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

Abstract

We employ the $\bar{\partial }$-steepest descent method in order to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\in L^2(\mathbb{R}):f^{\prime },xf\in L^2(\mathbb{R})\}$. The long time asymptotic behavior of the solution $u(x,t)$ is derived in a fixed space-time cone $S(x_1,x_2,v_1,v_2)=\{(x,t)\in {\mathbb{R}}^2:y=y_0+vt,y_0\in [y_1,y_2],v\in [v_1,v_2]\}$. 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 $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-1})$.

44 pages

References in corpus (5)

• The short pulse equation is integrable
• Long-time Asymptotics for the NLS equation via dbar methods
• Global well-posedness for the derivative nonlinear Schrödinger equation
• Soliton Resolution for the Short-pluse Equation
• Soliton resolution for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data

Cited by in corpus (2)

• Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data
• On the asymptotic stability of $N$-soliton solutions of the modified nonlinear Schrödinger equation