Solitary Wave Solutions Of Equations Research Articles

Optical Solitons of the Generalized Nonlinear Schrödinger Equation with Kerr Nonlinearity and Dispersion of Unrestricted Order

The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé test is applied for finding arbitrary constants in the expansion of the general solution into the Laurent series. It is shown that the equation does not pass the Painlevé test but has two arbitrary constants in local expansion. This fact allows us to look for solitary wave solutions for equations of unrestricted order. The main result of this paper is the theorem of existence of optical solitons for equations of unrestricted order that is proved by direct calculation. The optical solitons for partial differential equations of the twelfth order are given in detail.

New Symmetry Reductions, Dromions-Like and Compacton Solutions for a 2D BS(m,n) Equations Hierarchy with Fully Nonlinear Dispersion**The project supported by National Key Basic Research Development Project Program of China under Grant No. G1998030600, Doctoral Foundation of China under Grant No. 98014119 and National Natural Science Foundation of China under Grant No. 10072013

We have found two types of important exact solutions, compacton solutions, which are solitary waves with the property that after colliding with their own kind, they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction, in the and even models, and dromion solutions (exponentially decaying solutions in all direction) in many and models. In this paper, symmetry reductions in are considered for the break soliton-type equation with fully nonlinear dispersion (called equation) , which is a generalized model of break soliton equation , by using the extended direct reduction method. As a result, six types of symmetry reductions are obtained. Starting from the reduction equations and some simple transformations, we obtain the solitary wave solutions of equations, compacton solutions of equations and the compacton-like solution of the potential form (called ) . In addition, we show that the variable admits dromion solutions rather than the field itself in equation.

Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation

We obtain conditions for the nonlinear stability (Theorem 5.1) of solitary waves fro two classes of nonlinear dispersive equations which arise in the mathematical description of long wave propagation:(u is in general complex-valued), where the linear operator L, typically nonlocal, and the nonlinearity, are of some general class.For the case of homogeneous nonlinearities, A(u), a new variational characterization of solitary waves for I and II is presented (Theorem 3.1) and is exploited in the dynamic stability analysis. The existence of solitary waves in higher dimensions is also considered (Theorem 7.1).