Professor Avraham (Avy) Soffer was born in Israel on December 28, 1955. He earned his Bachelor’s, Master’s, and Doctor’s degrees in Science from Tel-Aviv University in 1976, 1980, and 1984, respectively. Between 1978 and 1983, he worked as an assistant at Tel-Aviv University.

This paper is concerned with the time decay estimates of the fourth order Schrödinger operator $H = ∆^2+V (x)$ in dimension three, where $V (x)$ is a real valued decaying potential. Assume that zero is a regular point or the first kind resonance of $H,$ and $H$ has no positive eigenvalues, we established the following time optimal decay estimates of $e^{−it H}$ with a regular term $H^{α/4}:$

When zero is the second or third kind resonance of $H,$ their decay will be significantly changed. We remark that such improved time decay estimates with the extra regular term $H^{α/4}$ will be interesting in the well-posedness and scattering of nonlinear fourth order Schrödinger equations with potentials.

The defocusing action ground state of the nonlinear Schrödinger equation can be characterized via three different but equivalent minimization formulations. In this work, we propose some deep neural network (DNN) approaches to compute the action ground state through the three formulations. We first consider the unconstrained formulation, where we propose the DNN with a shift layer and demonstrate its necessity towards finding the correct ground state. The other two formulations involve the $L^{p+1}$-normalization or the Nehari manifold constraint. We enforce them as hard constraints into the networks by further proposing a normalization layer or a projection layer to the DNN. Our DNNs can then be trained in an unconstrained and unsupervised manner. Systematical numerical experiments are conducted to demonstrate the effectiveness and superiority of the approaches.

This paper studies the $H^p-H^q$ estimates of a class of oscillatory integrals related to dispersive equations

under the assumption that the level hypersurfaces are convex and of finite type. As applications, we obtain the decay estimates for the solutions of higher order homogeneous and inhomogeneous Schrödinger equations.

The $r^p$ method, first introduced in [9], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the $r^p$-weighted energy using a novel Grönwall argument, which allows a larger range of $p$ than the standard method. A spherically symmetric version of our strategy first appeared in [22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.