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The focusing Kundu-Eckhaus (KE) equation with non-zero boundary conditions atinfinity, under two cases: simple zeros and double zeros, is investigatedsystematically via Riemann-Hilbert (RH) problem. We derive some new results forthe equation including the following seven parts.  (I) The analyticities and symmetries of the Jost function and the scatteringmatrix are analyzed with the help of the normalized Lax pair. (II) Based on theresulting symmetries, the corresponding discrete spectrum set and residueconditions of scattering coefficients are further obtained, which is veryimportant to construct the formulae of solution to the original equation. (III)A generalized RH problem is established by combining the analytic properties ofJost functions and modified eigenfunctions. (IV) The RH problem is solved bythe corresponding asymptotic behavior combined with the Plemelj's formulae andCauchy operator. The expression of the solution to the focusing KE equation isgiven under the condition of non-reflection. (V) From the reflectioncoefficients and discrete spectrums, the trace formula and the correspondingtheta condition are given to obtain the phase difference of the initial valueat the boundary. (VI) For the double zeros, there is a similar framework fromthe set of discrete spectral points, but the operation process is much morecomplicated than that of simple zeros, and new results and phenomena appear.(VII) Some interesting phenomena are obtained that one of the solutions isgradually to rouge waves when the spectrum points tend to singular points bychoosing appropriate parameters.

Inverse scattering transform and soliton solutions for the focusing Kundu-Eckhaus equation with nonvanishing boundary conditions