Long-time asymptotics for the Elastic Beam equation in the solitonless region via $\bar{\partial}$ methods

TL;DR

This paper analyzes the long-time behavior of solutions to the Elastic Beam equation in the solitonless region using the $ar{ ext{d}}$-steepest descent method, deriving asymptotics via Riemann-Hilbert problem techniques.

Contribution

It introduces a novel application of the $ar{ ext{d}}$-steepest descent method to the Elastic Beam equation, deriving its long-time asymptotics in the solitonless region.

Findings

Derived the Riemann-Hilbert problem for the Elastic Beam equation.

Analyzed the long-time asymptotics in the solitonless region.

Provided a framework for future asymptotic analysis of similar equations.

Abstract

In this work, we study the Cauchy problem of the Elastic Beam equation with initial value in weighted Sobolev space $H^{1,1}(\mathbb{R})$ via the $\bar{\partial}$-steepset descent method. Begin with the Lax pair of the Elastic Beam equation, we successfully derive the basic Riemann-Hilbert problem, which can be used to represent the solutions of the Elastic Beam equation. Then, considering the solitonless region and using the $\bar{\partial}$-steepset descent method, we analyse the long-time asymptotic behaviors of the solutions for the Elastic Beam equation.