Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions

Abstract

In this work, we employ the ∂¯-steepest descent method to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in weighted Sobolev space H(R). Firstly, we successfully derive the Hamiltonian function of the CSP equation based on its Lax pair. Furthermore, the long time asymptotic behavior of the solution u(x,t) is derived in a fixed space-time cone S(y1,y2,v1,v2)={(y,t)∈R2:y=y0+vt,y0∈[y1,y2],v∈[v1,v2]}. On the basis of the resulting asymptotic behavior, we prove the soliton resolution conjecture of the CSP equation which includes the soliton term confirmed by N(I)-soliton on discrete spectrum and the t−12 order term on continuous spectrum with residual error up to O(t−1).