Riemann–Hilbert problem and soliton solutions with their asymptotic analysis for the focusing nonlocal Hirota equation with step-like initial data

Abstract

The Riemann–Hilbert (RH) problem is developed to study the Cauchy problem of focusing space symmetric nonlocal Hirota (fsNH) equation with step-like initial data. Firstly, we analyze the characteristics of the Jost solutions and spectral functions, including analyticity, symmetry, and asymptotics as k→∞ and k→0. Then, we discuss three cases of zero point distributions of the spectral functions, and derive the expressions of spectral functions by constructing the special analytic functions and establishing the scalar RH problem. Furthermore, the constraint relations among the zeros of the spectral functions are analyzed. Next, the corresponding RH problem equipped with residual and singularity conditions is successfully derived. These singularity conditions are then transformed into a general residue condition, which reduces the problem of solving the RH problem with reflection-less potential to solving a system of algebraic equations. Ultimately, the corresponding soliton solutions of the fsNH equation and their asymptotics are derived from the solution of the RH problem.