Research On Numerical Solutions To Soliton Equations

In the research of mathematical theory, progress in soliton theory is mainly reflected inthat a large number of nonlinear soliton solutions of the partial differential equation have beenfound. The soliton theory has been formed in which the systematic nonlinear scienceintertwines with physics models. The important characteristics of soliton provide effectivesolutions for human’s understanding of nature and revealing many mysteries. The wideapplication prospect of soliton is showing its huge charm of science to the human.In this paper, N soliton solutions of Kdv equation are obtained by inverse scatteringmethod. Given amplitude, speed and phase difference as time tends to infinity, anothermethod is used to obtain a soliton solution of Kdv equation by introducing the Hopf-Colelikely transformation. Corresponding to N=2, a soliton solution to Kdv equation is obtainedand its motion state in the area is analyzed in detail in some cases. With the deepening of theresearch on numerical solution, the difference scheme satisfied discrete conservation law andimplicit difference scheme of quadratic conservation law are obtained by introducing thevirtual point. It is difficult to use inverse scattering method to solve the initial-boundary valueproblem, but the numerical research highlights its features and gives a kind of differencescheme which can satisfy the higher requirements of the conservation law.A difference operator is defined for the second order partial differential equation-S-Gequation. Then its difference scheme satisfied the conservation law is given. The solution toS-G equation under initial value condition is solved by using the inverse scattering method.Firstly, the change rule of scattering data with time is obtained when the scattering data in theequation where the initial conditions q(x), r(x) are potentials satisfy the S-G equation.Secondly, q(x,t), r(x,t) are recovered by T-L-M integral equations according to the scatteringdata. Thirdly, the three-dimensional image for the solution is simulated by virtue of maplesoftware program as the appropriate parameters selected. Finally, a new integrable generalizedNeumann system is introduced in this paper. The nonlinearization method is a very effectiveand important method by which the finite dimensional completely integrable Bargmannsystem and c. Neumann system are obtained through the nonlinearization of eigenvalueproblems associated with given soliton hierarchy. In recent years, the nonlinearization methodhas been extended to3×3and4x4spectral matrix problems. Some finite dimensional completely integrable systems in Liouville sense have been successfully gotten. In the lastchapter of this article, the nonlinearization method of the eigenvalue problem is applied to thesoliton hierarchy associated with a3x3eigenvalue problem and thus a new generalizedintegrable Neumann system and its n involutive conserved integrals are constructed.Moreover, the involutive solution of the soliton hierarchy is also given.