Spectral difference methods for solving equations of the KdV hierarchy

[摘要] The Korteweg-de Vries (KdV) hierarchy is an important class of nonlinear evolution equa-tions with various applications in the physical sciences and in engineering.In this thesis analytical solution methods were used to ¯nd exact solutions of the third and¯fth order KdV equations, and numerical methods were used to compute numerical solutionsof these equations.Analytical methods used include the Fan sub-equation method for constructing exact trav-eling wave solutions, and the simpli¯ed Hirota method for constructing exact N-solitonsolutions. Some well known cases were considered.The Fourier spectral method and the ¯nite di®erence method with Runge-Kutta time dis-cretisation were employed to solve the third and the ¯fth order KdV equations with periodicboundary conditions. The one soliton and the two soliton solutions were used as initialconditions. The numerical solutions are obtained and compared with the exact solutions.The propagation of a single soliton as well as the interaction of double soliton solutions ismodeled well by both numerical methods, although the Fourier spectral method performsbetter.The stability, consistency and convergence of these numerical methods were investigated.Error propagation is studied. The theoretically predicted quadratic convergence of the ¯nitedi®erence method as well as the exponential convergence of the Fourier spectral method iscon¯rmed in numerical experiments.

[发布机构] Stellenbosch University