On the asymptotic stability of N-soliton solution for the short pulse equation with weighted Sobolev initial data

Abstract

In this work, we are devoted to studying the Cauchy problem of the short pulse (SP) equation with weighted Sobolev initial data. By developing the ∂¯-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution q(x,t) in solitonic region of space-time, (yt)=ξ for any fixed ξ=ξ0∈(0,+∞), and give bounds for the error decaying as |t|→∞. Based on the resulting asymptotic behavior, the asymptotic approximation of the SP equation is characterized with the soliton term confirmed by N(I)-soliton on discrete spectrum with residual error up to O(t−1). Combining with previous results presented by Yang and Fan, that is the long time asymptotic expansion of the solution q(x,t) in solitonic region ξ=yt∈(−∞,0), our results show that the soliton resolution conjecture of the SP equation is fully solved in all space-time solitonic regions. Moreover, considering properties of the scattering map of SP equation, we derive an asymptotic stability result for a class of initial values.