Exact Solutions And Asymptotic Solutions Of Some Nonlinear Problems In Mathematical Physics

In this paper,we mainly study three nonlinear problems,namely,geodesics on n-dimensional quadrics,global asymptotic solutions of perturbed KdV equation and perturbed Burgers equation,and exact solutions to variant Bossinesq equations.In chapters 1 and 2,the Hamilton-Jacobi method is applied to study the problems of geodesics on n-dimensional quadrics.Firstly,the geodesics on three-dimensional quadrics are obtained,and the geodesics are the intersections of two three-dimensional surfaces,then these geodesics are proved indeed existing by implicit function theorem and numerical method.At last,the work to is extened to the n-dimensional condition and obtained the geodesics on n-dimensional quadrics,the existences of these geodesics are also proved.In chapter 3,the renormalization group method is applied to two famous equations in fluid mechanics(perturbed KdV equation and perturbed Burger’s equation)and obtain global asymptotic solutions.Then Kunihiro’s renormalization group method based on the theory of envelope in differential geometry is applied to eliminate the long-time terms and make them convergent at infinity,then the global asymptotic solutions are obtained.Lastly,the complete discrimination system for polynomial method is applied to solve variant Bossinesq equations,and give the classification of all single travelling wave solutions.