Pole Dynamics, Linearization, and Perturbations of the Satsuma--Mimura Equation

Abstract

This paper investigates the pole dynamics and perturbation theory of algebraic soliton solutions associated with the Satsuma--Mimura (SM) equation. First, we give a qualitative analysis of the pole system associated with algebraic soliton solutions, thereby completing a point left open in \cite{Yan_2025}. We then examine three perturbations of the SM equation. The $u_x$ perturbation preserves exact linearizability and leads to an explicit shifted algebraic soliton solution. The $u_{xx}$ perturbation can be reduced to the unperturbed SM equation by a scaling transformation, which yields the corresponding pole asymptotics and soliton profile. For the genuinely nontrivial $Hu_{xx}$ perturbation, we derive the first-order perturbation equations and present a short-time numerical simulation based on an explicit Euler discretization. These results clarify how algebraic solitons of the SM equation respond to different perturbative mechanisms.