Search

The boundary value problem for the focusing Kundu–Eckhaus equation with nonzero boundary conditions is studied by the Dbar dressing method in this work. A Dbar problem with non-canonical normalization condition at infinity is introduced to investigate the soliton solution. The eigenfunction of Dbar problem is meromorphic outside annulus with center 0, which is used to construct the Lax pair of the Kundu–Eckhaus equation with nonzero boundary conditions, which is a crucial step to further search for the soliton solution. Furthermore, the original nonlinear evolution equation and conservation law are obtained by means of choosing a special distribution matrix. Moreover, the N-soliton solutions of the focusing Kundu–Eckhaus equation with nonzero boundary conditions are discussed based on the symmetries and distribution. As concrete examples, the dynamic behaviors of the one-breather solution and the two-breather solution are analyzed graphically by considering different parameters. [ABSTRACT FROM AUTHOR]

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 β , 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. [ABSTRACT FROM AUTHOR]