Research On The Analytical Solutions And Evolution Characteristics Of Several Nonlinear Differential Equations

The study of analytical solutions for nonlinear differential equations is useful to explain important physical phenomena.In this work,we use Darboux transformation method and Riemann-Hilbert method to analyse the soliton solutions,breather wave solutions,rogue wave solutions and other nonlinear wave solutions for several nonlinear evolution equations.In addition,we study the propagation behavior of solutions in different cases.In chapter 1,we give a a brief introduction of the research background and significance of soliton theory and related methods.In chapter 2,we study the stability analysis and optical solitary wave solutions of a(2+1)-dimensional nonlinear Schr ¨odinger equation,which are derived from a multicomponent plasma with nonextensive distribution.Based on Ansatz and sub-equation theories,we employ a direct method to find stability analysis and optical solitary wave solutions of the(2+1)-dimensional equation.By considering the Ansatz method,we successfully construct the bright and dark soliton solutions of the equation.The subequation method is also extended to find complexitons solutions.Moreover,the explicit power series solution is also derived with its convergence analysis.Finally,the influences of each parameters on these solutions are discussed via graphical analysis.In chapter 3,we pay attention to a general coupled nonlinear Schr ¨odinger equation with zero boundary conditions and study it with Riemann-Hibert method.The analytical and asymptotic properties of Jost functions are obtained by the direct scattering analysis to establish a suitable RH problem.The Taylor series and Laurent expansion are used to solve the RH problem in the case of the order pole,and the expression of the -soliton solution of the equation in the case of the order pole is given.In chapter 4,we mainly use Darboux-dressing method to study the high order rogue wave and breather wave solutions of a three-component coupled nonlinear Schr ¨odinger equation.Then we obtain the temporally periodic breather wave solutions and the spatial periodic breather wave solutions by adjusting the spectrum parameter.We successfully get the second-order vector rogue waves.At last,we discuss the existence condition for the rogue waves which has a relation with modulation instability.In chapter 5,a brief summary of this thesis and related prospects are given for later work.