Soliton resolution for the generalized complex short pulse equation with the weighted Sobolev initial data

Abstract

In this work, the Cauchy problem for the generalized complex short pulse equation with initial conditions in the weighted Sobolev space H(R) is studied by using the Riemann-Hilbert method and the ∂‾-steepest descent method. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem can be expressed as solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, the long-time asymptotics and soliton resolution of the generalized complex short pulse equation in the soliton region are obtained by resorting to the ∂‾-steepest descent method. The results also indicate that the N-soliton solutions of the generalized complex short pulse equation are asymptotically stable.