Long time asymptotics behavior of the focusing nonlinear Kundu-Eckhaus equation

TL;DR

This paper analyzes the long-time behavior of solutions to the focusing nonlinear Kundu-Eckhaus equation using inverse scattering and Riemann-Hilbert methods, providing asymptotic expansions and soliton descriptions within specific space-time cones.

Contribution

It develops a detailed asymptotic analysis of the focusing nonlinear Kundu-Eckhaus equation using advanced inverse scattering techniques, including $ar{ ext{∂}}$ steepest descent, and characterizes soliton structures with precise error estimates.

Findings

Derived long-time asymptotics in fixed space-time cones.

Constructed soliton solutions with residual errors of order $ ext{O}(t^{-3/4})$.

Applied inverse scattering and Riemann-Hilbert methods to a nonlinear PDE.

Abstract

We study the Cauchy problem for the focusing nonlinear Kundu-Eckhaus equation and construct long time asymptotic expansion of its solution in fixed space-time cone with $C(x_1,x_2,v_1,v_2)=\{(x,t)\in\Re^2:x=x_0+vt$ $x_0\in[x_1,x_2],v\in[v_1,v_2] \}$. By using the inverse scattering transform, Riemann-Hilbert approach and $\overline\partial$ steepest descent method we obtain the lone time asymptotic behavior of the solution, at the same time we obtain the solitons in the cone compare with the all N-soliton the residual error up to order $\mathcal{O}(t^{-3/4})$.