Focusing and defocusing Hirota equations with non-zero boundary conditions: Inverse scattering transforms and soliton solutions

Abstract In this paper, we investigate the inverse scattering transforms and soliton solutions of both focusing and defocusing Hirota equations with non-zero boundary conditions (NZBCs). The inverse problems are solved via the study of the matrix Riemann-Hilbert problems. As a consequence, we present the general solutions for the potentials, and explicit expressions for the reflectionless potentials. Moreover, the trace formulae and theta conditions are also given. Particularly, we discuss the simple-pole and double-pole solutions for the focusing case, and the simple-pole solutions for the defocusing case. Moreover, the results of the focusing case with NZBCs can reduce to ones of the focusing case with zero-boundary condition (ZBC). These obtained solutions are useful to explain the related nonlinear wave phenomena.

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