New Similarity Solutions For Two Classes Of Nonlinear Wave Equations With Coefficients

Nonlinear partial differential equations with variable coefficients are widely applied in many mathematical and physical problems.Solving nonlinear partial differential equations with variable coefficients is an important subject for physicists and mathematicians.In this thesis,two classes of nonlinear wave equations with variable coefficients are studied:the(2+1)-dimensional nonlinear Schr?dinger equation with spatially modulated coefficients and the variable-coefficient forced Korteweg-de Vries equation.Applying the similarity construction method,CK direct reduction method and(G’/G)-expansion method,some new similarity solutions of the above two classes of equations are obtained.The organization of this thesis will be introduced as follows.In the first chapter,the methods,contents and main results of the thesis are introduced.In the second chapter,the(2+1)-dimensional nonlinear Schrodinger equation with spatially modulated coefficients is studied.First,after a transformation,the linear equation corresponding to the(2+1)-dimensional nonlinear Schr?dinger equation with spatially modulated coefficients is transformed into the Hukuhara equation and then it is solved by the similarity construction method.Then,the exact solutions of the(2+1)-dimensional nonlinear Schrodinger equation with spatially modulated coefficients,including bright and dark soliton solutions,hyperbolic function solution,trigonometric function solution and rational function solution,are obtained by the(G’/G)-expansion method and the ansatz method.Finally,the graphs of these solutions to the linear and nonlinear equations with k=1,2 are drew,respectively,which reveals that the evolution of solutions to the nonlinear Schr?dinger equation with spatially modulated coefficients is determined by the corresponding linear equation.In the third chapter,the variable-coefficient forced Korteweg-de Vries equation is studied.First,the variable-coefficient forced Korteweg-de Vries equation is transformed into an ordinary differential equation by using the CK direct reduction method.Then,the solutions of this ordinary differential equation are obtained by the(G’/G)-expansion method.Finally,the similarity solutions of the variable-coefficient forced Korteweg-de Vries equation are given and their images are plotted.In the fourth chapter,we summarize the work done in this thesis and look at future research questions.