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

Yong Zhang,Huanhe Dong,Shili Sun

doi:10.1155/2017/5453941

Yong Zhang, Huanhe Dong + Show 1 more

Open Access

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

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Abstract

The rational solutions, semirational solutions, and their interactions to the (3+1)-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 t≪0 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.

Highlights

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
We have shown that the line rogue waves possess a growing and decaying line profile that arises from a constant background and disappears in the initial constant background again
The obtained semirational solutions composed of solitons, breathers, lump solutions, and rogue waves exhibit a range of interesting and complicated dynamic behaviors

Summary

Introduction

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. The structure of the paper is as follows: In Section 2, we present the evolution breather to (3 + 1)-dimensional JimboMiwa equation by the parameter perturbation method, and their typical dynamics are analysed and illustrated. Similar to the skills of one breather, we can deal with the four-soliton solution by the following parameters choices: p1 = a2I, p2 = −a2I, q1 = b2 + c2I, q2 = b2 − c2I, m1 = m2 = d2, p3 = a3I,.

Lump Solution and Line Rogue Wave

The Interaction between Soliton and Other Localized Waves

Conclusion