[Paper Review] Soliton resolution for the Hirota equation with weighted Sobolev initial data

Jin‐Jie Yang, Shou‐Fu Tian+1 more|arXiv (Cornell University)|Jan 15, 2021

Nonlinear Waves and Solitons39 references4 citations

TL;DR

This paper establishes the long-time asymptotic behavior of solutions to the Hirota equation with initial data in the weighted Sobolev space $H^{1,1}(ℝ)$ using the $\overline{\partial}$ steepest descent method. It proves the soliton resolution conjecture: the solution decomposes into $\mathcal{N}(\mathcal{I})$ solitons, a radiative part decaying as $\mathcal{O}(t^{-1/2})$ in the continuous spectrum, and an error term of order $\mathcal{O}(t^{-3/4})$ from the $\overline{\partial}$ equation.

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 $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The long-time asymptotic behavior of the solution $q(x,t)$ is derived in any fixed space-time cone $C(x_{1},x_{2},v_{1},v_{2})=\left\{(x,t)\in \mathbb{R} imes\mathbb{R}: x=x_{0}+vt ~ ext{with}~ x_{0}\in[x_{1},x_{2}] ight\}$. 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.

Motivation & Objective

• To establish the long-time asymptotic behavior of solutions to the Hirota equation with initial data in the weighted Sobolev space $H^{1,1}(ℝ)$.
• To verify the soliton resolution conjecture for the Hirota equation in the context of finite mass initial data.
• To extend the $\overline{\partial}$ steepest descent method to the Hirota equation, which includes third-order dispersion and nonlinear terms.
• To derive precise decay rates for the radiative part and error terms in the asymptotic expansion of the solution.

Proposed method

• The $\overline{\partial}$ steepest descent method is applied to analyze the matrix Riemann-Hilbert problem associated with the Hirota equation.
• The solution is decomposed into a soliton part, a continuous spectrum contribution, and an error term arising from the $\overline{\partial}$ equation.
• A mixed $\overline{\partial}$-Riemann-Hilbert problem is formulated and decomposed into outer, local, and error subproblems.
• The outer model problem is solved explicitly, yielding the soliton contribution with $\mathcal{N}(\mathcal{I})$ solitons.
• A local Riemann-Hilbert problem near phase points is analyzed using parabolic cylinder model problems.
• The small-norm $\overline{\partial}$-problem for the error function is solved, yielding the $\mathcal{O}(t^{-3/4})$ error estimate.

Experimental results

Research questions

• RQ1 Does the soliton resolution conjecture hold for the Hirota equation with initial data in $H^{1,1}(ℝ)$?
• RQ2 What is the long-time asymptotic behavior of the solution $q(x,t)$ in fixed space-time cones?
• RQ3 What are the precise decay rates of the radiative part and error term in the asymptotic expansion?
• RQ4 Can the $\overline{\partial}$ steepest descent method be successfully applied to the Hirota equation, which includes third-order dispersion and nonlinear terms?

Key findings

• The solution $q(x,t)$ exhibits soliton resolution in any fixed space-time cone, decomposing into $\mathcal{N}(\mathcal{I})$ solitons from the discrete spectrum.
• The radiative part in the continuous spectrum decays as $\mathcal{O}(t^{-1/2})$ as $t \to \infty$.
• The error term arising from the $\overline{\partial}$ equation is bounded by $\mathcal{O}(t^{-3/4})$.
• The $\overline{\partial}$ steepest descent method successfully handles the Hirota equation without requiring additional decay or smoothness conditions on the initial data.
• The asymptotic analysis is valid for initial data in $H^{1,1}(ℝ)$, which includes finite mass and finite first moment.