In this talk, we will consider the NLS system of the third-harmonic generation. Our interest is in solitary wave solutions and their stability properties. The recent work of Oliveira and Pastor, discussed global well-posedness, finite time blow-up, as well as other aspects of the dynamics. These authors have also constructed solitary wave solutions, via the method of the Nehari manifold, in an appropriate range of parameters. Specifically, the waves exist only in spatial dimensions \(n=1,2,3\).

The G-transform to a data series is the extension of the Fourier transform from the unit circle to the entire complex plane.I shall introduce the Padé approximant to the G-transform and discuss some of its properties as regard its poles, zeros, and the residues. In particular, I’ll show examples of superresolution with respect to the Nyquist limit, numerical evidence of universality for the behavior of poles and zeros associated with noise and how the presence of signals alters that behavior. I’ll conclude showing a couple of applications.

In this talk, we discuss anti-plane shear deformations on a semi-infinite slab with a non-linear mixed traction displacement boundary condition. We apply global bifurcation theoretic methods and deduce extreme behavior at the terminal end solution curves. It is shown that arbitrarily large strains are encountered for a class of idealized materials. We also consider degenerate materials, prove that ellipticity breaks down, and show that a concurrent blow-up in the second derivative occurs.

While the research on water waves modeled by Euler's equations has a long history, mainly in the last two decades traveling periodic rotational waves have been constructed rigorously by means of bifurcation theorems. After introducing the problem, I will present a new reformulation in two dimensions in the pure-gravity case, where the problem is equivalently cast into the form “identity plus compact”, which is amenable to Rabinowitz's global bifurcation theorem.

This talk is based on the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct variationally special solutions of the two models, that is,  standing wave solutions of NLS and traveling waves for KDV, which turn out to have compact support, hence compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well.

A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid — think of it as a spinning bubble of air in water. In this talk, I will describe a general method for desingularizing non-degenerate translating, rotating, or stationary point vortex configurations into collections of steady hollow vortices. Through global bifurcation theory, moreover, these families can be extended to maximal curves of solutions that continue until the onset of a singularity.

We consider the dynamics of a boosted soliton evolving under the cubic NLS with an external potential. We show that for sufficiently large velocities, the soliton is effectively transmitted through the potential.

The Camassa-Holm-Kadomtsev-Petviashvili equation (CH-KP) is a two dimensional generalization of the Camassa-Holm equation which has been recently derived in the context of shallow water waves and nonlinear elasticity. In this talk we will discuss the stability of the one-dimensional traveling waves, solitary or periodic, with respect to two dimensional perturbations which are periodic in the transverse direction. We show that the stability or instability depends on a sign parameter of the transverse dispersion term.

We will review the use of Nussbaum-Szkola distributions in quantum information and in particular in computing quantum divergences. The talk will be based on joint works with T.C. John https://arxiv.org/abs/2308.02929, https://arxiv.org/abs/2203.01964, https://arxiv.org/abs/2303.03380.

We consider the 2-dim water wave problem -- the free boundary problem of the Euler equation with gravity and possibly surface tension -- of finite depth linearized at a uniformly monotonic shear flow \(U(x_2)\). Our main focuses are eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity waves, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation.