Interaction solutions of a (2+1)-dimensional dispersive long wave system

With the help of the consistent $tanh$ expansion, this paper obtains the interaction solutions between solitons and potential Burgers waves of a (2$+$1)-dimensional dispersive long wave system. Based on some known solutions of the potential Burgers equation, the multiple resonant soliton wave solutions, soliton–error function wave solutions, soliton–rational function wave solutions and soliton–periodic wave solutions are obtained directly.     

一般Hirota-Satsuma方程的多孤子解及孤子间的相互作用

用改进的齐次平衡法,首先把不可积的一般Hirota-Satsuma方程简化成可积模型—KdV方程,然后通过求解KdV方程得到了一般Hirota-Satsuma方程的多孤子解.利用得到的多孤子解分析了奇异孤子之间、钟型孤子与奇异孤子之间的相互作用,结果发现了相互作用的一些重要性质.     

The Jacobi elliptic function method and its application for two component BKP hierarchy equations

The periodic wave solutions for the two component BKP hierarchy are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

对称在破裂孤立子方程中的应用

首先对带有积分项的破裂孤立子方程（breaking soliton equation）进行变换,然后利用待定系数法求出它的对称,通过验证知道原方程的李群能构成李代数,再利用优化系统对原方程进行约化,求出了原方程的一些新解。     

Rational solutions and their interaction solutions of the (2+1)-dimensional modified dispersive water wave equation

A bilinear form for the modified dispersive water wave (mDWW) equation is presented by the truncated Painlevé series, which does not lead to lump solutions. In order to get lump solutions, a pair of quartic–linear forms for the mDWW equation is constructed by selecting a suitable seed solution of the mDWW equation in the truncated Painlevé series. Rational solutions are then computed by searching for positive quadratic function solutions. A regular nonsingular rational solution can describe a lump in this model. By combining quadratic functions with exponential functions, some novel interaction solutions are founded, including interaction solutions between a lump and a one-kink soliton, a bi-lump and a one-stripe soliton, and a bi-lump and a two-stripe soliton. Concrete lumps and their interaction solutions are illustrated by 3d-plots and contour plots.     

The N-soliton solution and localized wave interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation

In this paper, the $N$-soliton solution is constructed for the ($2+1$)-dimensional generalized Hirota–Satsuma–Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons and their interactions are obtained by choosing special parameters. Especially, by selecting appropriate parameters on the multi-soliton solutions, the two soliton can reduce to a periodic soliton or a lump soliton, the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton, while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons. Detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters. Finally, an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method, and the relevant interaction and propagation characteristics are discussed graphically. The results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two-dimensional nonlinear wave equations.     

The breather-type and periodic-type soliton solutions for the (2+1 )-dimensional breaking soliton equation

Exact breather-type and periodic-type soliton solutions including the double-breather-type soliton solutions, the breather-type periodic soliton solutions and breather-type two-soliton solutions, and the periodic-type two-soliton and three-soliton solutions for the (2+1)-dimensional breaking soliton equation are obtained using the extended three-wave method (ETM). The results show that the ETM may provide us with a straightforward and effective mathematical tool for seeking multi-wave solutions of higher dimensional nonlinear evolution equations.     

Solitary wave solutions of (3+1)-dimensional extended Zakharov–Kuznetsov equation by Lie symmetry approach

In this paper, the solitary wave solutions of (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) equation are constructed which appear in the magnetized two-ion-temperature dusty plasma and quantum physics. Lie group of transformation method is proposed to investigate the solution of (3+1)-dimensional eZK equation via Lie symmetry method. The optimal system of one dimensional Lie subalgebra is constructed by using Lie point symmetries. The three dimensional eZK equation reduced into number of ordinary differential equations (ODEs) by applying similarity reductions. Consequently, solutions so extracted are more general than erstwhile known results. We have obtained twenty one solutions in the explicit form, some of them are likewise general and some are new for the best study of us. Eventually, single soliton, quasi-periodic soliton, multisoliton, lump-type soliton, traveling wave and solitary wave-interaction behavior are illustrated graphically through numerical simulation for physical affirmation of the results. Please check whether the affiliations are correct.     

On closed form solutions of (2+1)-breaking soliton system by similarity transformations method

In the present research, similarity transformation method via Lie-group theory is proposed to seek some more exact closed form solutions of the (2+1)-dimensional breaking soliton system. The system describes the interactions of the Riemann wave along y-axis and long wave along x-axis. Some explicit solutions of breaking soliton system are attained with appropriate choices of the arbitrary functions and making use of arbitrary constants involved in the infinitesimals. In order to obtain physically meaningful solutions, numerical simulation is performed. On the basis of graphical representation, the physical analysis of solutions reveals into multi-solitons, periodic, quadratic, asymptotic and stationary profiles.     

含外力项时变系数KdV方程与时变系数耦合KdV方程组的孤子解

应用孤子拟解法研究了含外力项时变系数KdV方程与一类时变系数耦合KdV方程组.首先将方程经过变量代换转换为齐次方程,然后将孤子解假设为双曲正割函数的形式带入方程或方程组,最后借助Maple软件完成复杂的计算来确定假设的孤子解的待定系数,从而得到孤子解存在的条件及其孤子解.结果显示:孤子拟解法计算简便且能得到方程的亮孤子解.     

Some new solitonary solutions of the modified Benjamin-Bona-Mahony equation

In this paper, we use the exp-function method to construct some new soliton solutions of the Benjamin-Bona-Mahony and modified Benjamin-Bona-Mahony equations. These equations have important and fundamental applications in mathematical physics and engineering sciences. The exp-function method is used to find the soliton solution of a wide class of nonlinear evolution equations with symbolic computation. This method provides the concise and straightforward solution in a very easier way. The results obtained in this paper can be viewed as a refinement and improvement of the previously known results.     

New family of overturning soliton solutions for a typical breaking soliton equation

The breaking soliton equations are of current interest, while the application of computer algebra to sciences has a bright future. In this paper, a new family of overturning soliton solutions for a typical breaking soliton equation is obtained via a computer-algebra-based method. An example of explicit solutions from the family is given. Solitary waves are also shown to be merely a simple case belonging to the family.     

Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions

Systems of coupled non-linear Schrödinger equations with soliton solutions are integrated using the six-point scheme which is equivalent to the multi-symplectic Preissman scheme. The numerical dispersion relations are studied for the linearized equation. Numerical results for elastic and inelastic soliton collisions are presented. Numerical experiments confirm the excellent conservation of energy, momentum and norm in long-term computations and their relations to the qualitative behaviour of the soliton solutions.     

Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine–cosine method

In this paper, we establish exact solutions for coupled nonlinear evolution equations. The sine–cosine method is used to construct exact periodic and soliton solutions of coupled nonlinear evolution equations. Many new families of exact travelling wave solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equations and the coupled nonlinear Klein–Gordon and Nizhnik–Novikov–Veselov equations are successfully obtained. The obtained solutions include compactons, solitons, solitary patterns and periodic solutions. These solutions may be important and of significance for the explanation of some practical physical problems.     

Soliton fission and fusion of a new two-component Korteweg-de Vries (KdV) equation

In this paper, the Painlevé test is performed for a new two-component Korteweg-de Vries (KdV) equation proposed by Foursov. It is shown that this equation passes the integrability test and is P-integrable. By means of the truncated singular expansion, some explicit solutions from the trivial zero solution are derived. The phenomena of soliton fission and fusion are studied in detail.     

Symbolic computation of exact solutions for the compound KdV-Sawada–Kotera equation

The generalized F-expansion method is applied to construct the exact solutions of the compound KdV-Sawada–Kotera equation by the aid of the symbolic computation system Maple. Some new exact solutions which include Jacobi elliptic function solutions, soliton solutions and triangular periodic solutions are obtained via this method.     

Symbolic methods to construct exact solutions of nonlinear partial differential equations

Two straightforward methods for finding solitary-wave and soliton solutions are presented and applied to a variety of nonlinear partial differential equations. The first method is a simplied version of Hirota's method. It is shown to be an effective tool to explicitly construct. multi-soliton solutions of completely integrable evolution equations of fifth-order, including the Kaup-Kupershmidt equation for which the soliton solutions were not previously known. The second technique is the truncated Painlevé expansion method or singular manifold method. It is used to find closed-form solitary-wave solutions of the Fitzhugh-Nagumo equation with convection term, and an evolution equation due to Calogero. Since both methods are algorithmic, they can be implemented in the language of any symbolic manipulation program.     

A numerical study of compactons

The Korteweg–de Vries equation has been generalized by Rosenau and Hyman [Compactons: Solitons with finite wavelength, Phys. Rev. Lett. 70(5) (1993) 564] to a class of partial differential equations that has soliton solutions with compact support (compactons). Compactons are solitary waves with the remarkable soliton property that after colliding with other compactons, they re-emerge with the same coherent shape [Rosenau and Hyman, Compactons: Solitons with finite wave length, Phys. Rev. Lett. 70(5) (1993) 564]. In this paper finite difference and finite element methods have been developed to study these types of equations. The analytical solutions and conserved quantities are used to assess the accuracy of these methods. A single compacton as well as the interaction of compactons have been studied. The numerical results have shown that these compactons exhibit true soliton behavior.     

Solitons to rogue waves transition,lump solutions and interaction solutions for the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics

ABSTRACT

In this work, we investigate the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation in fluid dynamics, which plays an important role in depicting weakly dispersive waves propagated in a quasi-media and fluid mechanics. By employing Hirota's bilinear method, we derive the one- and two-soliton solutions of the equation. Moreover, we reduce those soliton solutions to the periodic line waves and exact breather waves by considering different parameters. A long wave limit is used to derive the rogue wave solutions. Based on the resulting bilinear representation, we introduce two types of special polynomial functions, which are employed to find the lump solutions and interaction solutions between lump and stripe soliton. It is hoped that our results can be used to enrich dynamic behaviours of the (3+1)-dimensional BKP-type equations.     

On the exact and numerical solution of the time-delayed Burgers equation

The time-delayed Burgers equation is introduced and the improved tanh-function method is used to construct exact multiple soliton and triangular periodic solutions. For an understanding of the nature of the exact solutions that contained the time-delay parameter, we calculated the numerical solutions of this equation by using the Adomian decomposition method to the boundary value problem.