EQUIVALENCE OF THE HUNTER-SAXON EQUATION AND THE GENERALIZED HEISENBERG FERROMAGNET EQUATION

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 𝑠𝑢ሺ𝑛 ൅ 1ሻ/𝑠ሺ𝑢ሺ1ሻ ⊕ 𝑢ሺ𝑛ሻሻ - symmetric space for the case 𝑛 ൌ 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.

Integrability structures of the generalized Hunter–Saxton equation

We consider integrability structures of the generalized Hunter–Saxton equation. We obtain the Lax representation with non-removable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give examples of employing the higher order symmetries to constructing exact globally defined solutions for the generalized Hunter–Saxton equation.

Lax Representation and Darboux Solutions of the Classical Painlevé Second Equation

In this article, we present Darboux solutions of the classical Painlevé second equation. We reexpress the classical Painlevé second Lax pair in new setting introducing gauge transformations to yield its Darboux expression in additive form. The new linear system of that equation carries similar structure as other integrable systems possess in the AKNS scheme. Finally, we generalize the Darboux transformation of the classical Painlevé second equation to the N -th form in terms of Wranskian.

Stable novel and accurate solitary wave solutions of an integrable equation: Qiao model

This article investigates the dynamical and physical behavior of the second positive member in a new, utterly integrable hierarchy. This investigation depends on constructing novel analytical and approximate solutions to the Qiao model. The model’s name is after the researcher who derived the mathematical formula of it in 2007. This model possesses a Lax representation and bi-Hamiltonian structure. This study employs the unified and variational iteration (VI) method to obtain analytical and numerical solutions to the considered model. The obtained analytical solutions are used to calculate the necessary conditions for applying the suggested numerical method that makes checking the obtained solutions’ accuracy a valuable option. The obtained solutions are sketched in different techniques to explain more physical and dynamics details of the Qiao model and show the matching between obtained solutions.

Inverse spectral problem for Jacobi operators and Miura transformation

We study a Miura-type transformation between Kac - van Moerbeke (Volterra) and Toda lattices in terms of the inverse spectral problem for Jacobi operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl function, can be used in solving initial-boundary value problem for both systems. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra lattices and the class of Toda lattices which is characterized by positivity of Jacobi operators in their Lax representation. Also, we discuss an implication of the latter result to the spectral theory.

Three Types Generalized Zn-Heisenberg Ferromagnet Models

By taking values in a commutative subalgebra gln,C, we construct a new generalized Zn-Heisenberg ferromagnet model in (1+1)-dimensions. The corresponding geometrical equivalence between the generalized Zn-Heisenberg ferromagnet model and Zn-mixed derivative nonlinear Schrödinger equation has been investigated. The Lax pairs associated with the generalized systems have been derived. In addition, we construct the generalized Zn-inhomogeneous Heisenberg ferromagnet model and Zn-Ishimori equation in (2+1)-dimensions. We also discuss the integrable properties of the multi-component systems. Meanwhile, the generalized Zn-nonlinear Schrödinger equation, Zn-Davey–Stewartson equation and their Lax representation have been well studied.

Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

Tensor fields associated with integrable systems of chiral

This article describes necessary conditions for chiral-type systems to admit Lax representation with values in simple compact Lie algebras. These conditions state that there exists a covariant constant tensor field with an additional property. It is proposed to construct in an invariant way some covariant tensor fields using the Lax representation of the system under consideration. These fields are constructed by taking linear differential forms with values in the Lie algebra that are constructed using the Lax representation of the system and substituting them into an arbitrary Ad-invariant form on the Lie algebra. The paper proves that such tensor fields are Killing tensor fields or covariant constant fields. The discovered necessary conditions for the existence of the Lax representation are obtained using a special case of such tensor fields associated with the Killing metric of the Lie algebra.