Studies On Symmetry Group,Invariant Solutions And Conservation Laws Of Nonlinear Partial Differential Equations

Nonlinear partial differential equations play an important role in many fields of science.In this thesis,the author mainly focus on the symmetry group,invariant solutions and conservation laws of nonlinear partial differential equations.The analysis depends mainly on the symmetry method.The emphases of this thesis are summarized as follows.1.The first chapter is the Introduction part.This chapter explains research background,research significance,research developments,current research situation of nonlinear partial differential equations in details.2.This chapter systemic addresses symmetry group and invariant solutions of(2 + 1)-dimensional nonlinear Schr?odinger equation(NLSE)with variables.For the special case,linear Schr?odinger equation(LSE),it can be transformed into the same form of equation.On the basis of different gauge constraint,we construct potential symmetries for the LSE.And then,we consider(2+1)-dimensional NLSE using Lie symmetry analysis.By means of similarity transformations,we study the(2+1)-dimensional NLSE with nonlinearities and potentials depending on time as well as on the spatial coordinates.At last,we present the rouge wave solutions of(2+1)-dimensional NLSE.3.In this chapter,on the basis of nonlocal symmetry,the third order,fourth order Burgers equation and the generalized fifth order KdV equation are investigated.First,nonlocal and potential symmetry of third-order Burgers equation is performed.On the basis of nonlocal symmetries,we linearize the nonlinear third-order Burgers equation to a linear third-order PDE.Furthermore,conservation laws are constructed of the equations.And then,based on the third-order Burgers equation,the fourth-order Burgers equation is studied.Nonlocal symmetry and explicit solutions of fourth-order Burgers equation are obtained.Furthermore,the nonlinear self-adjointness and conservation laws of the potential equation are presented in terms of the symmetries.At last,the generalized fifth order KdV equation is studied using group methods and conservation laws.The symmetry reductions and exact solutions to this equation are constructed.For some special cases,we obtain additional nontrivial conservation laws and scaling symmetries.4.This chapter mainly addresses some(2+1)-dimensional nonlinear partial differential equations with constant coefficients.First,an extended(2+1)-dimensional ZakharovKuznetsov-Burgers(ZKB)equation is studied.The Lie symmetry analysis leads to many plethora of solutions to the equation.The nonlinear self-adjointness and conservation laws of the equation are derived.And then,extended quantum Zakharov-Kuznetsov(QZK)equation is considered.Complete geometric symmetry and soliton solutions of extended quantum Zakharov-Kuznetsov(QZK)equation are investigated.5.In this chapter,a new(2+1)-dimensional sine-Gordon equation and a sinh-Gordon equation are investigated from the extended AKNS system.Based on the Hirota bilinear method,single,two,and N-solitary wave solutions of the new(2+1)-dimensional sineGordon equation are constructed.Symmetries are also considered.6.In this chapter,the complete algebra of Lie point symmetries for the class of time fractional nonlinear dispersive equation are derived.By means of the classical Lie symmetry method,the associated vector fields are obtained which in turn are utilized for the reduction of the equation.In particular,the conservation laws of the equation are obtained.7.The last chapter summarizes and concludes the thesis,and then,future direction and outlook of research projects was presented.