A renormalization method without spectrum theory for the perturbation of solitons

Abstract A direct renormalization method without spectrum theory is proposed to compute the perturbation of solitons in nearly integrable systems with multiple small parameters. The evolution equations of these parameters in unperturbed solitons are obtained as the renormalization equations. Compared with routine methods, the advantages of the renormalization method are that the formulation is only based on a clear and simple mathematical theory, namely the Taylor expansion at a general point, the secular terms in perturbation series are eliminated automatically, any priori physical assumption on the form of the solution is avoided, multiple time scales arise naturally from the final naive perturbation expansion, and the Green’s function and corresponding spectrum of linear differential operators are not needed. As applications, the perturbation of solitons for KDV, MKdV and nonlinear Schrodinger equations, are obtained.