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

We employ the ∂̄-steepest descent method in order to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in weighted Sobolev space H(R) = { f ∈ L(R) : f ′, x f ∈ L(R)}. The long time asymptotic behavior of the solution u(x, t) is derived in a fixed space-time cone S (x1, x2, v1, v2) = {(x, t) ∈ R : y = y0 + vt, y0 ∈ [y1, y2], v ∈ [v1, v2]}. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the CSP equation which includes the soliton term confirmed by N(I)-soliton on discrete spectrum and the t− 1 2 order term on continuous spectrum with residual error up to O(t−1).