Soliton resolution for the Hirota equation with weighted Sobolev initial data

Abstract

In this work, the $\overline{\partial }$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to 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 $q(x,t)$ is derived in any fixed space-time cone $C(x_1,x_2,v_1,v_2)=\left\{(x,t)\in \mathbb{R}\times \mathbb{R}:x=x_0+vt\text{with}{x}_0\in [x_1,x_2]\right\}$. We show that solution resolution conjecture of the Hirota equation is characterized by the leading order term $\mathcal{O}(t^{-1/2})$ in the continuous spectrum, $\mathcal{N}(\mathcal{I})$ soliton solutions in the discrete spectrum and error order $\mathcal{O}(t^{-3/4})$ from the $\overline{\partial }$ equation.

43 pages

References in corpus (3)

• 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

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