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

We consider the nonlinear Schrödinger equation on the half-line x ⩾ 0 with a Robin boundary condition at x = 0 and with initial data in the weighted Sobolev space H 1 , 1 ( R + ) . 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. In particular, by restricting our asymptotic result to solutions whose initial data are close to the initial profile of the stationary one-soliton, we obtain results on the asymptotic stability of the stationary one-soliton under any small perturbation in H 1 , 1 ( R + ) . In the focusing case, such a result was already established by Deift and Park using different methods, and our work provides an alternative approach to obtain such results. We treat both the focusing and the defocusing versions of the equation.

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 θ > 0 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. These asymptotics take the form P ( gap on [ 0 , s ] ) = C exp - a s 2 ρ + b s ρ + c ln s ( 1 + o ( 1 ) ) as s → + ∞ , where the constants ρ , a, and b have been derived explicitly via a differential identity in s and the analysis of a Riemann-Hilbert problem. Their method can be used to evaluate c (with more efforts), but does not allow for the evaluation of C. In this work, we obtain expressions for the constants c and C by employing a differential identity in θ . When θ is rational, we find that C can be expressed in terms of Barnes' G-function. We also show that the asymptotic formula can be extended to all orders in s.

Matrix Riemann-Hilbert problems with jumps across Carleson contours.

We develop a theory of n × n -matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour Γ 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. We introduce a notion of L p -Riemann-Hilbert problem and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.

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.