The Riemann-Hilbert approach for the focusing Hirota equation with single and double poles

Abstract

We study the Cauchy problem of the focusing Hirota equation with fastly decay initial value at infinity via Riemann-Hilbert (RH) approach. The Jost function and scattering matrix related to the initial value are obtained by spectral analysis, and their analytical, asymptotic and symmetric properties are analyzed. The solution of the Hirota equation is derived by solving the corresponding RH problem, which contains two cases of scattering data with single zeros and double zeros. The general form of solutions with no reflection potential is obtained. The two types of the discrete spectrum are selected for the purpose of comparing the influence of different eigenvalues on solution propagation behavior.