The Jackiw-Teitelboim gauge formulation of the 1+1 dimensional gravity allows us to relate different gauge fixing conditions with integrable hierarchies of evolution equations. We show that the equations for the Zweibein fields can be written as a pair of time reversed evolution equations of the reaction-diffusion type, admitting dissipative solutions. The spectral parameter for the related Lax...

We study a planar model of a nonrelativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lamé equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a specia...

In this article, we propose a new representation of Lax type for the sixth Painlevé equation. This representation, formulated in the framework of the loop algebra so(8)[z, z−1] of type D (1) 4 , provides a natural explanation of the affine Weyl group symmetry of PVI. After recalling a standard derivation of PVI, we describe in Section 2 fundamental Bäcklund transformations for PVI. In Section 3...

A discrete nonlinear system is analysed in case of open chain boundary conditions at the ends. It is shown that the integrability of the system remains intact, by obtaining a modified set of Lax equations which automatically take care of the boundary conditions. The same Lax pair also conforms to the conditions stipulated by Sklyanin [5]. The quantum inverse problem is set up and the diagonalis...

A generalized inverse scattering method has been developed for arbitrary n dimensional Lax equations. Subsequently, the method has been used to obtain N soliton solutions of a vector higher order nonlinear Schrödinger equation, proposed by us. It has been shown that under suitable reduction, vector higher order nonlinear Schrödinger equation reduces to higher order nonlinear Schrödinger equatio...

The relationship between solutions of the sinh-Laplace equation and the determination of various kinds of surfaces of constant Gaussian curvature, both positive and negative, will be investigated here. It is shown that when the metric is given in a particular set of coordinates, the Gaussian curvature is related to the sinh-Laplace equation in a direct way. The fundamental equations of surface ...

Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the leading-order asymptotics as t → ±∞ of the solution to the Cauchy problem for the modified nonlinear Schrödinger equation, i∂tu+ 1 2 ∂ xu+ |u|2u+ is∂x(|u|u) = 0, s ...

Abstract The Landau-Ginzburg-Higgs (LGH) equation explains the ocean engineering models, superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves. In this paper, with a simple modification of Ablowitz-Kaup-Newell-Segur (AKNS) formalism, integrability LGH is proved by deriving Lax pair. Hence that, inverse scattering transformation (IST) appli...

The cubic Szego equation has been studied as an integrable model for deterministic turbulence, starting with the foundational work of Gerard and Grellier. We introduce a truncated version this equation, wherein majority Fourier mode couplings are eliminated while signature features preserved, namely, Lax-pair structure nested hierarchy finite-dimensional dynamically invariant manifolds. Despite...

In this paper, we constructed relationships with the differents 2D elasticity tensor invariants. Indeed, let ${\bf A}$ be a tensor. Rotation group action leads to pair of Lax in linear elasticity. This five independent invariants chosen among six. The definite positive criteria are established determined We believe that approach finds interesting applications, as one elastic material classifica...