Conformal Field Theory, Solitons, and Elliptic Calogero-Sutherland Models.

We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.

The nonlinear Schrödinger equation on the half-line with homogeneous Robin boundary conditions.

We consider the nonlinear Schrödinger equation on the half-line with a Robin boundary condition at and with initial data in the weighted Sobolev space . We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann-Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution.

Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib-Borodin Ensembles.

We consider the limiting process that arises at the hard edge of Muttalib-Borodin ensembles. This point process depends on and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles.

Matrix Riemann-Hilbert problems with jumps across Carleson contours.

We develop a theory of -matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, unbounded contours with cusps, corners, and nontransversal intersections are allowed.

Classification of all travelling-wave solutions for some nonlinear dispersive equations.

We present a method for the classification of all weak travelling-wave solutions for some dispersive nonlinear wave equations. When applied to the Camassa-Holm or the Degasperis-Procesi equation, the approach shows the existence of not only smooth, peaked and cusped travelling-wave solutions, but also more exotic solutions with fractal-like wave profiles.