Related papers: Linear systems, Hankel products and the sinh-Gordo…
We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $\Gamma$ we reformulate the Dirichlet boundary value problem for the Helmholtz…
We consider the massive Klein-Gordon equation on a class of asymptotically static spacetimes. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods. Assuming in addition that the…
We investigate the representations of the solutions to Maxwell's equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the…
The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $\RR^d$ associated with a root system. The aim of the paper is to study local boundary behaviour of generalized harmonic functions…
This work investigates the long-time asymptotic behaviors of solutions to the initial value problem of the two-component nonlinear Klein-Gordon equation by inverse scattering transform and Riemann-Hilbert formulism. Two reflection…
We study a family of classical solutions of modified sinh-Gordon equation, $\partial_z\partial_{{\bar z}} \eta-\re^{2\eta}+p(z)\,p({\bar z})\ \re^{-2\eta}=0$ with $p(z)=z^{2\alpha}-s^{2\alpha}$. We show that certain connection coefficients…
We consider the discontinuous semi-classical Laguerre weight function with a jump $w(x;t,s)=\mathrm{e}^{-x^2+tx}(A+B\theta(x-s))$, where $x\in\mathbf{R}$, $t,s\ge0$, $A\ge0$, $A+B\ge0$, where $\theta(x)$ is 1 for $x > 0$ and 0 otherwise.…
We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of a hierarchy of higher order analogues to the Painlev\'{e} III equation. This Fredholm determinant describes the…
We continue with the study of the Hankel determinant, $$ D_{n}(t,\alpha):=\det\left(\int_{0}^{\infty}x^{j+k}w(x;t,\alpha)dx\right)_{j,k=0 }^{n-1}, $$ generated by singularly perturbed Laguerre weight, $$ w(x;t,\alpha):=x^{\alpha}{\rm…
In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic St{\"a}ckel manifolds having the topology of toric cylinders and satisfying the Robertson condition. On these manifolds the…
We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Pad\'{e} approximants and conformal mappings, for solving quasi-analytically a…
We consider the class of bounded self-adjoint Hankel operators $\mathbf H$, realised as integral operators on the positive semi-axis, that commute with dilations by a fixed factor. By analogy with the spectral theory of periodic…
By employing a novel generalization of the inverse scattering transform method known as the unified transform or Fokas method, it can be shown that the solution of certain physically significant boundary value problems for the elliptic…
Let $\Gamma$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $\L[u]=(-pu')'+qu$ acting on functions that are continuous on all of $\Gamma$, and twice continuously differentiable in the interior…
Based on the work of Chen and Its [{\em J. Approx. Theory} {\bf 162} ({2010}) {270--297}], we further study orthogonal polynomials with respect to the singularly perturbed Laguerre weight $w(x;t,\alpha) = {x^\alpha}{\mathrm e^{-…
It is well known that sigma models in symmetric spaces accept equivalent descriptions in terms of integrable systems such as the sine-Gordon equation through Pohlmeyer reduction. In this paper, we study the mapping between known solutions…
The diagonal matrix elements $\langle\theta_1,\theta_2|T_{2n}T_{-2n}|\theta_1,\theta_2\rangle$ between two-particle states in the sinh-Gordon model are computed analytically for all integers $n>0$. This confirms the proposal by F. Smirnov…
We formulate the generic $\tau$-function of the Painlev\'e II equation as a Fredholm determinant of an integrable (Its-Izergin-Korepin-Slavnov) operator. The $\tau$-function depends on the isomonodromic time $t$ and two Stokes' parameters,…
Using the relation established by Johnson--Zumbrun between Hill's method of aproximating spectra of periodic-coefficient ordinary differential operators and a generalized periodic Evans function given by the $2$-modified characteristic…
Let $j(z)$ be the modular $j$-invariant function. Let $\tau$ be an algebraic number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider and Siegel that if $\tau$ is not a CM point, i.e.,…