Degeneration of solitons for a ( $$3+1$$ 3 + 1 )-dimensional generalized nonlinear evolution equation for shallow water waves

Nonlinear Dynamics - A ( $$3+1$$ )-dimensional generalized shallow water waves equation is investigated with different methods. Based on symbolic computation and Hirota bilinear form, N-soliton...     

Wronskian,Pfaffian and periodic wave solutions for a $$(2 + 1)$$ ( 2 + 1 ) -dimensional extended shallow water wave equation

Nonlinear Dynamics - Under investigation in this paper is a $$(2 + 1)$$ -dimensional extended shallow water wave equation. Bilinear form is obtained via the generalized dependent variable...     

Multiwave interaction solutions for a (3+1)-dimensional nonlinear evolution equation

Nonlinear Dynamics - In this paper, by the direct algebraic method, together with the inheritance solving strategy, new types of interaction solutions among solitons, rational waves and periodic...     

Wronskian and Grammian solutions for the (2+1)-dimensional BKP equation

The (2 + 1)-dimensional BKP equation in the Hirota bilinear form is studied during this work. Wronskian and Grammian techniques are applied to the construction of Wronskian and Grammian solutions of this equation, respectively. It is shown that these solutions can be expressed as not only Pfaffians but also Wronskians and Grammians.     

Linear superposition formula of solutions for the extended (3+1)-dimensional shallow water wave equation

Nonlinear Dynamics - Active researches on the water waves have been done, and water waves are essentially complex waves controlled by gravity field and surface tension. Using the Hirota bilinear...     

A new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function

This paper proposes a new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Firstly, a Wronskian condition involving a free function is introduced for the equation. Secondly, by solving the Wronskian condition, some exact solutions are presented. Thirdly, the dynamical behaviors are analyzed by choosing specific functions in the Wronskian condition. In addition, some exact solutions are graphically illustrated by using Mathematica symbolic computations. The dynamical behaviors include stationary y-breather, line-soliton resonance, line-soliton-like phenomenon, parabola–soliton interaction, cubic–parabola–soliton resonance, kink behavior, and singular waves. These results not only illustrate the merits of the proposed method in deriving new exact solutions but also novel dynamical behaviors in the (3 + 1)-dimensional nonlinear evolution equation.

     

Multiple rogue wave,breather wave and interaction solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation

Nonlinear Dynamics - Based on a direct variable transformation, we obtain multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including...     

M-lump,rogue waves,breather waves,and interaction solutions among four nonlinear waves to new (3+1)-dimensional Hirota bilinear equation

In this research article, we study a new (3+1)-dimensional Hirota bilinear equation which can describe the dynamics of ion-acoustic wave and Alvin wave of small but finite amplitude in plasma physics and describe the propagation process of nonlinear waves in shallow water. First, we apply two methods to study the equation, namely the Hirota bilinear method and long-wave limit method M-lump solution, and line rogue waves are reported. Furthermore, we investigate the velocity, propagation trajectory, and interaction phenomenon of M-lump solution(M=2,3). Then, based on the multi-solitons, two cases of high-order breather solution are constructed by selecting some special parameters. Finally, four types interaction solutions are successfully obtained by employing long-wave limit method and selecting some special parameters. More importantly, we explore physical collision phenomenon of the interaction between nonlinear waves. In order to better illustrate the characteristics of the interaction solutions, the results are shown in three-dimensional plots and numerical simulation. To our knowledge, all of the obtained solutions in this article are novel. The results of this article may be provide an important theoretical basis for explaining some nonlinear phenomena in the field of fluid mechanics and shallow water.

     

M-lump,high-order breather solutions and interaction dynamics of a generalized $$(2 + 1)$$(2+1)-dimensional nonlinear wave equation

Nonlinear Dynamics - In this paper, a generalized $$(2+1)$$-dimensional nonlinear wave equation is obtained by extending the generalized $$(2+1)$$-dimensional Hirota bilinear equation into a more...     

Soliton-shape-preserving and soliton-complex interactions for a (1+1)-dimensional nonlinear dispersive-wave system in shallow water

Under investigation in this paper is a (1+1)-dimensional nonlinear dispersive-wave system for the long gravity waves in shallow water. With symbolic computation, we derive the multi-soliton solutions for the system. Four sorts of interactions for the system are discussed: (1) Soliton shape preserving, in which two solitons undergo the fusion behavior while the amplitudes and velocities of the other two remain unchanged during the interaction process; (2) Head-on collisions between the two-soliton complexes; (3) Overtaking collisions between the two-soliton complexes; (4) Two-soliton complexes formed by the inelastic collisions. Such soliton structures might be of certain value in fluid dynamics.     

State transition of lump-type waves for the (2+1)-dimensional generalized KdV equation

Nonlinear Dynamics - In this paper, we are mainly concerned with the (2+1)-dimensional generalized Korteweg–de Vries equation in fluid dynamics. Based on the translation transformation and...     

Riemann theta solutions and their asymptotic property for a (3 $$+$$ + 1)-dimensional water wave equation

Nonlinear Dynamics - In this paper, we consider the (3 $$+$$ 1)-dimensional water wave equation $$u_{yzt}+u_{xxxyz}-6u_{x}u_{xyz}-6u_{xy}u_{xz}=0.$$ Based on Bell polynomials, we obtain its Hirota...     

Rational solutions for a combined (3 + 1)-dimensional generalized BKP equation

Nonlinear Dynamics - In this paper, we focus on the rational solutions for a combined (3 + 1)-dimensional generalized BKP equation. By using the symbolic computation with Maple,...     

Rational and complexiton solutions of the (3+1)-dimensional KP equation

In this paper, the (3+1)-dimensional KP equation is mainly discussed. Based on the Wronskian technique, the double Wronskian solution is established. Generating functions for matrix entries satisfy the linear system of partial differential equations involving free parameters. Rational, Matveev, and complexiton solutions are obtained by taking special cases in a general double Wronskian solution.     

Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation

In the present paper,a general solution involving three arbitrary functions for the generalized(2+1)dimensional KdV-mKdV equation,which is derived fromthe generalized(1+1)-dimensional KdV-mKdV equation,is first introduced by means of the Wiess,Tabor,Carnevale(WTC) truncation method.And then multisymplectic formulations with several conservation lawstaken into account are presented for the generalized(2+1)dimensional KdV-mKdV equation based on the multisymplectic theory of Bridges.Subsequently,in order tosimulate the periodic wave solutions in terms of rationalfunctions of the Jacobi elliptic functions derived from thegeneral solution,a semi-implicit multi-symplectic schemeis constructed that is equivalent to the Preissmann scheme.From the results of the numerical experiments,we can conclude that the multi-symplectic schemes can accurately simulate the periodic wave solutions of the generalized(2+1)dimensional KdV-mKdV equation while preserve approximately the conservation laws.     

Integrable (3+1)-dimensional Ito equation: variety of lump solutions and multiple-soliton solutions

Nonlinear Dynamics - In this work, we study an extended integrable (3+1)-dimensional Ito equation, where its complete integrability is justified via Painlevé analysis. The simplified...     

Antidark solitons and soliton molecules in a (3 + 1)-dimensional nonlinear evolution equation

Nonlinear Dynamics - We investigate a (3 + 1)-dimensional nonlinear evolution equation which is a higher-dimensional generalization of the Korteweg–de Vries equation. On the...     

Nonlinear superposition between lump and other waves of the (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid dynamics

Nonlinear Dynamics - With the premise that N-soliton solutions are acquired, this paper will focus on the nonlinear superposition between one lump and other types of localised waves of the...