Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 5453941, 15 pages

https://doi.org/10.1155/2017/5453941

## Hybrid Solutions of (3 + 1)-Dimensional Jimbo-Miwa Equation

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Huanhe Dong

Received 20 July 2017; Accepted 28 September 2017; Published 28 November 2017

Academic Editor: Maria L. Gandarias

Copyright © 2017 Yong Zhang 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 rational solutions, semirational solutions, and their interactions to the ()-dimensional Jimbo-Miwa equation are obtained by the Hirota bilinear method and long wave limit. The hybrid solutions contain rogue wave, lump solution, and the breather solution, in which the breathers which are manifested as growing and decaying periodic line waves show different dynamics in different planes. Rogue waves are localized in time and are obtained theoretically as a long wave limit of breathers with indefinitely larger periods; they arise from a constant background at and then disappear in the constant background when time goes on. More importantly, the interactions between some hybrid solutions are demonstrated in detail by the three-dimensional figures, such as hybrid solution between the stripe soliton and breather and hybrid solution between stripe soliton and lump solution.

#### 1. Introduction

In soliton theory, the study of integrability to nonlinear evolution is always a hot topic of interest, which can be regarded as a key step of their exact solvability. Many areas of integrable systems are researched, such as Painlevé analysis [1], Hamiltonian structure [2–5], Lax pair [6–8], Bäcklund transformation (BT) [9–12], infinite conservation laws [13–16], and bilinear integrability [17–20]. Based on the bilinear methods, we have got many kinds of solutions, such as lump solutions [21–24] and Pfaffian solution [25]. Recently, rogue wave solutions of lots of nonlinear evolutional equations have been gained, for example, the Boussinesq equation [26] and KP equation [27, 28]. Rogue waves, which were originally coined for vivid description of transient gigantic ocean waves of extreme amplitudes that seem to appear out of nowhere and disappear without a trace, have taken the responsibility for numerous marine disasters. In recent year, the rogue wave phenomenon has appeared in a class of social and scientific contexts, ranging from geophysics and hydrodynamics [29] to oceanography, Bose-Einstein condensation, plasma physics [30], nonlinear optics [31, 32], and financial markets [33, 34]. Mathematically, rogue waves are a kind of rational solutions which are localized in both space and time [35]. The first-order or fundamental rogue waves of the nonlinear Schrödinger equation were first obtained by Peregrine in 1983 [36], and higher-order rogue waves of the NLS equation are presented recently. In addition to the NLS equations, a mass of complex systems possesses rogue wave solutions, such as the Hirota equation [37], the Sasa-Satsuma equation [38], multicomponent Yajima-Oikawa systems [39], and AB system [40]. More interestingly, the research on rogue waves varies from rational solutions to semirational solutions in the relevant studies [41, 42]. Whatever, the semirational solutions exhibit a range of more interesting and more complicated dynamic behavior, such as bright-dark rogue wave pair, rogue waves interacting with solitons [41], or breathers [42].

In this article, we focus on the ()-dimensional Jimbo-Miwa equationwhich is a second member in the entire Kadomtsev-Petviashvili (KP) hierarchy [43]. Although (1) is nonintegrable, many types of solutions have been given. In [44], Zhang and Chen have gained one kind of rogue waves by the interaction between positive quadratic function and hyperbolic cosine function. However, in order to boost the possible applications of the ()-dimensional Jimbo-Miwa equation in ocean studies and other fields, it is still necessary to find analytical form of the rogue waves for this equation.

The structure of the paper is as follows: In Section 2, we present the evolution breather to ()-dimensional Jimbo-Miwa equation by the parameter perturbation method, and their typical dynamics are analysed and illustrated. In Section 3, lump solution and line rogue wave have been gained by long wave limit which show different dynamics behaviors in each plane by choosing different parameters. In Section 4, we discuss the interaction between soliton and other localized waves, which includes the soliton and breather, the soliton and lump solution, and soliton and line rogue wave.

#### 2. The Evolution Breather to (3 + 1)-Dimensional Jimbo-Miwa Equation

Through the variable transformation (1) will be changed into its bilinear form aswhere is a real function with respect to , , , and and the operator is the classic Hirota bilinear operator defined as where is a polynomial of .

By the parameter perturbation method, the two-soliton solution to the Jimbo-Miwa equation can be written aswithwhereAs reported in the earlier work, the two-soliton solution will be reduced to the breather under appropriate constraints to the parameters, such as Davey-Stewartson (DS) equation and ()-dimensional Kadomtsev-Petviashvili (KP) equation and the similar breather also existed in the Jimbo-Miwa equation by choosingin (5), where indicates the conjugate operator. Without loss of generality, takingthe corresponding function can be written asFrom the framework of function , we know that the spatial variable is different from the spatial variable and in essence. When or , the solution will be changed into the breather and when , the solution will be transferred into the period line wave under a given time . Furthermore, the period line waves will go back to the constant uniformly when . So we can call these period line waves as line breather. The corresponding dynamics properties to solution are depicted in Figures 1 and 2.