The Study Of Nonlinear Evolution Equations And Solutions

In many natural science like the fluid dynamics, biology, mathematics, plasma physics, optics and communication, solitons have been well studied and widely applied, which have important physics significance. In this paper, we focus on some soliton solutions of the nonlinear evolution equations (NLEEs). The main results are briefly stated below:First, some new exact solutions (soliton-like solutions, similarity solutions etc.) of the generalized variable coefficient GL equation, including bright- and dark-soliton-like solutions are investigated. By putting a direct ansatz into the generalized variable coefficient GL equation, we obtain its some new exact solutions (soliton-like solutions, similarity solutions etc.), subsequently, we analyze and discuss the properties and peculiarities of those exact solutions.Second, we go on investigating the generalized complex GL (cGL) equation with variable-coefficient. In virtue of a special transformation, the cGL equation can be translated into some known classical equations with some constraints about the variable coefficients, such as KP equation. However, these known classical equations have been widely studied, and many kinds of solutions are obtained. Substituting these solutions into the special transformatin, we can obtain the correspond solutions of the cGL equation.Third, based on the truncate painleve expansion method and symbolic computation, the auto-Backlund transformation of the generalized variable-coefficient KP (GKP) equation is obtained, and some soliton-typed solutions can be got by using the auto-Backlund transformation.Fourth, using the extended homogeneous balance method, we derive the auto-Backlund transformation and more soliton-typed solutions of the (2+1)-dimensional generalized variable-coefficient KdV euation.Fifth, in virtue of the Classical Lie group method and the Extended tanh function method, we obtain the similarity reduction, multiple soliton-like and triangular periodic solutions of the spherical Kadomtsev-Petviashvili (SKP) equation, respectively.