Soliton resolution for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data

Abstract

In this work, we employ the $\bar{\partial }$ steepest descent method in order to study the Cauchy problem of the cgNLS equations 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 large time asymptotic behavior of the solution $u(x,t)$ and $v(x,t)$ are derived in a fixed space-time cone $S(x_1,x_2,v_1,v_2)=\{(x,t)\in {\mathbb{R}}^2:x=x_0+vt,x_0\in [x_1,x_2],v\in [v_1,v_2]\}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the cgNLS equations which contains the soliton term confirmed by $|\mathcal{Z}(\mathcal{I})|$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$.

37 pages

References in corpus (5)

• Long-time Asymptotics for the NLS equation via dbar methods
• Global well-posedness for the derivative nonlinear Schrödinger equation
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• Soliton Resolution for the Short-pluse Equation
• Long-time asymptotics in the modified Landau-Lifshitz equation with nonzero boundary conditions

Cited by in corpus (2)

• Soliton resolution for the complex short pulse equation with weighted Sobolev initial data
• Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data