Rogue wave and multi-pole solutions for the focusing Kundu-Eckhaus Equation with nonzero background via Riemann-Hilbert problem method

In this work, we use the inverse scattering transform method to consider the focusing Kundu-Eckhaus (KE) equation with nonzero background (NZBG) at infinity. Based on the analytical, symmetric, asymptotic properties of eigenfunctions, the inverse problem is solved via a matrix Riemann-Hilbert problem (RHP). The general multi-pole solutions are given in terms of the solution of the associated RHP. And the formula of the N simple-pole soliton solutions are obtained, too. We show that the Peregrine's rational solution can be viewed as some appropriate limit of the simple-pole soliton solutions at branch point. Furthermore, by taking some other proper limits, the two and three simple-pole soliton solutions can yield to the double- and triple-pole solutions for focusing KE equation with NZBG. The effect of the parameter beta, characterizing the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effect, and some typical collisions between solutions of focusing KE equation are graphically displayed.

Automated summary

In this study, the inverse scattering transform method is used to solve the inverse problem of the focusing Kundu-Eckhaus equation with nonzero background at infinity, obtaining general multi-pole solutions and formulas for the N simple-pole soliton solutions. Peregrine's rational solution is shown to be an appropriate limit of the simple-pole soliton solutions at branch points, and taking proper limits can yield double- and triple-pole soliton solutions for the equation. The effect of the parameter beta and typical collisions between solutions are also graphically displayed.