Study On The Stability And Exact Solutions To Some Nonlinear Evolution Equations

With the rapid development of science and technology, a large number ofnonlinear problems, which appeared in many fields, can be finally transformedto solving a class of nonlinear evolution equations and analyzing the propertiesof the solutions. Consequently, nonlinear evolution equations play more andmore important role in physics, mechanics, applied mathematics, geoscience,engineering science and technology, life sciences, and so on. It is considered asa hot and frontier subjects, the study of the exact solution of nonlinear equation,existence and stability of solutions caused lots of attention. Thus, solving andanalyzing stability of nonlinear evolution equation becomes a significant work.In this paper, we solved several important nonlinear partial differential equations,and studied the stability of some solutions. This dissertation mainly discussesthe following contents:1. The (2+1)-dimensional Zakharov-Kuznetsor (ZK) equation is reduced to(1+1)-dimensional Korteweg-de Vries (KdV) equation via introdcing a simplelinear transformation. The multi-solitary wave solutions of the (2+1)-dimensonalZK equation are derived. The results indicate that each of the solitary wave ofZK equation is line soliton and parallel with others.2. The reductive perturbation method is employed to describe the behaviourof ion-acoustic waves for plasmas in the absence of magnetic field, leading toa type of modified Kadomtsev-Petviashvili equation. The stability of a specialtype of solitary wave solutions for the modified Kadomtsev-Petviashviliequation is investigated with a finite difference scheme. The numerical resultsshow that this solitary wave is unstable under two particular initial perturbations.3. Based on the idea of the hyperbola function expansion method， someanalytical solutions of the modified Korteweg-de Vries (mKdV) equation areobtained by introducing new expansion functions．One of the single solitonsolutions has a kink-antikink structure and it reduces to a kink-like solution andbell-like solution under different limitation. The stability of the single solitonsolution with double kinks is investigated numerically．The results indicate that the soliton is stable under different disturbance．4. We obtained six classes of exact solutions for the coupled KdV equationby the extended hyperbola function expansion Method．One of the solutions is asolitary wave solution, which has two peaks．This solution is reduced to the kinkor bell-like soliton solution of the coupled KdV equation under differentlimitations. We also investigated the stability of the single solitary wave solutionwith double peaks numerically．The results indicate that the solution is stablewhen the amplitude of the disturbance, which has long wave length, and is verysmall.