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Liu Jian-Gen, Yang Xiao-Jun, Feng Yi-Ying. Resonant multiple wave solutions to some integrable soliton equations*

Project supported by the Yue-Qi Scholar of the China University of Mining and Technology (Grant No. 102504180004) and the 333 Project of Jiangsu Province, China (Grant No. BRA2018320).

. Chinese Physics B, 2019, 28(11): 110202  
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Resonant multiple wave solutions to some integrable soliton equations*

Project supported by the Yue-Qi Scholar of the China University of Mining and Technology (Grant No. 102504180004) and the 333 Project of Jiangsu Province, China (Grant No. BRA2018320).

Liu Jian-Gen1, 2, Yang Xiao-Jun1, 2, 3, †, Feng Yi-Ying2, 3
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China
School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China

 

† Corresponding author. E-mail: xjyang@cumt.edu.cn

Project supported by the Yue-Qi Scholar of the China University of Mining and Technology (Grant No. 102504180004) and the 333 Project of Jiangsu Province, China (Grant No. BRA2018320).

Abstract

To transform the exponential traveling wave solutions to bilinear differential equations, a sufficient and necessary condition is proposed. Motivated by the condition, we extend the results to the (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation, the (3+1)-dimensional generalized Kadomtsev–Petviashvili (g-KP) equation, and the B-type Kadomtsev–Petviashvili (BKP) equation. Aa a result, we obtain some new resonant multiple wave solutions through the parameterization for wave numbers and frequencies via some linear combinations of exponential traveling waves. Finally, these new resonant type solutions can be displayed in graphs to illustrate the resonant behaviors of multiple wave solutions.

PACS: 02.30.Ik;02.60.Cb;02.70.Wz
Keyword:linear superposition principle;resonant multiple wave solutions;(2+1)-dimensional Kadomtsev-Petviashvili (KP) equation;(3+1)-dimensional g-KP and BKP equations
1. Introduction

It is well known that the Hirota bilinear method is an efficient tool to construct exact solutions of nonlinear differential equations.[1,2] Based on Hirota bilinear forms, rational solutions, solitons solutions, rogue wave solutions, and periodic solutions could be obtained, e.g. in Refs. [3]–[7]. Meanwhile, the exact solutions play an essential role in proper understanding of qualitative features of the concerned phenomena and processes in various fields of science and engineering.[8–11] Recently, the lump solutions, lump type solutions and its various interaction solutions between lump solutions and other kinds of dispersive wave solutions are explored.[12–18] Meanwhile, using the Riemann–Hilbert method to construct N-soliton solutions has also attracted people's attention in the field of mathematical physics.[19,20] More recently, the linear superposition principle has been applied to exponential traveling waves of Hirota bilinear equations, and formed a specific sub-class of solutions from linear combinations of exponential wave solutions.[21,22] Some useful and extended results have been discussed in Refs. [23]–[27].

Let Υ be a polynomial in M variables, and then we consider a Hirota bilinear equation

where Dxi (1 ≤ i ≤ M) is the Hirota bilinear derivatives defined by[28] The polynomial Υ needs to satisfy and Next, we introduce N-wave variables and exponential wave functions where ki,j, 1 ≤ i ≤ N, 1 ≤ j ≤ M are all real constants. Based on Eq. (3), we can see that each of the exponential wave functions fi (1 ≤ i ≤ N) is a solution to the Hirota bilinear Eq. (1) with the following bilinear identity

Substitute the linear combination function into Eq. (1), with μi, 1 ≤ i ≤ N, being arbitrary real constants, which needs to satisfy the following bilinear identity which indicates that a linear combination function f defined by Eq. (8) solves Eq. (1) if and only if the following condition is satisfied.

For the above processes (2)–(9), the details can also refer to Refs. [21]–[27]. We now summarize it by two theorems as follows:

Theorem 1[23] Let Υ(x) be a polynomial in x ∈ RM and let ϖi, 1 ≤ i ≤ N, be the N-wave variables , for 1 ≤ i ≤ N, where bj’s are all real constants. Then any linear combination of the exponential waves fi = eϖi, 1 ≤ i ≤ N, solves linear differential equation Υ(Dx1, Dx2,…,DxM) f · f = 0 if and only if for 1 ≤ j ≤ i ≤ N.

Moreover, Eq. (11) is true for arbitrary ki,kj ∈ R if and only if it is also true for arbitrary constants . Then, we have the following Theorem 2.

Theorem 2[23] Let Υ(x) be a polynomial in x ∈ RM. Suppose that N1, N2 ∈ N, N = N1 + N2, bj ∈ R, αj ∈ Z for j = 1,2,…,M are all fixed. The N-wave variables , 1 ≤ i ≤ N with parameters If equation (11) is true for arbitrary kj,ki ∈ R, 1 ≤ j, i ≤ N, then any linear combination of the wave fi defined by with parameters θi, N1 ≤ i ≤ N, being arbitrary real constants, solves the bilinear differential equation

In what follows, we apply the linear superposition principle to extend the results for the (2+1)-dimensional KP equation, the (3+1)-dimensional g-KP equation and the BKP equation to obtain resonant multiple wave solutions in this paper.[29] To the best of our knowledge, they have not been obtained before.

2. Applications
2.1. The (2+1)-dimensional KP equation

In this section, we apply the linear superposition principle to solve the (2+1)-dimensional KP equation,[29] which is important because it serves as a kernel model in the universe Sato's theory,[30–32] that is, where δ = ± 1.

Equation (14) can be transformed into the following Hirota bilinear equation under the dependent variable transformation u = 2(ln F)xx.

By Eq. (5) and applying a kind of the parameterization for wave numbers and frequencies in Ref. [33], we have where a1, a2, a3 are constants to be determined later, and ϑi is arbitrary.

Applying Theorem 1 and Eq. (15), one can derive

Solving Eq. (17) we can obtain a set of solutions as follows: where a1 is a free parameter.

Suppose that N1, N2 in N, N = N1 + N2. The N-wave variables with parameters km 1 ≤ m ≤ N, defined by (5). Moreover, for N1 < m ≤ N, we have Using Theorem 2, any linear combination of the exponential and trigonometric waves is taken as follows:

Taking θm = 0, we have

Taking θm = 2kπ + π/2, k ∈ K, then we have Furthermore, we can obtain the following mixed type function solutions like complexiton solutions where ξ1m and ξ2m are constants.

2.2. The (3+1)-dimensional g-KP equation

In this section, we apply the linear superposition principle to solve the (3+1)-dimensional g-KP equation[29]

Equation (24) can be changed into the following Hirota bilinear equation under the dependent variable transformation u = 2(ln F)x.

Using Eq. (5) and applying a kind of the parameterization for wave numbers and frequencies in Ref. [33], we have where a1, a2, a3, a4 are constants to be determined later, and ϑi is arbitrary.

Applying Theorem 1 and Eq. (25), we have

Solving Eq. (27), we can obtain a set of solutions in the form of where a1 and a2 are arbitrary parameters.

Suppose that N1, N2 ∈ N, N = N1 + N2. We have the N-wave variables with parameters km, 1 ≤ m ≤ N, defined by (5). Moreover, for N1 < m ≤ N, we have

Using Theorem 2, any linear combination of exponential and trigonometric waves can be taken as follows:

Taking θm = 0, we have

Taking θm = 2kπ + π/2, k ∈ K, we have

Furthermore, we can obtain the following mixed-type function solutions like complexiton solutions where ξ1m and ξ2m are constants.

2.3. The (3+1)-dimensional BKP equation

In this section, we apply the linear superposition principle to solve the (3+1)-dimensional BKP equation[29]

Equation (34) can be transformed into the following Hirota bilinear equation under the dependent variable transformation u = 2(ln F)x.

Along with Eq. (5) and a kind of the parameterization for wave numbers and frequencies in Ref. [33], we have where a1, a2, a3, a4 are constants to be determined later, and αi, (i = 1,2,3,4) are integers which can be positive or negative, and ϑi is arbitrary.

Applying Theorem 1 and Eq. (35), we obtain

Solving Eq. (37), we can obtain a set of solutions to Eq. (37) in the following two cases:

Case (i) where a1 and a4 are free parameters.

Suppose that N1, N2 ∈ N, N = N1 + N2. We have the N-wave variables with parameters km, 1 ≤ m ≤ N, defined by (5). Moreover, for N1 < m ≤ N, we have Using Theorem 2, any linear combination of exponential and trigonometric waves gives

Taking θm = 0, we have

Taking θm = 2kπ + π/2, k ∈ K, we have

Furthermore, we can obtain the following mixed-type function solutions like complexiton solutions where ξ1m and ξ2m are constants.

Case (ii) where a1 and a3 are free parameters.

Suppose that N1, N2 ∈ N, N = N1 + N2. We have the N-wave variables with parameters km, 1 ≤ m ≤ N, defined by Eq. (5). Moreover, for N1 < m ≤ N, we have

Using Theorem 2, any linear combination of exponential and trigonometric waves reads

Taking θm = 0, we have

Taking θm = 2kπ + π/2, k ∈ K, we have

Furthermore, we can obtain the following mixed type function solutions like complexiton solutions where ξ1m and ξ2m are constants.

In order to better understand the resonant multiple wave solution, we display them by 3D-plot in Figs. 1–4.

Fig. 1. Characteristics of the resonant multiple wave solution (47) with N1 = 1, N = 3, a1 = a4 = 1, ϕ1 = ϕ2 = ϕ3 = 0, k1 = 1, ε1 = 1, ξ2 = ξ3 = 1, γ2 = γ3 = 1, σ2 = σ3 = 1, y = 0, z = 0. (a) 3D-plot (b) 2D-density plot.
Fig. 2. Characteristics of the resonant multiple wave solution (47) with N1 = 1, N = 3, a1 = a4 = 1, ϕ1 = ϕ2 = ϕ3 = 0, k1 = 1, ε1 = 1, ξ2 = ξ3 = 1, γ2 = γ3 = 1, σ2 = σ3 = 1, t = 0, z = 0. (a) 3D-plot (b) 2D-density plot.
Fig. 3. Characteristics of the resonant multiple wave solution (47) with N1 = 2, N = 4, a1 = a4 = 1, ϕ1 = ϕ2 = ϕ3 = ϕ4 = 0, k1 = k2 = 1, ε1 = ε2 = 1, ξ2 = ξ3 = ξ4 = 1, γ2 = γ3 = γ4 = 1, σ2 = σ3 = σ = 1, y = 0, z = 0. (a) 3D-plot (b) 2D-density plot.
Fig. 4. Characteristics of the resonant multiple wave solution (47) with N1 = 2, N = 4, a1 = a4 = 1, ϕ1 = ϕ2 = ϕ3 = ϕ4 = 0, k1 = k2 = 1, ε1 = ε2 = 1, ξ2 = ξ3 = ξ4 = 1, γ2 = γ3 = γ4 = 1, σ2 = σ3 = σ = 1, t = 0, z = 0. (a) 3D-plot (b) 2D-density plot.
3. Conclusions

In summary, resonant multiple-wave solutions including soliton and complexion type solutions have been generated for a generalized (3+1)-dimensional KP equation with the aid of the linear superposition principle in complex field sense. These type solutions could help us better understand resonant nonlinear phenomena. Meanwhile, we would like to point that this paper can be regarded as an extension to the article.[29] Furthermore, for other types of special solutions of soliton equations appearing in this paper, we would further consider them in the future work.

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