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

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.