Self-localized solutions of the Kundu-Eckhaus equation in nonlinear waveguides

In this paper we numerically analyze the 1D self-localized solutions of the Kundu-Eckhaus equation (KEE) in nonlinear wave guides using the spectral renormalization method (SRM) and compare our findings with those solutions of the nonlinear Schrödinger equation (NLSE). For cubic-quintic nonlinearity with Raman effect, as a benchmark problem we numerically construct single, dual and $N$-soliton solutions for the zero optical potential, i.e. $V$ = 0 , which are analytically derived before. We show that self-localized soliton solutions of the KEE with cubic-quintic nonlinearity and Raman effect do exist, at least for a range of parameters, for the photorefractive lattices with optical potentials in the form of $V$ = $I$$_{0}$cos$^{2}$($x$). Additionally, we also show that self-localized soliton solutions of the KEE with saturable cubic-quintic nonlinearity and Raman effect do also exist for some range of parameters. However, for all of the cases considered, these self-localized solitons are found to be unstable. We compare our findings for the KEE with their NLSE analogs and discuss our results.