MU Differential Equations Seminar Schedule

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 \(n=1,2,3\). They have also established some stability/instability results for these waves.

In this work, we systematically build and study solitary waves for this important model. We construct the waves in the largest possible parameter space, and we provide a complete classification of their spectral stability.

Finally, we showed instability by a blow-up, for dimension 3, and for a more restrictive set of parameters, we use virial identities methods to derive the strong instability, in the spirit of Ohta's approach. This is joint work with Atanas Stefanov.

In this talk, we will discuss some recent results on the structural implications of constant vorticity for steady three-dimensional internal water waves. It is known that in many physical regimes, water waves beneath vacuum that have constant vorticity are necessarily two dimensional. The situation is more subtle for internal waves that traveling along the interface between two immiscible fluids. When the layers have the same density, there is a large class of explicit steady waves with constant vorticity that are three-dimensional in that the velocity field and pressure depend on one horizontal variable while the interface is an arbitrary function of the other.

Our main theorem states that every three-dimensional traveling internal wave with bounded velocity for which the vorticity in the lower layer \(\boldsymbol{\omega}_1\) and vorticity in the upper layer \(\boldsymbol{\omega}_2\) are nonzero, constant, and parallel must belong to this family. If the densities in each layer are distinct, then in fact the flow is fully two dimensional. This result is obtained using a novel but fairly elementary argument based on unique continuation, the maximum principle, and an analysis of streamline patterns.

This is joint work with R. M. Chen, L. Fan, and M. H. Wheeler.

We study the microlocal properties of the scattering matrix associated to the semiclassical Schrödinger operator \(P=h^2\Delta_X+V\) on a Riemannian manifold with an infinite cylindrical end. We will show that under suitable hypotheses the scattering matrix "quantizes" the scattering map. The scattering map \(\kappa\) and its domain are determined by the Hamilton flow of \(|\xi|^2+V\upharpoonright_{h=0}\), the principal symbol of \(P\). As an application we prove that, under additional hypotheses on the scattering map, the eigenvalues of the associated unitary scattering matrix are equidistributed on the unit circle.

The goal of this talk will be to introduce the audience to the big picture: the setting, the objects of the interest, general questions, and some of the challenges involved, rather than giving proofs.

This talk is based on joint work with A. Uribe.

In this talk, we will consider the liner and nonlinear dynamics of perturbations of spectrally stable periodic stationary solutions of the Lugiato–Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in optics. It is known that spectrally stable \(T\)-periodic solutions are nonlinearly stable to subharmonic perturbations, i.e. to \(NT\)-periodic perturbations for some integer \(N\), with exponential decay rates. However, both the exponential rates of decay and the allowable size of initial perturbations both tend to zero as \(N \to \infty\), and hence such subharmonic stability results are not uniform in \(N\) and are, in fact, empty in the limit \(N=\infty\).

The primary goal of this talk is to introduce a methodology, in the context of the LLE, by which a uniform in \(N\) stability result for subharmonic perturbations may be achieved (at least at the linear level). The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e. integrable on the line, perturbations of such spectrally stable periodic solutions of LLE. If time permits, I will also discuss recent progress towards extending these results for the LLE to the nonlinear level.

This is joint with Mariana Haragus (Univ. Bourgogne Franche-Comté), Wesley Perkins (KU) and Bjorn de-Rijk (Stuttgart)

I will present background on oscillating heat pipes (OHPs), published results on an ODE model, and give a progress report on a new PDE model. In short, an OHP is a serpentine closed tube (toroidal geometry) partially filled with a liquid. When part of the boundary is heated and part is cooled, it is possible that the liquid separates into a two-phase flow consisting of vapor plugs separating fluid slugs that is set into oscillatory motion and serves as a device to efficiently transfer heat from the hot to the cold zone with no moving mechanical parts except for the fluid motion within the tube. At one level of modeling (usually with ODEs) fluid slugs are tracked and artificial means are used (if at all) to model nucleation and merging of fluid cells.

A more sophisticated approach, and the main focus of the talk, seeks to model the two-phase flow as a phenomenon that arises naturally from the underlying physics and eliminates the need to track slugs. This latter approach, called phase field modeling, is based on the ideas of Allen, Cahn, and Hilliard that lead to the Allen–Cahn and the Cahn–Hilliard equations, which form the basis of all such models. Roughly speaking, the dependent variable in these equations is a so-called order parameter, which is akin to a smoothed indicator function evolving in space and time, which gives the locations of the two phases of the flow. The underlying physics is thermodynamics; in particular, the minimization of Gibbs energy at equilibrium. An overview of this methodology, which has far reaching applications, will be discussed. Its application to OHPs is the motivation for a proposed PDE model whose predictive power has not yet been fully explored.

Most of the content of the talk is joint work with Frank Feng, Steve Lombardo, and Dave Retzloff all colleagues in the College of Engineering.

Dual quaternions are finding increasing use in the robotics and graphics industry, as a method to represent pose (position and orientation) of one frame with respect to another. There is a natural Lie-group/Lie algebra structure, and from this arises a need to compute the exponential and logarithm of a dual quaternion. Quite a few other authors have done this, but their formulas are either wrong, complicated, or hard to use.

In this talk we describe a general functional calculus for dual quaternions. The methods are quite elementary, and it nostalgically brings back an earlier time in our lives when mathematics was essentially simple, and about cute formulas.

In 1984, T. Kato showed that for sufficiently regular solutions, the vanishing viscosity limit is equivalent to having vanishing viscous dissipation in a boundary layer of width proportional to the viscosity. We prove that Kato's criterion holds for Hölder continuous weak solutions with the regularity index arbitrarily close to the Onsager's critical exponent through a new boundary layer foliation and a global mollification technique.

This is a joint work with Zhilei Liang and Dehua Wang.

The Hot Spots constant was recently introduced by Steinerberger as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We use a probabilistic technique to derive a general formula for a dimension-dependent upper bound that can be tailored to any specific class of bounded Lipschitz domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in \(\mathbb{R}^d\) for both small \(d\) and for asymptotically large \(d\) that significantly improve upon the existing results.

This is joint work with Phanuel Mariano and Jing Wang.