Inverse Scattering Transform and Soliton Solutions for the Hirota Equation with $N$ Distinct Arbitrary Order Poles

Abstract

We employ the Riemann-Hilbert (RH) method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions. Through the spectral analysis, the asymptoticity, symmetry, and analysis of the Jost functions are obtained, which play a key role in constructing the RH problem. Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem. Choosing some appropriate parameters of the resulting solutions, we further derive the soliton solutions with different order poles, including four cases of a fourth-order pole, two second-order poles, a third-order pole and a first-order pole, and four first-order points. Finally, the dynamical behavior of these solutions are analyzed via graphic analysis.