Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions
The Kundu-Eckhaus equation \[ iq_t+\frac{1}{2}q_{xx}+|q|^2q+2\beta |q|^4q-2i\beta(|q|^2)_x+q=0 \] is considered, with the initial condition \[ q(x,0)\sim Ae^{i(\mu x+\theta_\pm)} \text{ when } x\to\pm\infty. \] Here \(\beta, A(>0),\mu\), and \(\theta_\pm\) are real constants. The long-time asymptotics of the solution is established in three different sectors depending on the magnitude \(\xi=x/t\): plane wave sector \(\xi<\xi_1=\mu-2\beta A^2-2\sqrt{2}A\), plane wave sector \(\xi>\xi_2=\mu-2\beta A^2+2\sqrt{2}A\), and modulated genus 1 elliptic wave sector \(\xi_1<\xi<\xi_2\).
scientific article; zbMATH DE number 7041833
scientific article