EQUIVALENCE OF THE HUNTER-SAXON EQUATION AND THE GENERALIZED HEISENBERG FERROMAGNET EQUATION
Ключевые слова:
integrable systems, the Hunter-Saxton equation, generalized Heisenberg ferromagnet equation, the Lax representation, gauge equivalence.Аннотация
Integrable systems play an important role in modern mathematics, theoretical and mathematical
physics. The display of integrable equations with exact solutions and some special solutions can provide important
guarantees for the analysis of its various properties. The Hunter-Saxton equation belongs to the family of integrable
systems. The extensive and interesting mathematical theory, underlying the Hunter-Saxton equation, creates active
mathematical and physical research. The Hunter-Saxton equation (HSE) is a high-frequency limit of the famous
Camassa-Holm equation. The physical interpretation of HSE is the propagation of weakly nonlinear orientation
waves in a massive nematic liquid crystal director field. In this paper, we propose a matrix form of the Lax representation for HSE in symmetric space for the case 2. Lax pairs, introduced in 1968 by Peter Lax, are a tool for finding conserved
quantities of integrable evolutionary differential equations. The Lax representation expands the possibilities of the
equation we are considering. For example, in this paper, we will use the matrix Lax representation for the HSE to
establish the gauge equivalence of this equation with the generalized Heisenberg ferromagnet equation (GHFE). The
famous Heisenberg Ferromagnet Equation (HFE) is one of the classical equations integrable through the inverse
scattering transform. In this paper, we will consider its generalization. Andalso the connection between the decisions
of the HSE and the GHFE will be presented.
