Abstract
In this paper, we focus on the interaction solutions of a (2 + 1)-dimensional Vakhnenko equation. By using Hirota’s transformation combined with the three-wave method and with symbolic computation, some interaction solutions which include interaction solutions between exponential and trigonometric functions and interaction solutions between exponential and trigonometric and hyperbolic functions are presented.
1. Introduction
It is well known that nonlinear partial differential equations (NPDEs) and their solutions play a significant role in interpreting many important phenomena in nonlinear sciences. A variety of powerful methods are developed for finding the exact solutions of NPDEs, such as Hirota’s method [1, 2], simplified Hirota’s method [3, 4], the Lie symmetry analysis method [5, 6], the simplest equation method [5, 6], the invariant subspace method [7], and the nonlinear steepest descent method [8]. Very recently, the lump and interaction solutions [9–11] have attracted the attention of many scholars because of lump’s applications in nonlinear optics, physics, oceanography, etc, and the interaction solutions are valuable in analyzing the nonlinear dynamics of waves in shallow water and can be used for forecasting the appearance of rogue waves [12, 13].
In order to describe high-frequent wave propagations in a relaxing medium, the Vakhnenko equation [14], the generalized Vakhnenko equation [15], and the modified generalized Vakhnenko equation [16] were presented. Many different kinds of valuable results have been obtained [17–23]. Vakhnenko and Parkes [17] obtained the two-loop soliton solution for the Vakhnenko equation using Hirota’s bilinear method. Vakhnenko et al. [18] derived a Bäcklund transformation both in the bilinear and in ordinary form for the generalized Vakhnenko equation and found the exact N-soliton solution via the inverse scattering method. Wazwaz [19] derived multiple soliton solutions and multiple singular soliton solutions for the Vakhnenko equation, the generalized Vakhnenko equation, and the modified generalized Vakhnenko equation by the simplified form of the bilinear method. Wang and Chen [20] investigated the integrability of the modified generalized Vakhnenko equation and presented the quasiperiodic solution by applying Hirota direct method and Riemann theta function. Brunelli and Sakovich [21] obtained a bi-Hamiltonian formulation for the Vakhnenko equation via the Miura-type transformations. The dynamical behaviours and exact traveling wave solutions of the modified generalized Vakhnenko equation were studied in [22]. Hashemi et al. [23] determined the Lie symmetry group, the corresponding symmetry reductions, and invariant solutions of the modified generalized Vakhnenko equation by the Li group analysis method, and so on.
In 2008, Victor et al. [24] initially derived a (2 + 1)-dimensional Vakhnenko equationwhich is to model high-frequent wave perturbations in relaxing high-rate active barothropic media and involves two spatial variables and a temporal variable . With the aid of symbolic computation and Hirota’s method, Victor et al. [24] unearthed some typical solitary wave solutions to equation (1) and depicted single- and multivalued solutions depending on the dissipative parameter. Morrison and Parkes [15, 16] showed that under the following transformation:where and are two constants and , and are three independent variables. Li et al. [25] introduced a new function defined by
Then, and equation (1) becomes
Moreover, Li et al. [25] indicated that if is an explicit solution of equation (4), thenis an implicit solution of equation (1). Some 1-loop, 2-loop, and 3-loop soliton solutions were presented applying the improved Hirota method, and the traveling and interaction processes for the N-loop soliton solutions are explored in [25].
In this paper, we investigate interaction solutions of equation (1) via Hirota’s transformation [26] and three-wave methods [27–30].
2. Interaction Solutions of Equation (1)
Under the transformation , equation (4) becomes the Hirota bilinear equation:where is a real function, and the Hirota bilinear differential operator was defined by [26]
In fact, equation (6) is equivalent to one special case of a generalized Bogoyavlensky–Konopelchenko equation [31] upon combining and as .
Consider equation (6) as well as the following novel test function:which is a combination of [28] and [29], where are unknown constants to be determined later.
Substituting (8) into (6), we can obtain an algebraic system of .
Case 1. Choosing and with the aid of Maple, we present some solutions of the algebraic system as follows:where .where .where .
The parameters in set (9) generate the following class of interaction solutions to equation (6) aswhich further leads to furnish a class of interaction solutions to equation (1) as follows:where are arbitrary constants andThe parameters in set (10) generate the following class of interaction solutions to equation (6) aswhich further leads to furnish a class of interaction solutions to equation (1) as follows:where are arbitrary constants andSetting , solution (15) becomeswhich further leads to furnish a class of interaction solutions to equation (1) as follows:where are arbitrary constants andThe parameters in set (11) generate the following class of interaction solutions to equation (6) aswhich further leads to furnish a class of interaction solutions to equation (1) as follows:where are arbitrary constants and
Case 2. . Choosing and with the aid of Maple, we present some solutions of the algebraic system as follows:where .where .
The parameters in set (24) generate the following class of interaction solutions to equation (6) aswhich further leads to furnish a class of interaction solutions to equation (1) as follows:where are arbitrary constants andThe parameters in set (25) generate the following class of interaction solutions to equation (6) aswhich further leads to furnish a class of interaction solutions to equation (1) as follows:where are arbitrary constants andSetting , solution (29) becomeswhich further leads to furnish a class of interaction solutions to equation (1) as follows:where are arbitrary constants andSetting , solution (29) becomeswhich further leads to furnish a class of interaction solutions to equation (1) as follows:where are arbitrary constants and
3. Conclusions
In this paper, we explore a (2 + 1)-dimensional Vakhnenko equation by Hirota’s transformation combined with the three-wave method. With symbolic computation, five types of interaction solutions are obtained. It should be pointed out that some similar types of interaction solutions as that of the solution presented in this paper are not shown here for brevity. Future investigation may be applied to the method in this paper to search for lump solutions and interaction solutions between lump and other functions for the equation.
Data Availability
The data used to support the findings of this study are included within the article. For more details, they are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant no. 11461022 and the Natural Science Foundation of Yunnan Province, China, under Grant no. 2014FA037.