Families of orthogonal Laurent polynomials,hyperelliptic lie algebras and elliptic integrals

We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties are listed in Section 3. We show that these families of polynomials in the variable t satisfy certain second-order linear differential equations that may be of interest to mathematicians in conformal field theory and number theory. We also prove that these families of polynomials in the setting of Date–Jimbo–Kashiwara–Miwa algebras when multiplied by a suitable power of t are orthogonal with respect to explicitly described kernels. Particular cases lead to new identities of elliptic integrals (see Section 5).     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

Multiple Periodic-Soliton Solutions for （3＋1）-Dimensional Jimbo-Miwa Equation

2N line-soliton solutions of the （3＋1）-dimensional Jimbo-Miwa equation can be presented by resorting to the Hirota bilinear method. In this paper, N periodic-soliton solutions of the （3＋1）-dimensional Jimbo-Miwa equation are obtained from the 2N line-soliton solutions by selecting the parameters into conjugated complex parameters in pairs.     

Classification des Objets Galoisiens de U q (𝔤) à Homotopie PrèS

Pour toute algèbre enveloppante quantique U_q(𝔤) de Drinfeld–Jimbo et toute famille λ = (λ_ij)1≤i ∈ k^? d'éléments inversibles du corps de base, nous construisons explicitement par générateurs et relations un objet galoisien A_λ de U_q(𝔤) et nous montrons que tout objet galoisien de U_q(𝔤) est homotope à un unique objet de la forme A_λ.

For any Drinfeld–Jimbo quantum enveloping algebra U_q(𝔤) and for any family λ = (λ_ij)_{1≤i ∈ k^? of invertible elements of the base field, we explicitly construct a Galois object Aλ of Uq(𝔤) by generators and relations and we prove that any Galois object of Uq(𝔤) is homotopic to a unique object of type Aλ.}     

CONDITIONAL SIMILARITY REDUCTION APPROACH: JIMBO－MIWA EQUATION

The direct method developed by Clarkson and Kruskal (1989 J. Math. Phys. 30 2201) for finding the symmetry reductions of a nonlinear system is extended to find the conditional similarity solutions. Using the method of the Jimbo－Miwa (JM) equation, we find that three well-known (2+1)-dimensional models－the asymmetric Nizhnik--Novikov－Veselov equation, the breaking soliton equation and the Kadomtsev-Petviashvili equation－can all be obtained as the conditional similarity reductions of the JM equation.