The renormalization method for singular perturbation of solitons

Abstract

By the renormalization method based on the Taylor expansion (TR method, for simplicity), a new direct scheme to compute the singular perturbation of solitons is proposed. The proposed method can also be applied to perturbation of other solvable nonlinear evolution equations. The biggest advantage of the method is that the secular terms in perturbation series need not to be considered at all in our method. But, in other perturbation methods, the most important task is to eliminate the secular terms. In addition, comparing with the other direct approaches, we need not the multi-scale procedure. As an illustration example, the perturbation of shock wave for the perturbed Burgers equation is solved in detail, and a uniform asymptotical solution is obtained to describe the dynamics of the shock wave under the small perturbation and long time. By taking a concrete perturbation and computing the exact solution, we show that the renormalization solution gives exact correction of velocity of shock wave. In final, we compute the perturbation of the single soliton of the damping KdV equation.