Solutions of Kadomtsev–Petviashvili equation with power law nonlinearity in 1+3 dimensions

This paper studies the solution of the Kadomtsev–Petviasvili equation with power law nonlinearity in 1+3 dimensions. The Lie symmetry approach as well as the extended tanh‐function and G′/G methods are used to carry out the analysis. Subsequently, the soliton solution is obtained for this equation with power law nonlinearity. Both topological as well as non‐topological solitons are obtained for this equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Symmetry reduction,exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.     

Dynamics of solitons in plasmas for the complex KdV equation with power law nonlinearity

This paper obtains the 1-soliton solution of the complex KdV equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration. The soliton perturbation theory for this equation is developed and the soliton cooling is observed for bright solitons. Finally, the dark soliton solution is also obtained for this equation.     

1-Soliton solution of the coupled KdV equation and Gear–Grimshaw model

This paper carries out the integration of the coupled KdV equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration. The domain restrictions of the coefficients of nonlinear and dispersion terms fall out. The results are then supplemented by numerical simulations.     

Topological and non-topological exact soliton solution of the power law KdV equation

This paper obtains the exact 1-soliton solution of the perturbed Korteweg–de Vries equation with power law nonlinearity. Both topological as well as non-topological soliton solutions are obtained. The solitary wave ansatz is used to carry out this integration. The domain restrictions are identified in the process and the parameter constraints are also obtained. Finally, the numerical simulations are implemented in the paper.     

Topological 1-soliton solution of the Biswas–Milovic equation with power law nonlinearity

This paper studies the topological or dark solitons due to the Biswas–Milovic equation with power law nonlinearity. The coefficients of the dispersion and nonlinearity terms are time dependent. The damping (gain) term with time-dependent coefficient is also taken into consideration. The solitary wave ansatz is used to carry out the integration. The only requirement is that the coefficient of the damping term is Riemann integrable.     

Stationary solutions for the Biswas-Milovic equation

This paper studies the Biswas-Milovic equation by the aid of Lie symmetry analysis. Four types of nonlinearity are being studied for this equation. They are Kerr law, power law, parabolic law and the dual-power law. A closed form stationary solution is obtained for each case.     

Solitary waves of Boussinesq equation in a power law media

In this paper, the solitary wave solution of the Boussinesq equation, with power law nonlinearity, is obtained by virtue of solitary wave ansatze method. The numerical simulations are obtained to support the theory.     

Exact 1-soliton solution of the Zakharov equation in plasmas withpower law nonlinearity

This paper studies the Zakharov equation with power law nonlinearity. An exact 1-soliton solution is obtained by the ansatz method. The parameter regimes are identified in the process. The numerical simulation is also given to complete the study.     

1-Soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity

This paper obtains the 1-soliton solution of the Kadomtsev-Petviasvili equation with power law nonlinearity using the solitary wave ansatz. An exact soliton solution is obtained and a couple of conserved quantities are also computed.     

1-Soliton solution of the D(m, n) equation with generalized evolution

This paper obtains the 1-soliton solution of the nonlinear dispersive Drinfel’d-Sokolov equation with power law nonlinearity. In the first case the soliton solution is without the generalized evolution. The solitary wave ansatz method is used to carry out the integration. Subsequently, the He’s semi-inverse variational principle is used to integrate the equation with power law nonlinearity. Parametric conditions for the existence of envelope solitons are given.     

利用Darboux变换对KdV方程孤子解普遍性质的讨论

该文指出：利用Darboux变换不但可以非常简洁地得到文献［1］关于KdV方程单孤子解和双孤子解，而且便于讨论KdV方程的任意孤子解的性质．通过对KdV方程三孤子解的重点讨论，以及对KdV方程多孤子解的解析分析，得到了关于KdV方程任意阶孤子解的一些非常有意义的普遍结果．这些结果对于人们深入了解孤子相互作用规律具有重要的现实意义．     

1-Soliton solution of the Klein-Gordon-Zakharov equation with power law nonlinearity

This paper obtains the 1-soliton solution of the Klein-Gordon-Zakharov equation with power law nonlinearity. The solutions are obtained both in (1+1) and (1+2) dimensions. The solitary wave Ansatz method is applied to obtain the solution. The numerical simulations are included that supports the analysis.     

Soliton solution and bifurcation analysis of the Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation with power law nonlinearity

This paper addresses the Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation with power law nonlinearity. First the soliton solution is obtained by the aid of traveling wave hypothesis and along with it the constraint conditions fall out naturally, in order for the soliton solution to exist. Subsequently, the bifurcation analysis of this equation is carried out and the fixed points are obtained. The phase portraits are also analyzed for the existence of other solutions.     

Topological 1-soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity

This paper studies the Kadomtsev-Petviasvili equation with power law nonlinearity. Topological 1-soliton solution is obtained and the parameter domain is identified for these solitons to exist. The solitary wave ansatz is used to obtain this solution.     

Bright soliton solution to a generalized Burgers–KdV equation with time-dependent coefficients

We consider a generalized Burgers–KdV type equation with time-dependent coefficients incorporating a generalized evolution term, the effects of third-order dispersion, dissipation, nonlinearity, nonlinear diffusion and reaction. The exact bright soliton solution for the considered model is obtained by using a solitary wave ansatz in the form of sech^s function. The physical parameters in the soliton solution are obtained as functions of the time varying coefficients and the dependent exponents. The dependent exponents and the temporal variations of the model coefficients satisfy certain parametric conditions as shown by the obtained soliton solution. This solution may be useful to explain some physical phenomena in genuinely nonlinear dynamical systems that are described by Burgers–KdV type models.     

关于发展方程的守恒率的一个标记

本文使用微分代数的技巧，研究了发展方程的守恒率与对应的行波所满足方程的首次积分之间的关系．通过文本给出的结果，我们研究了Burgers方程和Burgers—KdV方程的可积性，证明了这两类方程都只有一个守恒率．利用本文给出的方法，可以通过常微分方程的研究方法来研究某些非线性发展方程．     

A Lie symmetry approach to nonlinear Schrödinger’s equation with non-Kerr law nonlinearity

This paper obtains the stationary 1-soliton solution of the nonlinear Schrödinger’s equation in non-Kerr law media. The types of nonlinearity that are considered are Kerr law, power law, parabolic law and the dual-power law. The technique that is used to carry out the integration of this equation is the Lie symmetry analysis.     

Topological Soliton and Other Exact Solutions to KdV–Caudrey–Dodd–Gibbon Equation

This paper studies the KdV–Caudrey–Dodd–Gibbon equation. The modified F-expansion method, exp-function method as well as the G′/G method are used to extract a few exact solutions to this equation. Later, the ansatz method is used to obtain the topological 1-soliton solution to this equation. The constraint conditions are also obtained that must remain valid for the existence of these solutions.     

On a KdV equation with higher-order nonlinearity: Traveling wave solutions

In this work, the improved tanh-coth method is used to obtain wave solutions to a Korteweg-de Vries (KdV) equation with higher-order nonlinearity, from which the standard KdV and the modified Korteweg-de Vries (mKdV) equations with variable coefficients can be derived as particular cases. However, the model studied here include other important equations with applications in several fields of physical and nonlinear sciences. Periodic and soliton solutions are formally derived.