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We compute instanton corrections to the low energy effective prepotential of N=2 supersymmetric theories in a variety of cases, including all classical gauge groups and even number of fundamental matter hypermultiplets. To this end, we take…
The problem of a spinless particle subject to a general mixing of vector and scalar inversely linear potentials in a two-dimensional world is analyzed. Exact bounded solutions are found in closed form by imposing boundary conditions on the…
The Riemann-Hilbert (RH) problem is first developed to study the focusing nonlinear Schr\"{o}dinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace…
Approximate analytical solutions of a two-term potential are studied for the relativistic wave equations, namely, for the Klein-Gordon and Dirac equations. The results are obtained by solving of a Riemann-type equation whose solution can be…
We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the…
We probe the application of the calculus of conormal distributions, in particular the Pull-Back and Push-Forward Theorems, to the method of layer potentials to solve the Dirichlet and Neumann problems on half-spaces. We obtain full…
The homogeneous Lippmann-Schwinger integral equation is solved in momentum space by using confining potentials. Since the confining potentials are unbounded at large distances, they lead to a singularity at small momentum. In order to…
We address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice. As Kay and Moses showed in 1956, reflectionless potentials in one dimensional quantum mechanics…
The Korteweg-De Vries (KdV) equation is a paradigmatic model of integrable classical fields, admitting solitoning solutions. When many solitons are near to each other, their shapes are modified, and it is not manifest, from the KdV field,…
We investigate Hamiltonian aspects of the integro-differential kinetic equation for dense soliton gas which results as a thermodynamic limit of the Whitham equations. Under a delta-functional ansatz, the kinetic equation reduces to a…
Spectrum of the Dirac Equation is obtained algebraically for arbitrary combination of Lorentz-scalar and Lorentz-vector Coulomb potentials using the Witten's Superalgebra approach. The result coincides with that, known from the explicit…
The effective and efficient numerical solution of Riemann-Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann-Hilbert problems, the resulting numerical…
Original English Summary. - A systematic method of constructing potentials, for which the one-variable Schroedinger equation can be solved in terms of the hypergeometric (HGM) function, is presented. All the potentials, obtained by…
We consider slow motion of a pointlike topological defect (vortex) in the nonlinear Schrodinger equation minimally coupled to Chern-Simons gauge field and subject to external uniform magnetic field. It turns out that a formal expansion of…
Statistical mechanics of 1D multivalent Coulomb gas may be mapped onto non-Hermitian quantum mechanics. We use this example to develop instanton calculus on Riemann surfaces. Borrowing from the formalism developed in the context of…
A simple and general approach for calculating the elliptic finite-gap solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of the finite-gap equations and the general representation of these…
In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation of the Schroedinger operator on a bounded interval as a finite three-diagonal symmetric Jacobi matrix. This…
Many properties of current \emph{ab initio} approaches to the quantum many-body problem, both perturbational or otherwise, are related to the singularity structure of Rayleigh--Schr\"odinger perturbation theory. A numerical procedure is…
We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue…
We propose a numerical method for finding solitary wave solutions of generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue problem on an unbounded domain. The artificial boundary conditions are obtained to make the…