Complexity
Volume 2019 (2019), Article ID 8034731, 11 pages
https://doi.org/10.1155/2019/8034731

Research Article

Resonant Soliton and Soliton-Cnoidal Wave Solutions for a (3+1)-Dimensional Korteweg-de Vries-Like Equation

Zheng-Yi Ma,1,2 Jin-Xi Fei,³ Jun-Chao Chen,¹ and Quan-Yong Zhu¹

¹Institute of Nonlinear Analysis and Department of Mathematics, Lishui University, Lishui 323000, China
²Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China
³Department of Photoelectric Engineering, Lishui University, Lishui 323000, China

Correspondence should be addressed to Zheng-Yi Ma; ma-zhengyi@163.com

Received 4 May 2019; Revised 12 July 2019; Accepted 15 July 2019; Published 28 July 2019

Academic Editor: Mohamed Boutayeb

Copyright © 2019 Zheng-Yi Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The residual symmetry of a (3+1)-dimensional Korteweg-de Vries (KdV)-like equation is constructed using the truncated Painlevé expansion. Such residual symmetry can be localized and the (3+1)-dimensional KdV-like equation is extended into an enlarged system by introducing some new variables. By using Lie’s first theorem, the finite transformation is obtained for this localized residual symmetry. Further, the linear superposition of multiple residual symmetries is localized and the -th Bäcklund transformation in the form of the determinants is constructed for this equation. For illustration more detail, the first three multiple wave solutions-the collisions of resonant solitons are depicted. Finally, with the aid of the link between the consistent tanh expansion (CTE) method and the truncated Painlevé expansion, the explicit soliton-cnoidal wave interaction solution containing three kinds of Jacobian elliptic functions for this equation is derived.

1. Introduction

In scientific and engineering fields, nonlinear evolution equations have been studied in wide applications, such as in the nonlinear optics [1–7], plasma physics [8, 9], fluid mechanics [10, 11], textile engineering [12], and wave propagation phenomena [13–16]. Explicitly, for finding solutions, which including solitons, cnoidal waves, Painlevé waves, Airy waves, Bessel waves, etc., people often take the symmetry reduction approach with nonlocal symmetries with the aid of Darboux transformation, Bäklund transformation, and residual symmetry [17–21]. Methodology, for finding nonlinear evolution equations having infinitely many symmetries or flows, Olver proposed a general method which preserve them and it was employed to the KdV, modified Korteweg-de Vries (mKdV), Burgers’, and sine-Gordon equations [22, 23]. Integrable nonlinear evolution equations possessing a remarkably rich algebraic structures, which including infinitely many symmetries and conserved quantities, existence of a bi-Hamiltonian formulation through a recursion operator were reviewed [24]. After that, Lou concluded that the residues of the truncated Painlevé expansions were all nonlocal symmetries of the original system for any Painlevé integrable systems [25]. This criterion was applied some well-known concrete examples, such as the KdV equation, the fifth order KdV equation, the Sawada-Kotera (SK) system, the Kaup-Kupershmidt (KK) equation, the Boussinesq equation, and the Kadomtsev-Petvishvili (KP) system. At the same time, a consistent Riccati expansion (CRE) method, which can be considered as an extension of the usual Riccati equation method and the tanh function expansion method was proposed for some integrable systems [26]. This method was systematic applied to many CRE solvable systems, such as the KdV equation and the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov (ANNV) model, and the derived exact solution shared a similar determining equation which describing the interaction between a soliton and a cnoidal wave.

A (3+1)-dimensional KdV-like equation can be described as

Its soliton solutions were constructed by means of the simplified Hirota’s method [27]. The bilinear form, Bäklund transformations, Lax pairs and infinite conservation laws of Eq. (1) by means of the Bell polynomial method were constructed [28]. Its N-soliton solutions and periodic wave solutions were also presented with the aid of the bilinear formula and Riemann theta function.

This paper is organized as follows. In Section 2, with the aid of the truncated Painlevé expansion, the residual symmetry of the (3+1)-dimensional KdV-like equation is derived, and this nonlocal symmetry is localized by introducing eight auxiliary variables. Subsequently, we can obtain the finite symmetry transformation by solving the initial value problem. In Section 3, through localizing the linear superposition of multiple residual symmetries and construct the infinite transformation for this equation, the multiple residual symmetries and the -th Bäcklund transformation are obtained. A direct result shows that one can derive special soliton solutions-the collisions of resonant solitons. In Section 4, using the CTE method, the explicit soliton-cnoidal wave interaction solution of the variables and with the Jacobian elliptic functions for this equation is obtained. A brief summary is given in Section 5.

2. Residual Symmetry and Finite Transformation of the (3+1)-Dimensional KdV-Like Equation

The (3+1)-dimensional KdV-like equation is integrability and its equivalent form has

three parameters are real constants.

The truncated Painlevé expansion of (2) and (3) reads

where are functions of and is a real function. After substituting (4) into (2) and (3), one can derive the following results through vanishing the coefficients of powers and solving an over-determined equations ( is a arbitrary parameter)

which should hold on two consistent conditions

where the four variable quantities

Indeed, (8) and (9) possess Schwarzian structure for the (3+1)-dimensional KdV-like equation (1) and have invariant form under the Möbious transformation [29–31].

It is all know that the set of solutions and of (5)-(7) establish a link between the (3+1)-dimensional KdV-like equation (1) or (2) and (3) and their consistent conditions (8) and (9). That is to say, if the function satisfies the consistent conditions (8) and (9), then (4) with (5)-(7) is an auto-Bäcklund transformation between the solutions and of (2) and (3). According to the residual symmetry theorem, the residuals and are just the nonlocal symmetry with respect to the solutions and of (5)-(7) and one should transform them to a local Lie point symmetry for studying this nonlocal symmetry [25].

For the above purpose, the eight auxiliary variables , ,, , , , , and are introduced, which should obey the rule

Therefore, the local Lie point symmetry of the prolonged system of (2), (3), (8), (9), and (11) for the residual symmetry becomes

Correspondingly, by solving the initial value problem

and

with the infinitesimal parameter , one can write down the following finite transformation theorem.

Theorem 1. If is a solution of the extended system (2), (3), (8), (9), and (11), so are , , , and  , where

3. -th Bäcklund Transformation Related to Multiple Residual Symmetries

In fact, one can further study the -th Bäcklund transformation related to multiple residual symmetries of (2) and (3). The reason is that these residual symmetries just depend on the solution of the Schwarzian structure (8) and (9). The infinite residual symmetries of the fields and have

where is an arbitrary positive integer, are arbitrary real constants, and are different solutions of the Schwarzian Equations (8) and (9) with different parameters , which need to satisfy

and three variable quantities

Correspondingly, the eight auxiliary variables , , , , , , , and are renew introduced, which have the relation

in order to localize the residual symmetry and in (16).

After writing the linearized form of some enlarged system, the symmetry (16) is localized to a Lie point symmetry

For the Lie point symmetry (23), the following -th Bäcklund transformation theorem can be summarized according to Lie’s first principle with the aid of its initial value problem.

Theorem 2. If is a solution of the prolonged (3+1)-dimensional KdV-like system (2), (3) and (17)-(21), so are , and , here

where and are two determinants of the matrices and , which satisfy

and

From the above Theorem 2, one can derive an infinite number of explicit solutions from a suitable seed solution of (2) and (3) under some special circumstances. Especially, we have the recursive soliton solutions from the known one for this system. For example, when the seed solution takes for (2) and (3), it is not difficult to verify that the prolonged (3+1)-dimensional KdV-like system (2), (3), and (17)-(21) possesses the following soliton function :

The corresponding first three multiple wave solutions of (2) and (3) are explicitly written

and

For illustration of more detail, the parameters are all taken , but . Figure 1 displays the bell-like bright and dark solitons for the above condition of (28). Figures 1(a) and 1(b) are two line solitons for and with the amplitude and , respectively, while Figure 1(c) is a dark one for with the amplitude . Figures 2 and 3 are the collisions of two-resonant solitons and three-resonant solitons of (29) and (30), respectively.

Figure 1: The plots of three line solitons expressed by (28) for the solution of (2) and (3) with the parameters are all taken , but .

Figure 2: Three interactions of two-resonant solitons expressed by (29) for the solution of (2) and (3) with the parameters are all taken , but .

Figure 3: Three interactions of three-resonant solitons expressed by (30) for the solution of (2) and (3) with the parameters are all taken , but .

4. CTE and Soliton-Cnoidal Wave Interaction Solution

With the help of a Riccati equation, Lou proposed the CRE for solving nonlinear systems [26]. While the CTE method is a special form of CRE and this approach is a generalization of the traditional tanh function expansion method [32]. For the (3+1)-dimensional KdV-like equation (2) and (3), through introducing the tangent transformation

the truncated Painlevé expansion of (2) and (3) can be reduced

where are three constants and is a real function.

At the same time, the Schwarzian structure (8) and (9) owns the following consistent condition:

In order to derive the soliton-cnoidal wave solution, the solution of (35) and (36) should assume as the following form:

After substituting (37) into the consistent condition (35) and (36), the elliptic equation is satisfied

where the coefficients are

and are two arbitrary constants.

One concrete solution of (37) is

with is a Jacobian elliptic sine function with modulus . For this time, the relations of the coefficients are

As the result, the soliton-cnoidal wave interaction solution of (2) and (3) is

with , and , while , and are three kinds of Jacobian elliptic functions with modulus . Figure 4 shows that one bright/dark soliton resides on a cnoidal wave instead of constant background for the above solution , and of (2) and (3) when the coefficients are taken For this soliton-cnoidal wave, the influence of the electron superthermality, positron concentration, and magnetic field obliqueness was investigated in detail [8].

Figure 4: The , and plots of the soliton-cnoidal wave interaction solution (42), (43), and (44), respectively.

5. Summary

The KdV equation is well known and it higher dimensional extension is important. For example, initial-boundary value problems of the coupled mKdV equation on the half-line through the Fokas method [33, 34], analysis on lump, lumpoff and rogue waves with predictability to the (2+1)-dimensional B-type KP equation [35], rogue waves, bright-dark solitons, and traveling wave solutions of the (3+1)-dimensional generalized KP equation [36] were studied in detail. In this paper, the nonlocal symmetry of the (3+1)-dimensional KdV-like equation is obtained with the aid of its the truncated Painlevé expansion. After introducing the auxiliary variables and , an enlarged system which possesses a Lie point symmetry for the nonlocal symmetry is taken. By applying Lie’s first theorem for the localized point symmetries, we obtain the corresponding finite transformation. Further, we can localize the linear superposition of multiple residual symmetries and construct the infinite transformation for this equation. From Theorem 2, the -th Bäcklund transformation can be expressed in a compact way of determinants. According to this conclusion, one can derive special soliton solutions-the collisions of resonant solitons from some seed solutions. At the same time, the explicit soliton-cnoidal wave interaction solution of the variables , and with three Jacobian elliptic functions for this (3+1)-dimensional KdV-like equation is also derived using the CTE method.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11705077, 11775104, and 11447017), the Natural Science Foundation of Zhejiang Province (Grant No. LY14A010005), and the Scientific Research Foundation of the First-Class Discipline of Zhejiang Province (B) (No. 201601).

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