Solitary Wave Solutions of High Order Scalar Fields and Coupled Scalar Fields

1999 ◽  
Vol 54 (3-4) ◽  
pp. 195-203 ◽  
Author(s):  
Jian-song Yang ◽  
Sen-yue Lou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
General Solution ◽  
Solitary Wave ◽  
Arbitrary Function ◽  
Scalar Fields ◽  
High Order ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Klein Gordon

An arbitrary Klein-Gordon field with a quite general constrained condition (which contains an arbitrary function) can be used as an auxilialy field such that some special types of solutions of high order scalar fields can be obtain by solving an ordinary differential equation (ODE). For a special type of constraint, the general solution of the ODE can be obtained by twice integrating. The solitary wave solutions of the model ф5 are treated in an alternative simple way. The obtained solutions of the ф5 model can be changed to those of the ф8 field and coupled scalar fields.

scholarly journals A Note on Solitary Wave Solutions of the Nonlinear Generalized Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Lei Zhang ◽  
Xing Tao Wang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Solitary Wave ◽  
Numerical Simulations ◽  
Solitary Wave Solutions ◽  
Simple Method ◽  
Wave Solutions ◽  
Holm Equation

We give a simple method for applying ordinary differential equation to solve the nonlinear generalized Camassa-Holm equation ut+2kux−uxxt+aumux−2uxuxx+uuxxx=0. Furthermore we give a new ansätz. In the cases where m=1,2,3, the numerical simulations demonstrate the results.

A Generalized Sub-Equation Expansion Method and Some Analytical Solutions to the Inhomogeneous Higher-Order Nonlinear Schrödinger Equation

2008 ◽  
Vol 63 (12) ◽  
pp. 763-777 ◽  
Author(s):  
Biao Li ◽  
Yong Chen ◽  
Yu-Qi Li
Keyword(s):  
Solitary Wave ◽  
Analytical Solutions ◽  
Elliptic Function ◽  
Schrodinger Equation ◽  
Expansion Method ◽  
High Order ◽  
Solitary Wave Solutions ◽  
Exact Analytical Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

On the basis of symbolic computation a generalized sub-equation expansion method is presented for constructing some exact analytical solutions of nonlinear partial differential equations. To illustrate the validity of the method, we investigate the exact analytical solutions of the inhomogeneous high-order nonlinear Schrödinger equation (IHNLSE) including not only the group velocity dispersion, self-phase-modulation, but also various high-order effects, such as the third-order dispersion, self-steepening and self-frequency shift. As a result, a broad class of exact analytical solutions of the IHNLSE are obtained. From our results, many previous solutions of some nonlinear Schrödinger-type equations can be recovered by means of suitable selections of the arbitrary functions and arbitrary constants. With the aid of computer simulation, the abundant structure of bright and dark solitary wave solutions, combined-type solitary wave solutions, dispersion-managed solitary wave solutions, Jacobi elliptic function solutions and Weierstrass elliptic function solutions are shown by some figures.

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