Alexander Tovbis

The eigenvalue problem for the focusing nonlinear Schrödinger equations: new solvable cases

Abstract In this paper, we study the semi-classical limit of the Zakharov–Shabat eigenvalue problem for the focusing of NLS with some specific initial data. In all these cases, the eigenvalue problem is reduced to connection problems for the hypergeometric equation and for other classical equations. The special initial data [Suppl. Prog. Theor. Phys. 55 (1974) 284] is contained in our family of initial data, parameterized by a real parameter μ , as a particular case μ =0. We find that beyond a certain value of the parameter μ , the pure-point spectrum becomes empty and all the scattering information is contained in the reflection coefficient.

Asymptotics of Orthogonal Polynomials with Complex Varying Quartic Weight: Global Structure, Critical Point Behavior and the First Painlevé Equation

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials

Finite Hilbert transform with incomplete data: null-space and singular values

\pi _n(z)

Nonlinear Steepest Descent Asymptotics for Semiclassical Limit of Integrable Systems: Continuation in the Parameter Space

πn(z) with the quartic exponential weight

Breaking Homoclinic Connections for a Singularly Perturbed Differential Equation and the Stokes Phenomenon

\exp \left[-N\left(\frac{1}{2}z^2 + \frac{1}{4}t z^4\right)\right]

Semiclassical dynamics of quasi-one-dimensional, attractive Bose–Einstein condensates

exp-N12z2+14tz4, where