Long Time Asymptotics Behavior of the Focusing Nonlinear Kundu-Eckhaus Equation

Abstract

The authors study the Cauchy problem for the focusing nonlinear Kundu-Eckhaus (KE for short) equation and construct the long time asymptotic expansion of its solution in fixed space-time cone with C(x ₁, x ₂, v ₁, v ₂) = {(x, t) ∈ ℝ² : x = x ₀ + vt, x ₀ ∈ [x ₁, x ₂], v ∈ [v ₁, v ₂]}. By using the inverse scattering transform, Riemann-Hilbert approach and $\overline{\partial}$ steepest descent method, they obtain the lone time asymptotic behavior of the solution, at the same time, they obtain the solitons in the cone compare with the all N-soliton the residual error up to order $\cal{O}(t^{-{3\over 4}})$.

Keywords

Focusing Kundu-Eckhaus equation / Riemann-Hilbert problem / $\overline{\partial}$ steepest descent method

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