Related papers: Resolution of the Cauchy problem for the Toda latt…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial…
We consider solutions of the Cauchy problem for semilinear equations with (possibly) different L\'evy operators. We provide various results on their convergence under the assumption that symbols of the involved operators converge to the…
Solutions for all Adler-Bobenko-Suris equations excluding Q4 and several lattice Boussinesq-type equations are reconsidered by employing the Cauchy matrix approach. Through introducing a ``fake'' nonautonomous plane wave factor, we derive…
Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic Toda lattice of N particles. These conditions involve the Nekrasov-Shatashvili free energy,…
We continue the study of the theory of scattering for some long range Hartree equations with potential |x|^-gamma, performed in a previous paper, denoted as I, in the range 1/2 < gamma < 1. Here we extend the results to the range 1/3 <…
We consider a periodic Jacobi operator $H$ with finitely supported perturbations on ${\Bbb Z}.$ We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of…
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential \[iu_{t} +\Delta u-c|x|^{-2}u=\lambda|x|^{-b} |u|^{\sigma } u,\; u(0)=u_{0} \in…
A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problems with the initial condition the delta function concentrated at a single plane (i.e. the plane…
While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible…
The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary…
In this article, we consider the generalised two-parameter Cauchy two-matrix model and corresponding integrable lattice equation. It is shown that with parameters chosen as $1/k_i$ when $k_i\in\mathbb{Z}_{>0}$ ($i=1,\,2$), the average…
A system of equations consisting of an infinite string coupled to a nonlinear oscillator is considered. The Cauchy problem for the system with the periodic initial data is studied. The main goal is to prove the convergence of the solutions…
We consider the inverse problem for the wave equation which consists of determining an unknown space-dependent force function acting on a vibrating structure from Cauchy boundary data. Since only boundary data are used as measurements, the…
We consider the soliton solutions in 1- and (1+1)-dimensional Toda lattice models with a boundary. We make use of the solutions already known on a full line by means of the Hirota's method. We explicitly construct the solutions satisfying…
We use the combinatorics of toric networks and the double affine geometric $R$-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this…
It has been observed that the dynamics of the Toda lattice can be well described by solutions of the Korteweg-de Vries (KdV) equation in the continuum limit. We show that, under the KdV scaling and a suitable translation, the solution of…
We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schr\"odinger (NNLS) equation $ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with the step-like initial data close to the ``shifted step function''…
We investigate scattering properties of a Moyal deformed version of the nonlinear Schr\"odinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general…
We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudodifferential Lax operator, can be…