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Bell polynomials approach applied to (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation.

1. Introduction

It is well known that investigation of integrable properties of nonlinear evolution equations (NEEs) can be considered as a pretest and the first step of its exact solvability. The integrability features of soliton equations can be characterized by Hirota bilinear form, Lax pair, infinite symmetries, infinite conservation laws, Painleve test, Hamiltonian structure, Backlund transformation (BT), and so on. The bilinear form of a soliton equation can not only be used to produce many of the known families of multisoliton solutions, but also be employed to derive the bilinear BT, Lax pair, and infinite sets of conserved quantities [1-6]. However, it relies on a particular skill and tedious calculation. In the early 1930s, the classical Bell polynomials were introduced by Bell which are specified by a generating function and exhibiting some important properties [7]. Recently, Lambert and coworkers have proposed a relatively convenient procedure based on Bell polynomials which enables us to obtain bilinear forms, bilinear BTs, Lax pairs, and Darboux covariant Lax pairs for NEEs [8-11]. It is shown that Bell polynomials play an important role in the characterization of bilinearizable equations and a deep relation between the integrability of an NEE and the Bell polynomials. Furthermore, Fan [12], Fan and Chow [13], and Wang and Chen [14, 15] developed the approach to construct infinite conservation laws by decoupling binary-Bell-polynomial-type BT into a Riccati type equation and a divergence type equation. Afterwards, Fan [16] and Fan and Hon [17] extended this method to supersymmetric equations. On the basis of their work, we apply the bell polynomials approach to the high-dimensional variable-coefficient NEEs.

Many physical and mechanical situations are governed by variable-coefficient NEEs, which might be more realistic than the constant coefficient ones in modeling a variety of complex nonlinear phenomena in physical and engineering fields [1820],

The (2 + 1)-dimensional analogue of the Caudrey-DoddGibbon-Kotera-Sawada (CDGKS) equation is in the form of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

with [[partial derivative].sup.-1.sub.x] = [integral] x dx. Equation (1) is first proposed by Konopelchenko and Dubrovsky [21] and then considered by many authors in various aspects such as its quasiperiodic solutions [22], algebraic-geometric solution [23], IV-soliton solutions [24], nonlocal symmetry [25], and symmetry reductions [26]. Based on (1), we will consider a (2 + 1)-dimensional variable-coefficient CDGKS equation as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [a.sub.i] = [a.sub.i](t), i = 1, ..., 9, are analytic functions with respect to t. The aim of this paper is applying the Bell polynomials approach to systematically investigate the integrability of (2), which includes bilinear form, bilinear BT, Lax pair, and infinite conservation laws.

The layout of this paper is as follows. Basic concepts and identities about Bell polynomials will be briefly introduced in Section 2. In Section 3, by virtue of Bell polynomials and the Hirota bilinear method, the bilinear form and IV-soliton solutions of (2) are obtained. In Sections 4 and 5, with the aid of Bell polynomials, the bilinear BT, Lax pair, and infinite conservation laws of (2) are systematically presented, respectively. Section 6 will be our conclusions.

2. Bell Polynomials

The Bell polynomials [7, 9, 10] used here are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where f(x) is a C[infinity] function and [f.sub.rx] = [[partial derivative].sup.r.sub.x]f; according to formula (3), the first three are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Based on one-dimensional Bell polynomials, the multidimensional Bell polynomials are expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

with f = f{[x.sub.1], ..., [x.sub.l]) being a C[infinity] function and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; the associated two-dimensional Bell polynomials can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

The most important multidimensional binary Bell polynomials, namely, Y-polynomials, can be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

with the first few lowest order binary Bell polynomials being

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

The Y-polynomials can be linked to the standard Hirota expressions through the identity [10]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

in which [[summation].sup.l.sub.i=1] [n.sub.i] [greater than or equal to] 1 and the operators [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are classical Hirota bilinear operators defined by [1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Introducing a new field q = w-v, in the particular case F = G one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

in which the even-order P-polynomials is called P-polynomials; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Moreover, the binary Bell polynomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be written as the combination of P-polynomials and Y-polynomials:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Under the Hopf-Cole transformation v = ln [psi], the Y-polynomials can be linearized into the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

which provides a straightforward way for the related Lax systems of NEEs.

3. Bilinear Form and Y-Soliton Solutions for (2)

Firstly, introduce a dimensionless potential field q by setting

u = c[q.sub.2x], (17)

with c = c(t) to be determined. Substituting (17) into (2), integration with respect to x yields the following potential version of (2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (18)

on account of the dimension of u (dim u = -2), we find that setting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [c.sub.0] is an arbitrary constant. In order to write (18) in local bilinear form, here are two cases which are considered to eliminate the effect of the integration [[partial derivative].sup.-1.sub.x]. The bilinear form and N-soliton solutions for each case will be discussed by selecting appropriate constraints on variable coefficients [a.sub.i], I = 1, ..., 9.

3.1. Case 1. Let [a.sub.7] = [a.sub.8]; (18) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

This equation can be viewed as a homogeneous P-condition [8] of weight 6 (the weight of each term being defined as minus its dimension, a weight 3 to y). That means (19) can be written as a linear combination of P-polynomials of weight 6:

[P.sub.x,t] (q) + [a.sub.1][P.sub.6x](q) + [a.sub.5][P.sub.3x,y](q) + [a.sub.6][P.sub.2y](q) = 0; (20)

under the following constraint condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

According to the property (12), via the following transformation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

P-polynomials expression (20) produces the bilinear form of (2) as follows:

([D.sub.x][D.sub.t] + [a.sub.1][D.sup.6.sub.x] + [a.sub.5][D.sup.3.sub.x][D.sub.y] + [a.sub.6][D.sup.2.sub.y]) G x G = 0. (24)

Starting from this bilinear equation, the one-soliton solution of (2) can be easily obtained by regular perturbation method

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

However, the multisoliton solutions cannot be derived by means of bilinear equation (24). For the sake of obtaining multisoliton solutions of (2), we take

[a.sub.5] = 5[c.sub.1][a.sub.1], [a.sub.6] = 5[c.sub.2.sup.1][a.sub.1], (27)

where [c.sub.1] is an arbitrary constant; the bilinear equation can be expressed as

([D.sub.x][D.sub.t] + [a.sub.1][D.sup.6.sub.x] + 5[c.sub.1][a.sub.1][D.sup.3.sub.x][D.sub.y] - 5[c.sup.2.sub.1][a.sub.1][D.sup.2.sub.y]) G x G = 0, (28)

with the conditions (22) and (27); that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Based on the bilinear equation (28), the IV-soliton solutions for (2) can be constructed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

with [k.sub.j], [l.sub.j], and [[xi].sub.j] (j = 1, 2, ..., N) being arbitrary constants; [[summation].sub.[mu]=j0,1] indicates a summation over all possible combinations of [[mu].sub.j] = 0,1 (j = 1,2, ..., N). For N = 1, the one-soliton solution for (2) can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

For N = 2, we can obtain the two-soliton solution for (2) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Based on solutions (32) and (33), we present some figures to describe the propagations and collisions of the solitary waves. Figure 1 shows the propagation of one-soliton solution via solution (32) when t = -2, t = -1, and t = 2, which maintains its shape except for the phase shift, and the propagation direction can be changed. Figures 2 and 3 illustrate the oblique collision between the two solitons, which keep their original shapes invariant except for phase shifts as mentioned above. It is obvious that the large-amplitude soliton moves faster than the small one. Different from Figure 2, Figure 3 displays that both solitons change their directions during the collision.

3.2. Case 2. As another case, we introduce an auxiliary variable s and a subsidiary condition

[q.sub.4x] + 3[q.sup.2.sub.2x] + [q.sub.x,s] = 0, (34)

in virtue of which, similarly, (18) can be written as a linear combination of P-polynomials of weight 6 (a weight 3 to s):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

with the following constraint condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

Solving for (36) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

Thus, the P-polynomials expression of (2) and (34) reads

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38)

in which [alpha] = [alpha](t) is an arbitrary function.

System (38) produces the bilinear form of (2) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (39)

by property (12) and transformation (23). From the bilinear equation (39), we can only get the one-soliton solution which is the same as the above formulae (25) and (26). Therefore, (2) under the constraint conditions (37) is not integrable since its multisoliton solutions cannot be obtained.

4. Bilinear BT and Lax Pair for (2)

In order to search for the bilinear BT and Tax pair of (2), under the integrable constraint condition (29) in case 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Tet

q = 2 ln G, q' = 2 ln F (41)

be two solutions of (40), respectively. On introducing two new variables

v = [q' - q]/2 = ln(F/G), w = [q' + q]/2 = ln(FG), (42)

one has the corresponding two-field condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

The simplest possible choice is a homogeneous Y-constraint[8] of weight 2; it can only be of form

[Y.sub.2x](v, w) + a[Y.sub.y](v) = [lambda]. (45)

It is easy to find that eliminating [w.sub.2x] (and its derivatives) by means of form (45) does not enable one to express the remainder R(v, w) as the v-derivative of a linear combination of (Y-polynomials. However, a homogeneous (Y-constraint of weight 3

[Y.sub.3x] (v, w) + [c.sub.1][Y.sub.y] (v) = [lambda], (46)

[lambda] = arbitrary parameter of weight 3,

can be used to express R(v, w) as follows:

R(v,w) = -5/2 [a.sub.1][[partial derivative].sub.x][[Y.sub.5x](v,w) - [c.sub.1][Y.sub.2x,y](v,w) + 3[lambda][Y.sub.2x] (v,w)]. (47)

Thus, the two-field condition (43) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

where we prefer the equation in the conserved form, which is useful to construct conservation laws later. It is seen that the two-field condition (43) can be decoupled into a pair of parameter-dependent (Y-constraints (of weight 3 and weight 5):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

In view of (10), the bilinear BT for (2) is obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

By application of formulae (15) and (16), the system (50) is linearized to be the Tax pair of (2) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (51)

Starting from this Tax pair with [a.sub.1] = -1, [a.sub.9] = 0, [c.sub.0] = 3, and [c.sub.1] = 1, the Darboux transformation and nonlocal symmetry of the equation can be established [25]. Checking that the compatibility condition of system (51) is just the potential of (40).

5. Infinite Conservation Laws for (2)

In what follows, we present the infinite conservation laws by recursion formulae for (2). The conservation laws actually have been hinted in the binary-Bell-polynomial-type BT (46) and (48), which can be rewritten in the conserved form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (52)

by using the relation

[[partial derivative].sub.t]([v.sub.x]) = [[partial derivative].sub.x]([v.sub.t]) = [v.sub.x,t], [[partial derivative].sub.y]([v.sub.x]) = [[partial derivative].sub.x]([v.sub.y]) = [v.sub.x,y]. (53)

By introducing a new potential function

[eta] = [q'.sub.x] - [q.sub.x]/2, (54)

in this way, there are

[v.sub.x] = [eta], [w.sub.x] = [q.sub.x] + [eta]. (55)

Substituting (55) into system (52), we obtain

[[eta].sub.2x] +3[eta]([q.sub.2x] + [[eta].sub.x]) + [[eta].sup.3] + [c.sub.1][[partial derivative].sup.- 1.sub.x][[eta].sub.y] = [lambda] = [[epsilon].sup.3], (56)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (57)

It maybe noticed that (56) is not a Riccati-type equation. Similar to [27], inserting expansion

[eta] = [epsilon] + [[infinity].summation over (n=1)][I.sub.n](q, [q.sub.x], [q.sub.y], ...) [[epsilon].sup.-1] (58)

into (56) would lead to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (59)

collecting the coefficients for the power of [epsilon], we explicity obtain the recursion relations for the conserved densities [I'.sub.n]s:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (60)

Applying (58) to divergence-type equation (57) and comparing the power of e provide us with an infinite sequence of conservation laws:

[I.sub.n,t] + [F.sub.n,x] + [G.sub.n,y] = 0, (n = 1, 2, ...), (61)

where the first fluxes [F'.sub.n]s are given explicitly by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (62)

and the second flues [G'.sub.n]s are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (63)

With the recursion formulae (60), (62), and (63) presented above, the infinite conservation laws for (2) can be constructed. In particular, the first conservation law is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (64)

or equivalently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (65)

which is exactly (2) under the constraint conditions (29).

6. Conclusion

In this paper, a (2 + l)-dimensional variable-coefficient CDGKS equation has been investigated by the Bell polynomials approach. For case 1, the CDGKS equation is completely integrable in the sense that it admits bilinear BT, Lax pair, and infinite conservation laws which are derived in a direct and systematic way. By means of the bilinear equation, the Nsoliton solutions for the variable-coefficient CDGKS equation are presented. Different parameters and functions are selected to obtain some soliton solutions and also analyze their graphics in Figures 1-3. However, for case 2, the variablecoefficient CDGKS equation under the constraint conditions (37) is not integrable since its multisoliton solutions cannot be obtained. In addition, the integrable constraint conditions on variable coefficients of the equation can be naturally found in the procedure of applying the Bell polynomials approach.

http://dx.doi.org/10.1155/2014/523136

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant nos. 11271211, 11275072, and 11435005 and K. C. Wong Magna Fund in Ningbo University.

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Wen-guang Cheng, (1) Biao Li, (1) and Yong Chen (1,2)

(1) Nonlinear Science Center, Ningbo University, Ningbo 315211, China

(2) Shanghai Key Laboratory of Trustworthy Computing East China Normal University, Shanghai 200062, China Correspondence should be addressed to Biao Li; libiao@nbu.edu.cn

Received 28 May 2014; Revised 12 August 2014; Accepted 18 August 2014; Published 14 October 2014

Academic Editor: Changbum Chun
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