A space-time fully decoupled wavelet Galerkin method for solving a class of nonlinear wave problems

A space–time fully decoupled formulation for solving wave problems with various nonlinearities is proposed based on sampling approximation for a function defined on a bounded interval by combining techniques of boundary extension and Coiflet-type wavelet expansion. By using a wavelet Galerkin approach for spatial discretization, nonlinear wave equations are first transformed into a system of ordinary differential equations, in which all matrixes are completely independent of time and never need to be updated in the time integration. Then, the classical fourth-order explicit Runge–Kutta method is employed to solve the resulting semi-discretization system. Several widely considered test problems are solved, and results demonstrate that the proposed wavelet algorithm has a much better accuracy and a faster convergence rate than many existing numerical methods. Finally, interactions of breather with strong attractive impurity in the sine-Gordon model are investigated by using the proposed wavelet method. Results show that a low-frequency breather with intermediate initial velocity can either pass the impurity, be trapped, be reflected, or break into a kink–antikink pair, depending sensitively on its initial phase. When the frequency of breather is the same as that of the impurity mode, it can always escape from the potential trap due to the severe resonance. These observations support that the proposed wavelet method is capable of capturing complex nonlinear phenomena, even those extremely sensitive to parameters.