Dispersionless Limits of Ma Equations

Abstract

At present, there is a great interest in the study of solitons, which are used in many fundamental
theories, such as mathematics, physics, and others. Solitons are called a structurally stable solitary
wave propagating in a nonlinear medium, which retains its structure when colliding with each
other. The theory of solitons is based on nonlinear integrable equations. The fundamental mathematical
mechanism for solving nonlinear integrable equations is the inverse scattering method.
This method establishes a connection between a nonlinear integrable equation with a linear system.
Dispersionless integrable equations are one of the new sections of the theory of integrable
equations. They gained considerable interest due to their extensive use in various applications of
natural science. In this paper, we investigated one of the generalizations of the Landau-Lifshitz
equation known from soliton theory, which is called the Ma equation. The Landau – Lifshitz equation
is the geometric equivalent of the nonlinear Schr¨odinger equation, and there is also a gauge
equivalence between them. The nonlinear equations of Ma describe the resonant interaction of
short and long waves in a plasma. Also, the dispersionless Ma equation was found and a Lax
representation was constructed for it, which proves its integrability.