Inverse scattering transform and soliton solutions for the focusing Kundu-Eckhaus equation with nonvanishing boundary conditions
The focusing Kundu-Eckhaus (KE) equation with non-zero boundary conditions at infinity, under two cases: simple zeros and double zeros, is investigated systematically via Riemann-Hilbert (RH) problem. We derive some new results for the equation including the following seven parts. (I) The analyticities and symmetries of the Jost function and the scattering matrix are analyzed with the help of the normalized Lax pair. (II) Based on the resulting symmetries, the corresponding discrete spectrum set and residue conditions of scattering coefficients are further obtained, which is very important to construct the formulae of solution to the original equation. (III) A generalized RH problem is established by combining the analytic properties of Jost functions and modified eigenfunctions. (IV) The RH problem is solved by the corresponding asymptotic behavior combined with the Plemelj's formulae and Cauchy operator. The expression of the solution to the focusing KE equation is given under the condition of non-reflection. (V) From the reflection coefficients and discrete spectrums, the trace formula and the corresponding theta condition are given to obtain the phase difference of the initial value at the boundary. (VI) For the double zeros, there is a similar framework from the set of discrete spectral points, but the operation process is much more complicated than that of simple zeros, and new results and phenomena appear. (VII) Some interesting phenomena are obtained that one of the solutions is gradually to rouge waves when the spectrum points tend to singular points by choosing appropriate parameters.