Georgia Institute of Technology College of Sciences

This talk concerns the Benjamin-Ono (BO) equation of internal wave theory, and properties of the solution of the Cauchy initial-value problem in the situation that the initial data is fixed but the coefficient of the nonlocal dispersive term in the equation is allowed to tend to zero (i.e., the zero-dispersion limit). It is well-known that existence of a limit requires the weak topology because high-frequency oscillations appear even though they are not present in the initial data.  Physically, this phenomenon corresponds to the generation of a dispersive shock wave. In the setting of the Korteweg-de Vries (KdV) equation, it has been shown that dispersive shock waves exhibit a universal form independent of initial data near the two edges of the dispersive shock wave, and also near the gradient catastrophe point for the inviscid Burgers equation from which the shock wave forms. In this talk, we will present corresponding universality results for the BO equation. These have quite a different character than in the KdV case; while for KdV one has universal wave profiles expressed in terms of solutions of Painlevé-type equations, for BO one instead has expressions in terms of classical Airy functions and Pearcey integrals. These results are proved for general rational initial data using a new approach based on an explicit formula for the solution of the Cauchy problem for BO. This is joint work with Elliot Blackstone and Matthew Mitchell, based on other work with Blackstone, Louise Gassot, and Patrick Gérard.

It has long been conjectured that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-time, due to the presence of stable (timelike) trapping and the manifestation of long-range scattering. We will present our recent resolution of this problem, establishing that, contrary to previous expectations, solutions with sufficiently localized initial data decay polynomially in time. Time permitting, we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums.

We consider Klein-Gordon equations with an external potential and cubic nonlinearity in three spatial dimensions. It is assumed that the linear operator has internal modes, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In 1999, Soffer and Weinstein treated the case when the linear operator has one large eigenvalue and proved the decay of the solution. In 2022, we solved the general one eigenvalue case. In our recent work, we solved the multiple internal modes case: the operator can has multiple and possibly degenerate eigenvalues. Indeed, we determined the sharp decay rate of the overall solution, as well as distinct decay rates for different modes of the solution. This is a joint work with Prof. Zhen Lei and Dr. Jie Liu.

We study the sharp asymptotics for a class of quasilinear wave equations satisfying a weak null condition but not the classical null condition in three spatial dimensions. We prove that the asymptotics are very different from those for the equations satisfying the classical null condition. In particular, at leading order, the solution displays a continuous superposition of decay rates.

Moreover, we show that any solution that decays faster than expected in a compact spatial region must vanish identically. The talk is based on joint work in progress with Jonathan Luk and Dongxiao Yu.