Advances in Mathematical Physics

Volume 2018, Article ID 9178480, 8 pages

https://doi.org/10.1155/2018/9178480

## Abundant Lump-Type Solutions and Interaction Solutions for a Nonlinear (3+1) Dimensional Model

^{1}Department of Mathematics and Physics, Faculty of Engineering, Zagazig University, Egypt^{2}Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA^{3}Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia^{4}College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China^{5}International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

Correspondence should be addressed to R. Sadat; moc.oohay@tadasleamhar_gne

Received 15 October 2018; Accepted 29 November 2018; Published 10 December 2018

Academic Editor: Qin Zhou

Copyright © 2018 R. Sadat 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

We explore dynamical features of lump solutions as diversion and propagation in the space. Through the Hirota bilinear method and the Cole-Hopf transformation, lump-type solutions and their interaction solutions with one- or two-stripe solutions have been generated for a generalized (3+1) shallow water-like (SWL) equation, via symbolic computations associated with three different ansatzes. The analyticity and localization of the resulting solutions in the , and space have been analyzed. Three-dimensional plots and contour plots are made for some special cases of the solutions to illustrate physical motions and peak dynamics of lump soliton waves in higher dimensions. The study of lump-type solutions moderates the visuality of optics media and oceanography waves.

#### 1. Introduction

It is very important to control the physical mechanisms of rough waves and interaction waves specially with lump-type waves. The significance of nonlinear waves of these types appears from natural disasters. Many physical phenomena need analytical approaches to classify the physical dynamics of nonlinear evaluation equations. The Darboux transformation (DT) and the Lie symmetry (LS) method [1–3] are efficient approaches to obtaining closed-form solutions. However, some problems occur in applying those methods, such as how to find Lax pairs in the DT method and how to carry out the back-substitution procedure in the LS method. There are also new types of closed-form solutions, for example, positions and complexions [4–8], and even new collision phenomena including fissions and fusions [9–14]. The Hirota bilinear method plays an influential role in discovering all the mentioned types of solutions to overcome a lot of analytic problems. Most studies apply the Hirota method to completely integrability nonlinear problems as in [10, 15–23]. We would like to demonstrate that the Hirota method can be used to explore various types of closed-form solutions: interaction solutions of lumps with solitons, kinks, line-solitons, resonance solutions, and one- or two-stripe solitons; and two classes of breather solutions (time periodic or space periodic solutions). Our analysis will show that those solutions can predicate the characteristics and physical significance of nonlinear problems.

Consider the following generalized (3+1) SWL equation [24, 25]: There are a few studies on this equation. For example, Tian et al. in [26] generated a traveling wave solution via the tanh method. In 2010, Zayed [27] used method to obtain some traveling wave solutions by reducing the independent variables using the linear D’lambert transformation.

In what follows, we investigate lump soliton solutions and their dynamics and the susceptibility of their interactions with other types of solutions using the Hirota method for (1). By using the singular manifold method (SMM) with two-term truncated series, one derives the same ansatz in [24, 25]This is called the Cole-Hopf transformation, where is an auxiliary or test function that will be determined later. Starting by substituting (2) into (1), one getsThe transformation increases the nonlinearity but allows us to work with the test function. In [24], Zhang used Bell polynomial theories to generate lump-kink solutions, lumps with one-stripe solitons and lumps with two-stripe solitons for (1), but he supposed that to minimize the number of independent variables and so studied the equation in a (2+1)-dimensional domain.

#### 2. Lump Soliton Solutions

To generate single-lump solutions, we suppose thatwhere , are real unknowns that will be found subsequently. We carry out a direct substitution of (4) into (3) and gather the coefficients of the resulting polynomial in , and , to obtain a nonlinear algebraic system in . By solving this system of nonlinear algebraic equations with the aid of Maple, we acquire some sets of solutions for the parameters. Avoiding the redundancy, we surpass one studying case as follows:Using the aggregation equation (4), one can represent the auxiliary function aswhereBy using (2), the solution of (1) has the formIncorporating (6) and (5) into (8), one gets a class of lump solutions of (1) depicted in Figure 1.