On the integrable perturbations of the Camassa-Holm equation

R. A. Kraenkel, M. Senthilvelan, and A. I. Zenchuk

Citation: Journal of Mathematical Physics 41, 3160 (2000); doi: 10.1063/1.533298 View online: http://dx.doi.org/10.1063/1.533298

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/41/5?ver=pdfcov Published by the AIP Publishing

On the integrable perturbations of the Camassa–Holm

equation

R. A. Kraenkel, M. Senthilvelan, and A. I. Zenchuk Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil

共Received 12 August 1999; accepted for publication 27 December 1999兲

We present an investigation of the nonlinear partial differential equations共PDE兲 which are asymptotically representable as a linear combination of the equations from the Camassa–Holm hierarchy. For this purpose we use the infinitesimal trans-formations of dependent and independent variables of the original PDE. This ap-proach is helpful for the analysis of the systems of the PDE which can be asymp-totically represented as the evolution equations of polynomial structure. © 2000 American Institute of Physics.关S0022-2488共00兲02605-0兴

I. INTRODUCTION

The Camassa–Holm equation 共sometimes also called Fuchssteiner–Fokas–Camassa–Holm equation兲 has been found as an equation describing the propagation of long-waves in shallow-water1,2when higher-order terms are taken into account. A remarkable fact is that, like, say, KdV, it is an integrable equation.1,3This equation reads

m_t

1共u兲⫽CH

共0兲_{共u兲⫽⫺2ku}

x⫺umx⫺2uxm, m⫽u⫺uxx. 共1兲

Many of its properties have been studied recently in Refs. 3–9. Being a completely integrable equation, a hierarchy in the Lax sense can be constructed.4 We will use the following symbolic representation for this hierarchy:

mt_m共u兲⫽CH共m兲共u兲, m⫽u⫺uxx. 共2兲

The simplest equations of this hierarchy are

m_t ⫺1⫽CH 共⫺1兲_⫽⫺m x⫺ 1 2共⳵x 3_⫺⳵ x兲 1

冑

m⫹k, 共3兲 m_t 2⫽CH 共2兲_⫽⫺Vm x⫺2kVx⫺2Vxm, m⫽u⫺uxx, 共4兲

where V is defined by the following equation:

2kux⫹2uxm⫹umx⫹共⳵x 3_⫺

⳵x兲V⫽0.

On the other hand, Eq.共1兲 has been obtained as the leading order term in the expansion of the shallow water system in the powers of small parameters characterizing scales in amplitude and wavelength. Thus, in a physical sense, it is only an approximate equation. We are thus taken naturally to consider the next terms in the perturbative expansions. These terms appear in the CH as the corrections of small order. This leads us to the study the perturbed CH equation.

A rather classical question to ask about the perturbations of an integrable equation is the determination of the perturbations that do not destroy the integrability of the system. Among them are the perturbations which are a superposition of the higher order equations of the corresponding hierarchy. However, we should recall that the equation under consideration is to describe a

physi-JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 5 MAY 2000

3160

cal system. Therefore it comes from the perturbative series in small parameter⑀which has to be truncated at a certain order⑀N. We are, thus, taken to consider as equivalent共up to this order兲 all equations which can be obtained one from the other by infinitesimal transformations up to the same order ⑀N. We say then that the two equations are asymptotically equivalent.10,11 If an equation is asymptotically equivalent to an integrable equation we say that it is asymptotically integrable.12 In the same order of ideas we say that a perturbation of a completely integrable equation is asymptotically integrable, if the resulting equation is asymptotically integrable. With this in mind, the relevant question becomes: what are the perturbations of the CH equation that are asymptotically integrable?. This is the problem we investigate in this paper.

The analogous problem to the one formulated above has been considered for a number of integrable evolution-type equations such as the nonlinear Schro¨dinger equation 共NLS兲,11–15 Korteweg–de Vries equation共KdV兲,10,11,16Kadomtsev–Petviashvili equation共KP兲,12and the Bur-gers equation.17 However, the CH equation is of the nonevolutionary type. This introduces a considerable complication in the problem and will call for several adaptations of the techniques used previously.

The paper is organized as follows. In Sec. II we remind the definition of infinitesimal trans-formation and define the class of nonlinear PDE asymptotically equivalent to the superposition of the equations from the CH-hierarchy共asymptotic integrability兲. In Sec. III we give the multiscale decomposition of the equations considered in Sec. II and consider the first obstacle to the asymptotic integrability of the CH-type equation with perturbations, based on the symmetry approach.12,18Then we discuss different types of infinitesimal transformations共transformations of dependent and independent variables兲. In Sec. IV we formulate the definition of an approximate symmetry for the nonintegrable PDE of the evolutionary type. Finally, we present the general conclusions.

II. INFINITESIMAL TRANSFORMATIONS

We say that a PDE,

m_␶共v兲⫽P共v,⑀兲, ⑀Ⰶ1, m共v兲⫽v⫺vxx 共5兲

is asymptotically equivalent to the CH-hierarchy up to the order⑀N, iff

共i兲 after the expansion in powers of small parameter⑀it becomes m_␶共v兲⫽

兺

k a_k共⑀兲CH共k兲共v兲⫹Q共⑀,v兲, Q⫽

兺

m⫽M₀ M ⑀m_Q m共v兲 共6兲 and

共ii兲 under the infinitesimal transformations of dependent and/or independent variables of the general form,

v共␰,␶n兲⫽u共x,tn兲⫹⑀0F0共x,tn,⑀兲, 共7兲

␰⫽x⫹⑀1F1共x,tn,⑀兲, 共8兲

␶n⫽tn⫹⑀n_⫹1Fn_⫹1共x,tn,⑀兲, n⫽1,2,..., ⑀k⫽⑀k共⑀兲Ⰶ1, 共9兲 Eq.共6兲 gets the following structure:

mt共u兲⫽

兺

k

ak共⑀兲CH共k兲共u兲⫹O共⑀共N兲兲, ⳵t⬅

兺

k

ak共⑀兲⳵t_k⫽⳵␶. 共10兲

m_␶共v兲⫹2kv_␰⫹vm_␰共v兲⫹2v_␰m共v兲⫺Q⫽0, m共v兲⫽v⫺v_␰␰ 共11兲 with perturbationsQ in first order in⑀k,

Q⬅

兺

k ⑀k P_k, F_k⫽F_k共␰,␶_k兲, 共12兲 P0⫽2kF0␰⫹F0␶₁⫺F0␰␰␶₁⫹共3F0v兲␰⫺共2F0␰v␰兲␰⫺F0v␰␰␰⫺F0␰␰␰v, 共13兲 P1⫽⫺2kv␰F1␰⫺F1␶₁v␰⫺3vv␰F1␰⫹6v␰v␰␰F1␰⫹2v␰␰F1␰␶⫹2v_␰ 2 F1␰␰ ⫹v␰␶F1_␰␰⫹3vv_␰␰F1_␰␰⫹2v␰␰␶₁F1_␰⫹v_␰F1␰␰␶₁⫹v_␰␰␰F1␶₁⫹3vv_␰␰␰F1_␰⫹vv_␰F1_␰␰␰, 共14兲 P₂⫽⫺2kF_2␰v_␶ 1⫺v␶1F2␶1⫺3vv␶1F2␰⫹4v␰F2␰v␰␶1⫹2v␰␶1F2␰␶1⫹2F2␰v␰␶1␶1⫹2v␶1F2␰v␰␰ ⫹v␶1␶1F2␰␰⫹2v␶1v␰F2␰␰⫹3vv␰␶1F2␰␰⫹F2␶1v␰␰␶1⫹3vF2␰v␰␰␶1⫹v␶F2␰␰␶1⫹vv␶1F2␰␰␰, 共15兲 Pk⫽⫺2kFk␰v␶_k⫺v␶_kFk␶₁⫺3vv␶_kFk␰⫹4v␰Fk␰v␰␶_k⫹2v␰␶_kFk␰␶₁⫹2Fk␰v␰␶₁␶_k ⫹2v␶kFk␰v␰␰⫹v␶1␶kFk␰␰⫹2v␶kv␰Fk␰␰⫹3vv␰␶kFk␰␰⫹Fk␶1v␰␰␶k⫹3vFk␰v␰␰␶k ⫹v␶kFk␰␰␶1⫹vv␶kFk␰␰␰, k⬎2, 共16兲

to the CH equation up to the order O(⑀2).

In what follows we discuss briefly some features associated with the infinitesimal transforma-tions 共7兲–共9兲. For simplicity let us consider the first order infinitesimal transformations 共7兲–共9兲 with Fk not depending on⑀j( j⫽0,1,...), that is Fk⫽Fk(x,tn). First of all note that the transfor-mation associated with v, Eq. 共7兲, takes a predominant position since it allows us to eliminate

from Eq. 共6兲 the perturbations of the form,

Q⫽⑀0共共1⫺⳵␰ 2_兲⳵

␶1F0共␰,␶1兲⫹G共F0共␰,␶1兲,v兲兲, 共17兲 where G is the nonlinear part of the correction and its structure is defined by the operators CH(n) from Eq. 共6兲 and does not contain the ␶ derivative of the function v. So if one considers the

function F0 of the form,

F₀共v兲⫽ f 共m共v兲兲⫹L共v兲, 共18兲

where f is a local function of m and its␰-derivatives and L is a linear differential operator in␰with constant coefficients then the first term in the perturbation共17兲 has the form,

兺

k

冉

共1⫺⳵␰ 2_兲⳵f共m共v兲兲 ⳵mk共v兲 ⫹ ⳵L共m共v兲兲 ⳵mk共v兲

冊

⳵␶1 mk共v兲, mk共v兲⬅ ⳵k_m共v兲 ⳵␰k ,

and the time derivatives can be eliminated from the perturbation of this order by using the main order of Eq. 共6兲.

As a result, transformation 共7兲 allows us to treat local perturbations in Eq. 共6兲 which are polynomials inv and its␰derivatives. We wish to stress that only the infinitesimal transformation

共7兲 possesses this property whereas the other ones 共8兲, 共9兲 after substitution into Eq. 共6兲 lead to the

t derivatives of the function u which can be eliminated only by introducing a nonlocal operator (1⫺⳵xx)⫺1. This happens due to the fact that the CH is a nonevolutionary type equation which differs from other integrable systems like KdV, NLS, and so on.

Now let us discuss the relations between the solution u(x,t) andv(␰,␶) of Eqs.共10兲 and 共6兲. One can construct without any problem the solution v of Eq. 共6兲 which comes from the given

solution u of Eq.共10兲 by performing the direct calculations through formulas 共7兲–共9兲. But some times one needs to reverse the infinitesimal transformations, which is not a trivial problem. An interesting case of this kind is related with the initial value problem for Eq.共6兲. For example, let us consider a function v0(␰)⫽v(␰,␶)兩␶⫽0 and using this we try to construct the corresponding initial data u0(x)⫽u(x,t)兩t⫽0 for Eq.共10兲. By doing this we relate the initial value problems of Eqs.共6兲 and 共10兲. To perform this algorithm first of all one needs to solve the system of equations for function u0 and variable x,

v0共x⫹⑀1F1共x,0兲兲⫽u0共x兲⫹⑀0F0共x兲, ␰⫽x⫹⑀1F1共x,0兲 共19兲

共one should remember that the functions Fj are the given functions of arguments兲 to find u0(x). It is not difficult to do if all the functions F_kin Eqs.共7兲–共9兲 are given functions of the independent variables x, tn and do not depend explicitly on the solution u itself. In this case the perturbation in Eq.共6兲 has inhomogeneous, linear, and nonlinear terms with variable coefficients. But of particu-lar interest are the situations when the functions Fk’s are the functions of u and its derivatives and do not depend on independent variables explicitly. In this case Eq.共6兲 has the form of a nonlinear equation with constant coefficients and Eq. 共19兲 for determining u0 is a differential equation

共algebraic in particular cases兲 on the function u0and can be integrated both numerically and by the perturbation method, as far as the small parameter⑀is involved in these equations. To clarify the last statement let us note that Eq.共19兲 can be expanded in powers of the parameters⑀kup to the first order

v0共x兲⫺u0共x兲⫺⑀0F0共x,0兲⫹⑀1⳵xv0共x兲F1共x,0兲⫹O共⑀j 2_兲⫽0

so that one can look for the solution u0(x) of the form,

u0共x兲⫽v0共x兲⫹

兺

j⫽0 1

兺

k⬎0⑀j k_u jk共x兲.

We will come back to the problem of inversion of the infinitesimal transformations共7兲–共9兲 in Sec. III B.

Now let us come to the general algorithm to construct the solutions of Eq.共6兲 with an arbitrary perturbation which comes from the physical point of view.

Let us assume for simplicity that the equation under consideration has the form共6兲 with first order perturbation of the formQ⫽⑀Pˆ (␰,␶1). To construct the solutions of these equations up to the order⑀one needs to consider the transformations共7兲–共9兲 which reduces this equation to the form共10兲 up to the order⑀. To do this let us substitute Eqs.共7兲–共9兲 into 共6兲, then the first order correction has the following form⑀(⫺P(Fk(x,t),u)⫹Pˆ(x,t)). To eliminate it one needs to solve the equation

Pˆ共x,t兲⫽P共Fk共x,t兲,u兲 共20兲

with respect to the functions Fk.

This equation is a PDE which can be solved, generally speaking, numerically for any given solution u of Eq. 共10兲. The only requirement is that the functions Fk should be restricted for all values of the parameters␰and␶. Only under these circumstances the transformations共7兲–共9兲 can be considered as the infinitesimal ones. There are many types of the infinitesimal transformations which remove the correction of the above form and one can choose a suitable one according to the convenience. After this the solution of the original equation can be calculated by using Eqs.

共7兲–共9兲 for any given solution u of the transformed Eq. 共10兲.

This general discussion of the first order perturbations can be straightforwardly extended to the perturbations of higher order.

III. MULTISCALE DECOMPOSITION AND ASYMPTOTICALLY EQUIVALENT EQUATIONS

In the previous section we have discussed the possibility for Eq. 共6兲 to be considered as asymptotically equivalent to the CH-hierarchy共10兲. However, we have not specified the role of the small parameter⑀there. Of particular interest is Eq.共6兲 with the parameter⑀which appears from the rescaling of v and x 共multiscale decomposition兲. For example, CH can be derived from the

shallow water equation by introducing the small scales in x and u. So we wish to investigate system共6兲 under scaling. In order to do this let us substitute the rescaled variables,

u→␦1u, v→␦1v, ␰→␦2, ␶→␶/␦2, 共21兲 where ␦k, k⫽1,2 are small parameters, into systems 共2兲 and 共6兲 and expand the equations in powers of these parameters. For convenience let us choose␦1⫽⑀2,␦2⫽⑀. The main advantage of this rescaling is that it allows us to rewrite system共2兲 in an evolutionary form represented by the infinite series ut_k⫽

兺

m⭓0⑀ m_CH m 共k兲_{, 共22兲}

whereas Eq. 共6兲 transforms to

v_␶⫽

兺

k m

兺

⭓0␣k

CH_m共k兲⫹Q共⑀,v兲. 共23兲

We say that Eq.共23兲 is equivalent to the hierarchy of CH given by the above Eq. 共22兲 up to the order⑀Nif there exists infinitesimal transformations of the form共7兲–共9兲 such that Eq. 共23兲 can be brought to the form,

ut⫽

兺

k m

兺

_⭓0 ⑀

k_a

kCHm共k兲⫹O共⑀

N⫹1_{兲. 共24兲}

In the following we write down a few equations explicitly which comes from the CH hierarchy under the scaling共21兲 共up to the order⑀6_),

ut₁⫽⫺2kux⫺⑀2共3uux⫹2kuxxx兲⫺⑀4共7uxuxx⫹2uuxxx⫹2kuxxxxx兲

⫺⑀6_共2ku

7⫹23uxxuxxx⫹11uxuxxxx⫹2uuxxxxx兲⫹O共⑀8兲, 共25兲

ut₃⫽⫺ux⫺ ux 2k3/2⫺⑀ 2

冉

uxxx 2k3/2⫹ 3uux 4k5/2

冊

⫺⑀ 4

冉

15u 2_u x 16k7/2 ⫹ 3uxuxx 4k5/2 ⫹ 3uuxxx 4k5/2

冊

⫹⑀6

冉

35u 3_u x 32k9/2 ⫹ 15uuxuxx 8k7/2 ⫹ 15u2uxxx 167/2 ⫹ 3uxxuxxx 4k5/2

冊

⫹O共⑀ 8_{兲, 共26兲} ut_⫺2⫽⫺4k2ux⫺k⑀2共12uux⫹8kuxxx兲 ⫺⑀4

冉

15u 2_u x 2 ⫹48kuxuxx⫹18kuuxxx⫹12k 2_u 5

冊

⫺⑀6

冉

_16k2_u 7⫹ 35u_x3 2 ⫹55uuxuxx⫹10u 2_u xxx⫹202kuxxuxxx

⫹108kuxuxxxx⫹26kuuxxxxx

冊

⫹O共⑀8兲. 共27兲

We wish to mention that only the above equations from the CH-hierarchy have a completely local polynomial structure under the multiscale transformation, whereas all other equations contain the nonlocal terms.

With these ideas in mind let us now proceed to investigate Eq.共23兲 with corrections by using the concept of asymptotic integrability of the PDE of the evolutionary type which was discussed in detail in Ref. 12. Even though the main ideas of our algorithm are the same, we find certain important differences related with the particular structure of the CH-hierarchy.

A. Asymptotically commuting flows and obstacles to the integrability

To begin with let us recall some facts from the symmetry approach to the integrability of the nonlinear PDE in (1⫹1)-dimensions.12,18Suppose that the evolutions in t1 and t2 are described by the equations,

ut₁⫽F共u兲, ut₂⫽G共u兲, 共28兲

where F(u) and G(u) are functions of u and its x-derivatives, are considered as commuting flows if ut₁t₂⫽ut₂t₁. In other words,

K共F,G兲⬅

兺

k

冉

⳵F ⳵u_k⳵x k G⫺⳵G ⳵u_k⳵x k F

冊

⫽0. 共29兲

Analogously if G and F are represented in the form of a series in small parameter⑀,

F⫽

兺

k ⑀ k_F k, G⫽

兺

k ⑀ k_G k, 共30兲

then we can call these flows are commuting up to the order⑀Nwhen the commutator共29兲 is of the order O(⑀(N⫹1)): K(F,G)⫽O(⑀(N⫹1)). It is evident that the infinitesimal transformations共7兲–共9兲 do not disturb the integrability.

In the following we focus our attention on evolutionary type equations having local polyno-mial type perturbations, i.e., the functions Fkand Gkin共30兲 have local polynomial structure in u and its x derivatives.

We wish to construct a general polynomial type Eqs. 共28兲–共30兲 which would completely commute with CH. If it so then the equation we have constructed can said to be a local polynomial symmetry for the CH-equation. By the direct commutation one can find this symmetry of the form

共we give several terms兲

ut⫽␣01ux⫹⑀2共␣21uxxx⫹␣22uux兲⫹⑀4共␣41u5⫹␣42uuxxx⫹␣43uxuxx⫹␣44u2ux兲 ⫹⑀6_共␣ 61u7⫹␣62uu5⫹␣63uxu4⫹␣64uxxuxxx⫹␣65u2uxxx⫹␣66uuxuxx⫹␣67ux 3_⫹␣ 68u3ux兲 ⫹⑀8_共␣ 81u9⫹␣82uu7⫹␣83uxu6⫹␣84uxxu5⫹␣85uxxxu4⫹␣86u2u5⫹␣87uuxu4⫹␣88uuxxuxxx ⫹␣89ux 2_u xxx⫹␣8共10兲uxuxx 2 _⫹␣ 8共11兲u3uxxx⫹␣8共12兲u2uxuxx⫹␣8共13兲uux 3_⫹␣ 8共14兲u4ux兲⫹O共⑀10兲, 共31兲

where all coefficients␣n jare fixed in terms of the coefficients␣k1(k⭐n) which are left arbitrary. We represent them in the following form:

␣22⫽共1/2k兲共3,0,0,0兲, ␣42⫽共1/2k兲共⫺3,5,0,0兲, ␣43⫽共1/2k兲共⫺3,10,0,0兲, ␣44⫽共15/8k2兲共⫺1,1,0,0兲, ␣62⫽共1/2k兲共0,⫺5,7,0兲, ␣63⫽共1/2k兲共0,⫺10,21,0兲,

␣66⫽共5/4k2兲共3,⫺17,14,0兲, ␣67⫽共35/8k2兲共0,⫺1,1,0兲, ␣68⫽共35/16k3兲共1,⫺2,1,0兲, ␣82⫽共1/2k兲共0,0,⫺7,9兲, ␣83⫽共3/2k兲共0,0,⫺7,12兲, ␣84⫽共1/2k兲共0,5,⫺42,84兲, ␣85⫽共1/k兲共0,5,⫺28,63兲, ␣86⫽共7/8k2兲共0,5,⫺14,9兲, ␣87⫽共7/4k2兲共0,10,⫺37,27兲, ␣88⫽共⫺15/4k2兲共1,⫺8,28,⫺21兲, ␣89⫽共21/8k 2_{兲共0,5,⫺28,23兲, ␣} 810⫽共1/8k 2_{兲共⫺15,155,⫺791,651兲,} ␣811⫽共35/16k3兲共⫺1,5,⫺7,3兲, ␣812⫽共105/16k3兲共⫺1,8,⫺13,6兲, ␣813⫽共315/16k3兲共0,1,⫺2,1兲, ␣814⫽共315/128k4兲共⫺1,3,⫺3,1兲, 共32兲 where we adopted the designation

共a1,a2,a3,a4兲⫽共a1␣21⫹a2␣41⫹a3␣61⫹a4␣81兲.

It follows from the above form of corrections that if the first nontrivial term in the symmetry共31兲 is of the order⑀N, then all higher order terms cannot be equal to zero. From this it follows the most important property of the symmetry共31兲; this symmetry is represented by an infinite series in powers of the parameter ⑀.

Now let us construct a differential equation which has a general polynomial structure and asymptotically equivalent to the general symmetry共31兲 of the CH. For this purpose let us consider an infinitesimal transformation of the form,

v⫽u⫹⑀2_共a

21u2⫹a22u2⫹a23⳵x⫺1共u兲u1兲⫹⑀3共a31uu1⫹a32u3兲

⫹⑀4_共a

41u3⫹a42u1 2_⫹a

43uu2⫹a44u4⫹a45u3⳵x⫺1u⫹a46u2⳵x⫺1u2

⫹a47uu1⳵x⫺1u⫹a48u1⳵x⫺1u2兲⫹¯ 共33兲

which transforms Eq.共23兲 into an another polynomial equation. If this equation coincides with the symmetry共31兲 up to the order⑀N, then the original Eq.共23兲 is asymptotically equivalent to the CH-symmetry up to the order ⑀N. We suppose that the transformation 共33兲 covers all possible transformations共7兲–共9兲, which are related with the perturbations of the local polynomial type, but we leave the proof of this statement beyond the scope of this paper. Also we consider only perturbations up to the order⑀6. Let us substitute the transformation共33兲 into Eq. 共23兲 and take the coefficients akn in formula 共33兲 from the condition that the final transformed equation coin-cides with the symmetry 共31兲. This can be done if the original Eq. 共23兲 has the form,

vt⫽␣01vx⫹⑀2共␣21vxxx⫹␣22vvx兲⫹⑀4共␣41v5⫹␤42vvxxx⫹␤43vxvxx⫹␤44v2vx兲 ⫹⑀5_␤ 51共vxx 2 _⫹v xvxxx兲⫹⑀6共␣61u7⫹␤62vv5⫹␤63vxv4⫹␤64vxxvxxx ␤65v2vxxx⫹␤66vvxvxx⫹␤67vx 3_⫹␤ 68v3vx兲⫹O共⑀7兲. 共34兲

It is worth noting that the order⑀5appears in the above Eq.共34兲, while it does not exist in the symmetry共31兲. The correction up to the order⑀4 does not give any obstacle to the integrability while the correction of the order⑀6 requires the coefficients␤knto satisfy the following relation:

3共70␣61⫺k共85␤62⫺30␤63⫹12␤64兲⫹k2共16␤65⫹18␤66⫺48␤67兲⫺24k3␤68兲␣21 ⫺共300␣41⫺335k␣41␤42⫹k2共⫺12␤43 2 _⫹60␤ 42␤43⫹8␤42 2_⫹120␣ 41␤44兲 ⫹k3_共24␤ 43␤44⫹8␤42␤44兲兲⫽0. 共35兲

The above Eq.共35兲 represents an obstacle to the integrability for Eq. 共34兲 up to the order⑀6. It is the same as the one derived in Refs. 10 and 16.

B. Different types of infinitesimal transformations

In the previous discussion we have considered an equation of the general form共34兲 which is reducible to an integrable equation up to the order⑀6through an infinitesimal gauge transforma-tion 共7兲, 共33兲. However, we have already seen in Sec. II. 关vide Eq. 共19兲兴 that it is not so easy to reverse this transformation, if required. That is why for the practical purpose of constructing solutions to the equation with perturbation it can be useful to consider infinitesimal transformation of the independent variables as far as each of Eqs. 共7兲–共9兲 has its own distinct structures.

In this section we demonstrate the advantage of the infinitesimal transformation共8兲 by con-sidering a simple example.

Let us consider an equation of the form,

vt⫽

兺

k CH_k共0兲共v兲⫺Q共⑀,v兲, Q⫽⑀5_共9a 31v2 2_⫺6ka 32v2 2_⫹9a 31vxv3⫺6ka32vxv3兲⫹⑀6共3a41vx 3_⫺12ka 41v2v3兲. The perturbation Q can be treated by two manners; either through the transformation

共i兲 v⫽u⫹⑀3_共a

31uxxx⫹a32uux兲⫹⑀4a41ux 2

, ␰⫽x, ␶⫽t, v⫽v共␰,␶兲, u⫽u共x,t兲, 共36兲 or through the transformation,

共ii兲 v共␰,␶兲⫽u共x,t兲, ␰⫽x⫹⑀2b21u⫹⑀2b31u␰, ␶⫽t, v⫽v共␰,␶兲, u⫽u共x,t兲, 共37兲 where

b21⫽ 1

2k共3a31⫺2ka32兲, b31⫽⫺a41.

Each of these transformations leads to CH up to the order⑀6. Of course it is more convenient to consider Eq. 共36兲 if one needs to construct the solution v(␰,␶) related with given u(x,t). How-ever, if one needs to reverse the infinitesimal transformation and find function u which is related with the givenv共for instance to solve the initial value problem兲 the second transformation 共37兲 is

more preferable since it involves only one differentiation with respect to x. However, we will not consider an explicit example for this kind of solutions here.

IV. GENERALIZATIONS OF THE ASYMPTOTIC INTEGRABILITY

Generally speaking, the asymptotic integrability is not related with complete integrability. For instance, it is possible to consider the equation of the general form, which does not possess an exact symmetry but can possess an approximate symmetry,

ut_j⫽␣01共 j兲u1⫹⑀共␣11共 j兲u2⫹␣12共 j兲u 2_兲⫹⑀2_共␣ 21 共 j兲_u 3⫹␣22共 j兲uu1兲 ⫹⑀3_共␣ 31 共 j兲_u 4⫹␣32共 j兲u1 2_⫹␣ 33 共 j兲_uu 2⫹␣34共 j兲u 3_兲⫹⑀4_共␣ 41 共 j兲_u 5⫹␣42共 j兲uu3⫹␣共 j兲43u1u2⫹␣44共 j兲u 2_u 1兲 ⫹⑀5_共␣ 51 共 j兲_u 6⫹␣52共 j兲u2 2_⫹␣ 53 共 j兲_u 1u3⫹␣54共 j兲uu4⫹␣55共 j兲uu1 2_⫹␣ 56 共 j兲_u2_u 2⫹␣共 j兲57u4兲⫹⑀6共␣61共 j兲u7 ⫹␣62共 j兲uu5⫹␣63共 j兲u1u4⫹␣64共 j兲u2u3⫹␣65共 j兲u 2_u 3⫹␣66共 j兲uu1u2⫹␣67共 j兲u1 3_⫹␣ 68 共 j兲_u3_u 1兲. 共38兲 To find an approximate symmetry, let us consider the commutator of two equations of the above form. It is represented by the infinite series in powers of ⑀. One can find the restrictions on the coefficients of Eq. 共38兲 from the conditions that this commutator is of the order ⑀(N⫹1). Direct calculations give the following results. The commutator is of the order⑀3 if

⫺2␣12共 j兲␣11共i兲⫹2␣11共 j兲␣12共i兲⫽0, of the order⑀4 if

⫺2␣22共 j兲␣11共i兲⫺6␣12共 j兲␣21共i兲⫹6␣21共 j兲␣12共i兲⫹2␣11共 j兲␣22共i兲⫽0, ␣22共 j兲␣12共i兲⫺␣12共 j兲␣22共i兲⫽0,

of the order⑀5 _if

⫺2␣32共 j兲␣共i兲11⫺3␣共 j兲22␣21共i兲⫺6␣12共 j兲␣31共i兲⫹6␣31共 j兲␣12共i兲⫹3␣21共 j兲␣22共i兲⫹2␣11共 j兲␣32共i兲⫽0,

⫺2␣33共 j兲␣共i兲11⫺3␣共 j兲22␣21共i兲⫺8␣12共 j兲␣31共i兲⫹8␣31共 j兲␣12共i兲⫹3␣21共 j兲␣22共i兲⫹2␣11共 j兲␣33共i兲⫽0, ␣33共 j兲␣12共i兲⫺␣12共 j兲␣33共i兲⫽0,

⫺6␣34共 j兲␣共i兲11⫹2␣共 j兲32␣12共i兲⫹2␣33共 j兲␣12共i兲⫺2␣12共 j兲␣32共i兲⫺2␣12共 j兲␣33共i兲⫹6␣11共 j兲␣34共i兲⫽0, ␣34共 j兲␣12共i兲⫺␣12共 j兲␣34共i兲⫽0,

of the order⑀6 if

⫺2␣43共 j兲␣共i兲11⫺6␣共 j兲32␣21共i兲⫺3␣33共 j兲␣21共i兲⫺10␣22共 j兲␣共i兲31⫺20␣12共 j兲␣41共i兲⫹20␣41共 j兲␣12共i兲

⫹10␣31共 j兲␣22共i兲⫹6␣21共 j兲␣32共i兲⫹3␣21共 j兲␣33共i兲⫹2␣11共 j兲␣43共i兲⫽0,

⫺2␣42共 j兲␣11共i兲⫺3␣33共 j兲␣21共i兲⫺4␣22共 j兲␣共i兲31⫺10␣共 j兲12␣41共i兲⫹10␣41共 j兲␣12共i兲⫹4␣31共 j兲␣22共i兲⫹3␣21共 j兲␣33共i兲

⫹2␣11共 j兲␣42共i兲⫽0, ␣42共 j兲␣12共i兲⫺␣12共 j兲␣42共i兲⫽0,

⫺2␣44共 j兲␣11共i兲⫺6␣34共 j兲␣21共i兲⫹2␣43共 j兲␣12共i兲⫹␣32共 j兲␣22共i兲⫺␣22共 j兲␣共i兲32⫺2␣12共 j兲␣43共i兲⫹6␣21共 j兲␣34共i兲⫹2␣11共 j兲␣44共i兲⫽0,

⫺4␣44共 j兲␣11共i兲⫺18␣共 j兲34␣21共i兲⫹2␣43共 j兲␣12共i兲⫹6␣42共 j兲␣12共i兲⫹2␣33共 j兲␣22共i兲⫺2␣22共 j兲␣33共i兲

⫺2␣12共 j兲␣43共i兲⫺6␣12共 j兲␣42共i兲⫹18␣21共 j兲␣34共i兲⫹4␣11共 j兲␣44共i兲⫽0, 2␣₄₄共 j兲␣₁₂共i兲⫺␣₃₄共 j兲␣₂₂共i兲⫹␣₂₂共 j兲␣₃₄共i兲⫺2␣₁₂共 j兲␣₄₄共i兲⫽0,

and so on. Equation共38兲, with different i, j, whose coefficients satisfy the above conditions up to the order⑀Nare said to be commuting up to the order⑀N. Analogously one can consider a class

of equations, equivalent to the given one, as it was done in Sec. II for the CH-type equations. We are not considering this question here. The important fact is that none of Eq.共38兲 with different j does necessarily belong to a completely integrable hierarchy. There is no regular way to construct, in general, the solutions of this type of equations except by the perturbation method.

V. CONCLUDING REMARKS

The PDE of the polynomial structure共38兲 can be considered as the perturbations of the KdV-or CH-type equations. The obstacles to integrability do not depend on what kind of the integrable equation we are considering. But these equations have quite different solutions, for example, cuspon and peakon of CH. And both of these equations possess soliton solutions. We have seen that the CH hierarchy under the rescaling共21兲 gets the structure of the evolution equations of the polynomial type which are not scale invariant and are represented as infinite series in powers of ⑀关see Eqs. 共25兲–共27兲兴. This is the main point of the CH hierarchy.

We have considered a class of equations which is asymptotically equivalent to the CH-hierarchy up to the order⑀6.

We have discussed the advantages of different types of the infinitesimal transformations

共7兲–共9兲. We have shown that along with gauge transformation 共7兲 the transformation of

indepen-dent variables are useful in the cases when one needs to reverse the infinitesimal transformations

共for example, for solving the initial value problem兲.

Generally speaking, the asymptotic integrability is not related with complete integrability. One can consider the approximate symmetries of the nonintegrable Eq. 共38兲 and definition of the equation which is asymptotically equivalent to the given共nonintegrable兲 one.

ACKNOWLEDGMENTS

The authors thank the referee for calling their attention to the fact that the equation studied in this paper is sometimes called the Fuchssteiner–Fokas–Camassa–Holm equation. The authors M.S. and A.Z. thank FAPESP共Brazil兲 for financial support.

1_{R. Camassa and D. D. Holm, Phys. Rev. Lett. 71, 1661共1993兲.} 2_{A. S. Fokas, Physica D 87, 145共1995兲; Phys. Rev. Lett. 77, 2347 共1996兲.} 3

B. Fuchssteiner and A. S. Fokas, Physica D 4, 47共1981兲.

4

M. S. Alber, R. Camassa, D. D. Holm, and J. E. Marsden, Lett. Math. Phys. 32, 137共1994兲.

5_{B. Fuchssteiner, Physica D 95, 229共1996兲.}

6_{Y. A. Li and P. J. Olver, Discrete Cont. Dyn. Syst. 3, 419共1997兲.} 7_{Y. A. Li and P. J. Olver, Discrete Cont. Dyn. Syst. 4, 159共1998兲.} 8

A. Zenchuk, JETP Lett. 68, 715共1998兲.

9_{R. A. Kraenkel and A. I. Zenchuk, J. Phys. A 32, 4733共1999兲.} 10_{Y. Kodama, Phys. Lett. A 112, 193共1985兲.}

11_{Y. Kodama, Phys. Lett. A 107, 245共1985兲.} 12

Y. Kodama and A. V. Mikhailov, ‘‘Obstacles to Asymptotic Integrability,’’ in Algebraic Aspects of Integrability, edited by I. M. Gelfand and A. Fokas共Birkhauser, Basil, 1996兲.

13_{T. Kano, J. Phys. Soc. Jpn. 58, 4322共1989兲.}

14_{A. Hasegawa and Y. Kodama, Opt. Lett. 15, 1443共1990兲.} 15_{A. Hasegawa and Y. Kodama, Phys. Rev. Lett. A 66, 161共1991兲.} 16

Y. Kodama, Phys. Lett. A 123, 276共1987兲.

17_{R. A. Kraenkel, J. G. Pereira, and E. C. de Rey Neto, Phys. Rev. E 58, 2526共1998兲.}

18_{A. V. Mikhailov, A. B. Shabat, and V. V. Sokolov, ‘‘The Symmetry Approach to Classification of the Integrable}