Related papers: Schlesinger system, Einstein equations and hyperel…
The Schr\"odinger equation for two and tree-body problems is solved for scattering states in a hybrid representation where solutions are expanded in the eigenstates of the harmonic oscillator in the interaction region and on a finite…
We present auto and hetero B\"acklund transformations of the nonholonomic Veselova system using standard divisor arithmetic on the hyperelliptic curve of genus two. As a by-product one gets two natural integrable systems on the cotangent…
Methods developed for the analysis of integrable systems are used to study the problem of hyperK\"ahler metrics building as formulated in D=2 N=4 supersymmetric harmonic superspace. We show, in particular, that the constraint equation…
Various types of Lagrange and Finsler geometries and the Einstein gravity theory, and modifications, can be modelled by nonholonomic distributions on tangent bundles/ manifolds when the fundamental geometric objects are adapted to nonlinear…
In part I we introduced modified Landweber-Kaczmarz methods and have established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse…
The complete integrability of the hyperbolic Gaudin Hamiltonian and other related integrable systems is shown to be easily derived by taking into account their sl(2,R) coalgebra symmetry. By using the properties induced by such a coalgebra…
We propose quantum Hamiltonians of the double elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the…
In this work numerical methods for solving Einstein's equations are developed and applied to the study of inhomogeneous cosmological models. A two-dimensional computer code is described which implements two advanced numerical methods:…
We study integrable Euler equations on the Lie algebra $\mathfrak{gl}(3,\mathbb{R})$ by interpreting them as evolutions on the space of hexagons inscribed in a real cubic curve.
A Teichm\"uller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichm\"uller curves of positive genus. Our methods are based on…
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schr\"odinger…
The non-linear sigma model of the dimensionally reduced Einstein (-Maxwell) theory is diagonally embedded into that of the two-dimensional heterotic string theory. Consequently, the embedded string backgrounds satisfy the (electro-magnetic)…
We show how the Einstein equations with cosmological constant (and/or various types of matter field sources) can be integrated in a very general form following the anholonomic deformation method for constructing exact solutions in four and…
We consider a class of axially symmetric solutions to Einstein's equations incorporating a $\theta$-dependent scalar field and extend these solutions by introducing electric and magnetic charges via Harrison transformations. Subsequently,…
We develop a unified Courant--Hilbert framework for constructing two-dimensional integrable sigma models deformed by two couplings: a marginal one $\gamma$ and an irrelevant one $\lambda$. The integrability condition is encoded in a…
We associate with each simple Lie algebra a system of second-order differential equations invariant under a non-compact real form of the corresponding Lie group. In the limit of a contraction to a Schr\"odinger algebra, these equations…
We present a useful proposition for discovering extended Laplace-Runge-Lentz vectors of certain quantum mechanical systems. We propose a new family of superintegrable systems and construct their integrals of motion. We solve these systems…
We consider the quantum inverse scattering method for several mixed integrable models based on the rational SU(N) R-matrix with general toroidal boundary conditions. This includes systems whose Hilbert spaces are invariant by the discrete…
Surfaces of revolution in three-dimensional Euclidean space are considered. Several new examples of surfaces of revolution associated with well-known solvable cases of the Schoedinger equation (infinite well, harmonic oscillator, Coulomb…
This paper proposes a Cartesian grid-based boundary integral method for efficiently and stably solving two representative moving interface problems, the Hele-Shaw flow and the Stefan problem. Elliptic and parabolic partial differential…