The Study Of The Analytical Solutions,Symmetries And Conservation Laws For Several Types Nonlinear Differential Equations

The study of the analytical solutions,symmetries and conservation laws for non-linear differential equations plays a vital role in scientifically explaining of the cor-responding physical phenomenon and engineering application.In this thesis,firstly,Hirota bilinear method,Bell polynomials and Riemann theta function are clarified.Then we extend them to fifth-order KdV type equation,the periodic wave solution-s of the equation are systematically investigated and the asymptotic analysis of the periodic wave solutions are also proved.Meanwhile,breather wave and rogue wave so-lutions of(3+1)-dimensional Kadomtsev-Petviashvili equation are obtained by devel-oping the homoclinic breather limit approach.Moreover,we extend the Lie symmetry analysis method and the adjoint equation method to the time fractional Kolmogorov-Petrovskii-Piskunov equation,then symmetries and conservation laws of the equation are constructed.Finally,based on the Lie symmetry analysis method,the adjoint e-quation method and linear stability analysis are clarified,then we expand them to the(3+1)-dimensional nonlinear Schrodinger equation,symmetries,conservation laws and modulation instability analysis of the equation are investigated,respectively.In the first chapter,the research background and significance of soliton theory,Lie group and conservation laws are briefly introduced,then the main work of this thesis is also illustrated.In the second chapter,based on the Hirota bilinear method,we focus on the bi-nary Bell polynomials and Riemann theta functions,and extend them to the fifth-order KdV type equation,its bilinear represention is obtained.Then the Riemann theta func-tion periodic wave solutions and soliton solutions of the equation are constructed.In addition,the asymptotic properties of the periodic wave solutions are analyzed in de-tail,which strictly shows that the periodic wave solutions can be degenerated into the soliton solutions under the small amplitude limit.In the third chapter,homoclinic breather limit approach and solitary ansatz method are extended to(3+1)-dimensional generalized Kadomtsev-Petviashvili equation.Its breather wave,rogue wave solutions and bright-dark soliton solutions are obtained.In addition,rogue wave comes from the extreme behaviour of breather wave are also carried out in detail.In the fourth chapter,by extending Lie symmetry method to the time fraction-al Kolmogorov-Petrovskii-Piskunov equation,its infinitesimal symmetric vector fields and similarity reductions are obtained.Then,by using the power series method,the power series solutions of the equation are derived.Moreover,the convergence of pow-er series solution is also analyzed.In addition,the adjoint equation method is extended to the equation,the conservation laws of the equation are obtained.In the fifth chapter,linear stability analysis is extended to(3+1)-dimensional non-linear Schrodinger equation,the modulation instability analysis of the equation is stud-ied.Then,Lie symmetry method is also extended to the equation,the vector fields of the infinitesimal symmetry for the equation are obtained.In addition,according to the associated theory of adjoint equation method,the conservation laws of the equation are systematically constructed.In the sixth chapter,we give some conclusions and prospects of this thesis.