Related papers: Elliptic solutions to difference non-linear equati…
We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials which can be considered as a generalization of the Stieltjes-Carlitz elliptic polynomials.…
Two Lie algebraic forms of the 2-body Elliptic Calogero model are presented. Translation-invariant and dilatation-invariant discretizations of the model are obtained.
Algebro-geometric solutions of the lattice potential modified Kadomtsev-Petviashvili (lpmKP) equation are constructed. A Darboux transformation of the Kaup--Newell spectral problem is employed to generate a Lax triad for the lpmKP equation,…
A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple…
In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \left\{\begin{aligned} -\Delta u & =\lambda |u|^{p-2}u…
We study elliptic families of solutions to the recently introduced constrained Toda hierarchy, i.e., solutions which are elliptic functions of some linear combination of the hierarchical times. Equations of motion for poles of such…
A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient $G(t)=\alpha_1(t)I+\alpha_2(t)Q(t)$, $\alpha_1(t), \alpha_2(t)\in H(L)$, $Q(t)$ is a $2\times 2$ zero-trace…
This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Camassa-Holm Dym (CHD2) hierarchy. Our main tools include the polynomial recursive…
We mainly construct and analyze the multi elliptic-localized solutions under the background of elliptic function solutions for the focusing modified Korteweg-de Vries (mKdV) equation. Based on the Darboux-B\"{a}cklund transformation, we…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
We consider elliptic solutions of the semi-discrete BKP equation and derive equations of motion for their poles. The basic tool is the auxiliary linear problem for the wave function.
We outline an approach to a theory of various generalizations of the elliptic Calogero-Moser (CM) and Ruijsenaars-Shneider (RS) systems based on a special inverse problem for linear operators with elliptic coefficients. Hamiltonian theory…
The paper gives a survey of the modern results on elliptic problems on the H\"ormander function spaces. More precisely, elliptic problems are studied on a Hilbert scale of the isotropic H\"ormander spaces parametrized by a real number and a…
We derive theta function representation of algebro-geometric solution of a Coupled Burgers equation which the second nonlinear evolution equation in a hierarchy. We also derive the algebro-geometric characters of the meromorphic function…
We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral…
We propose the algebro-geometric mothod of construction of solutions of the discrete KP equation over a finite field. We also perform the corresponding reduction to the finite field version of the discrete KdV equation. We write down…
Using the matrix-resolvent method and a formula of the second-named author on the $n$-point function for a KP tau-function, we show that the tau-function of an arbitrary solution to the Toda lattice hierarchy is a KP tau-function. We then…
In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…
Resorting to the Lax matrix and elliptic variables, the discrete Chen-Lee-Liu hierarchy is decomposed into solvable ordinary differential equations. Based on the theory of algebraic curve, the continuous flow and discrete flow related to…