1 + εη ) u 0 + ε A 2 η 2 + ε β α cu 2 0 i x = 0 . (19) One can eliminate the u 0 xxx -term by choo s ing γ α = 1 6( c − A ) . (20) Equation (19) b ec o mes ρ t + Aρ x αε + h 1 + ε (1 + Ac 2 + β α ) η  u 0 i x = 0 . (21) With the choice α = 1 + Ac 2 + β α (22) we can write (21) in the form ρ t + Aρ x + αε ( ρu 0 ) x = 0 , (23) which contains only the v ariables ρ a nd u 0 but not η . 5 4 Tw o comp onen t Camassa-Holm system In this sectio n we pro ceed with our der iv a tion in a direc tion that leads to a tw o comp onent Camassa-Holm system. Expressing η in terms of ρ in (11) we obtain (matchin g ter ms of o r der O ( ε , δ 2 )): m t + Am x − Au 0 x + δ 2  A 6 + κ ( A − c ) − γ α  u 0 xxx + ε  1 − α 2 + 2 β α c 2  u 0 u 0 x + 1 2 εα ( ρ 2 ) x = 0 , (24) where m = u 0 − δ 2 ( 1 2 + κ ) u 0 xx , κ is arbitra ry: we are adding and subtracting δ 2 κu 0 xxt , making us e of (16). Fixing κ = 1 A − c  γ α − A 6  (25) leads to the dis app earanc e of the u 0 xxx - term . The relations (2 0) and (25) g ive κ = 1 6( c − A )  A − 1 c − A  . (26) Thu s eq ua tion (2 4) can b e wr itten a s (matching only terms up to the or der O ( ε, δ 2 )) m t + Am x − Au 0 x + ε 1 3  1 − α 2 + 2 β α c 2  [2 mu 0 x + u 0 m x ] + ρρ x εα = 0 . (27) Recall that m = u 0 − δ 2 B u 0 xx , where (see (26) and (6 )) B = κ + 1 2 = A 2 − c 2 + 2 3( c − A ) 2 = 1 3 c 2 ( c − A ) 2 . (28) Note tha t B is a lwa ys p ositive and the denominator in (28) nonzero (since c 6 = A – see (15)). The rescaling u 0 → 1 αε u 0 , x → δ √ B x , t → δ √ B t in (2 7) and (23) is now only for the sake of mathematical clar ity and simplicity and gives: m t + Am x − Au 0 x + 1 3 α  1 − α 2 + 2 β α c 2  [2 mu 0 x + u 0 m x ] + ρρ x = 0 , m = u 0 − u 0 xx , ρ t + Aρ x + ( ρu 0 ) x = 0 . Finally , we choose 1 3 α  1 − α 2 + 2 β α c 2  = 1 (29) and thus m t + Am x − Au 0 x + 2 mu 0 x + u 0 m x + ρρ x = 0 , m = u 0 − u 0 xx , (30) ρ t + Aρ x + ( ρu 0 ) x = 0 . (31) 6 The constants α , β a nd γ can b e determined from the constraints (22), (29) and (20): α = 1 3(1 + c 2 ) + c 2 3 , (32) β = h 1 3(1 + c 2 ) − 3 + c 2 6 i α, (33) γ = 1 6( c − A ) α. (34) Note that from (32) it fo llows that α is always po sitive. Now let us express the origina l v ariable η in terms of the ’a uxiliary’ v ar iable ρ . Befor e the rescaling we had αε η = ρ − 1 − ε 2 β c 2 u 2 0 − εδ 2 γ u 0 xx . Since in the leading order η = cu 0 the resca ling of η is η → 1 αε η , thus in terms of the rescaled v ariables η = ρ − 1 − β c 2 α 2 u 2 0 − B γ α u 0 xx . With a Galilean transfo r mation (that we use only to simplify our equa tions and to bring them to the form that is widely used), such tha t ∂ t ′ = ∂ t + A∂ x , ∂ x ′ = ∂ x ( x ′ = x − At , t ′ = t ) we o bta in m t ′ − Au 0 x ′ + 2 mu 0 x ′ + u 0 m x ′ + ρρ x ′ = 0 , m = u 0 − u 0 x ′ x ′ (35) ρ t ′ + ( ρu 0 ) x ′ = 0 . (36) The system (35), (36) is an integrable 2-c omp onent Camassa -Holm system that app ear s in [28], ge neralizing the famo us Camassa- Holm equa tion [1]. Let us dro p the primes a nd write it in the for m m t − Au 0 x + 2 mu 0 x + u 0 m x + ρρ x = 0 , m = u 0 − u 0 xx (37) ρ t + ( ρu 0 ) x = 0 . (38) It gener alizes the Camas s a-Holm e q uation [1] in a sense that it can b e obtained from it via the obvious r e duction ρ ≡ 0. The system is integrable, s ince it can be written as a compatibility condition o f t wo line a r systems (Lax pair ) with a sp ectral par a meter ζ : Ψ xx =  − ζ 2 ρ 2 + ζ ( m − A 2 ) + 1 4  Ψ , Ψ t =  1 2 ζ − u 0  Ψ x + 1 2 u 0 x Ψ . The system is also bi-Hamiltonia n. The firs t Poisson bra ck et is { F 1 , F 2 } = − Z h δ F 1 δ m ( − A∂ + m∂ + ∂ m ) δ F 2 δ m + δ F 1 δ m ρ∂ δ F 2 δ ρ + δ F 1 δ ρ ∂ ρ δ F 2 δ m i d x for the Hamiltonia n H 1 = 1 2 R ( u 0 ( m − A 2 ) + ρ 2 )d x . The second Poisson brack et is { F 1 , F 2 } 2 = − Z h δ F 1 δ m ( ∂ − ∂ 3 ) δ F 2 δ m + δ F 1 δ ρ ∂ δ F 2 δ ρ i d x 7 for the Hamiltonian H 2 = 1 2 R ( u 0 ρ 2 + u 3 0 + u 0 u 2 0 x − Au 2 0 )d x. It has tw o Casimir s : R ρ d x and R m d x . The system has an int ere sting interpretation in group-theoretica l context. The first Poisson brack et gives rise to a Lie-algebr aic structure. This fact is well studied in the ca se ρ ≡ 0 when the system co insides with the Cama s sa-Holm equation [38, 39, 40, 4 1, 42]. T he n the corr esp onding Lie alg ebra is the Virasor o algebra. By considering the expansions m = 1 2 π X n ∈ Z L n e inx , ρ = 1 2 π X n ∈ Z ρ n e inx we obtain the following Lie-a lgebra with resp ect of the first Poisson bracket: i { L n , L k } = ( n − k ) L n + k − 2 π Anδ n + k , (39) i { ρ n , L k } = nρ n + k , (40) i { ρ n , ρ k } = 0 (41) The Lie algebra (39) – (41) is a semidirect product of the Vir a soro a lge- bra ( vir ) (39) with a central charge pr op ortiona l to A , and the ab elian algebr a C ∞ ( R ) (41) [3 7]. No te that the cen tra l extension o f the Virasor o alge br a co n- tains only An term but not An 3 term since the Hamilton op era to r contains only the first deriv ative A∂ . The sys tem (3 7 ), (38) r epresents the equatio ns of the geo desic motion on the corres p o nding Lie g roup Vir ⋉ C ∞ ( R ) for the metric, defined by H 1 : || ( u 0 , ρ ) || 2 = Z ( u 2 0 + u 2 0 x + ρ 2 )d x, which is r ig ht-in v ar iant under the natur al gro up action. 5 Zakharo v- Ito system In this section we describ e a deriv ation that ma tches the approximate eq uations (11), (23) (wher e ρ is given in (17)) to the int egr able Z ahk a rov-Ito sys tem [29, 30, 43, 4 4]: u 0 t − 4 k u 0 x + u 0 xxx + 3 u 0 u 0 x + ρρ x = 0 . (42) ρ t + ( u 0 ρ ) x = 0 , (43) where k is an arbitr ary consta nt . The s ystem is forma lly integrable by the virtue of the L a x pa ir [45, 4 6] Ψ xx =  ζ − u 0 2 + k − ζ − 1 ρ 2 16  Ψ , Ψ t = − (4 ζ + u 0 )Ψ x + 1 2 u 0 x Ψ . Equation (11) can b e written in the form (cf. (16) and (12)) 8 u 0 t + δ 2 K u 0 xxx + ε  1 − α 2 + 2 β α  u 0 u 0 x + ρρ x αε = 0 . (44) where K is a co nstant, given by [see (20) and (15)] K = c 2 − A 3 − γ α = 1 3 ( c − A ) . (45) F or o ne of the ro ots of the equation in (6), K is p os itive and for the o ther it is negative. W e can fix the co nstants α , β and γ by the conditions (22) a nd 1 3 α  1 − α 2 + 2 β α c 2  = 1 , which are for mally the s ame a s thos e for the Camassa- Holm s ystem, giving the same express ions (32) – (34). The rescaling u 0 → 1 αε u 0 , x → δ √ K x , t → δ √ K t in (44) and (23) gives: u 0 t + Au 0 x − Au 0 x + u 0 xxx + 3 u 0 u 0 x + ρρ x = 0 , ρ t + Aρ x + ( ρu 0 ) x = 0 . Note that a co ordinate change ( x, t ) → i ( x, t ) maps into a nother system with real v ariables and therefore if K < 0 w e still can apply formally the ab ov e rescaling. One can use a Galilean transfo rmation x ′ = x − At , t ′ = t to obtain u 0 t ′ − Au 0 x ′ + u 0 x ′ x ′ x ′ + 3 u 0 u 0 x ′ + ρρ x ′ = 0 , ρ t ′ + ( ρu 0 ) x ′ = 0 , that matches the Za kharov-Ito system (42), (43) if the constant is chosen to be k = A/ 4 . The ’physical’ v a r iable η in terms o f the ’auxiliary’ ρ and u 0 (for the resca led v ariables ) is η = ρ − 1 − β c 2 α 2 u 2 0 − K γ α u 0 xx . 6 Kaup-Boussinesq system Another in tegra ble system ma tching the water wa ves asymptotic equations to the first o rder of the s mall parameter s ε, δ is the K a up-Boussines q sys tem [3 1, 33]. In this section we describ e briefly its deriv ation. Introducing V = u 0 − δ 2  1 2 − A 3 c  u 0 xx ≡ u 0 − δ 2  1 6 + 1 3 c 2  u 0 xx the equation (1 1) can b e wr itten as V t + εV V x + η x = 0 . (46) Equation (10) in the first order in ε, δ 2 is η t + h Aη + (1 + εη ) u 0 + ε A 2 η 2 i x − δ 2 1 6 u 0 xxx = 0 9 and with a shift η → η − 1 ε it b ecomes η t + ε (1 + Ac 2 )( η u 0 ) x − δ 2 1 6 u 0 xxx = 0 , or η t + ε 1 + c 2 2 ( η V ) x − δ 2 1 6 V xxx = 0 . (47) F urther rescaling in (4 6) and (47) leads to the Kaup-Bouss inesq s ystem V t + V V x + η x = 0 , η t − 1 4 V xxx + 1 + c 2 2 ( η V ) x = 0 , which is integrable iff A = 0 ( c 2 = 1) with a La x pair Ψ xx = −  ( ζ − 1 2 V ) 2 − η  Ψ , Ψ t = − ( ζ + 1 2 V )Ψ x + 1 4 V x Ψ . The integrability of the system, as well as the Inv erse Scatter ing Metho d fo r it has b een inv estigated firstly by D.J. K aup [31]. His motiv atio n ha s b een to derive a n in tegr able w ater -wa ve system with a second-order eigenv alue problem, which is re a dily so lv a ble in compariso n to the third-or der eigenv alue problem for the Bous sinesq e quation. In our context, how ever, this system is r e lev a nt only in the case with zero vorticit y . 7 Discussion Apparently the descr ib e d metho d can be us e d for other tw o-co mp onent inte- grable sys tems with a s imilar structure, e.g. see the cla ssification in [46]. It is int ere s ting to investigate further which sp ecific pr op erties of the original gov- erning equations are pres e rved in the ’integrable’ approximate mo dels. F or example the 2-co mp onent Cama s sa-Holm system for cer tain initial data admits wa ve bre a king [32]. Peakons do not occur in the case A = 0 [32] and most certainly not in the ca s e A 6 = 0, due to the term with linear disp ersion Au 0 x . How ever, in the ’short-wav e limit’ where m = − u 0 xx and A = 0 p ea kon solu- tions ar e p os sible [32]. Recently a similar system with p eakon so lutions have bee n constr ucted in [47]. The case with − ρρ x term (instead of + ρρ x ) in (37) is also integrable [48] and it is s tudied in [49]. The tw o comp o nent Camassa -Holm system app ear s also in plasma theory mo dels [50, 51] and in the theory of metamor phosis [52]. Other in tegr able multi-compo nent gener alizations of the Ca massa-Ho lm equa - tion (including other tw o-comp one nt ones) a r e constructed in [48]. Ac kno wledgmen ts The author is thankful to Pro f. A . Consta ntin, P rof. R. Johns o n and Dr G. Grahovski for stim ulating discussio ns and to b oth refere e s for their co mments and suggestions. P ar t of this work has b een done during the w or kshop ’W av e Motion’ held in Ob erwolfach, Germany (8 – 14 F ebrua r y 20 09). Partial supp or t from INT AS gr ant No 05-10 00008 -788 3 is acknowledged. 10 References [1] R. Ca massa and D.D. Ho lm, An integrable shallow water equation with peaked solitons, Phys. Rev . Lett. 71 (1993 ) 1 661– 1664 ; a rXiv: patt-sol/9 3 0500 2 v 1 [2] A. Dega s p e ris and M. Pro cesi, Asymptotic integrability . In: Symmetry a nd per turbation theory (ed. A. 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