We investigate the effects of given pressure gradients on hydrodynamic flow equations. We obtain results in terms of implicit solutions and also in the framework of an extra-dimensional formalism involving the diffusion/Schrödinger equation.

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.

Based on the combined compact difference scheme, an alternating direction implicit method is proposed for solving two-dimensional cubic nonlinear Schrödinger equations. The proposed method is sixth-order accurate in space and second-order accurate in time. The linear Fourier analysis method is exploited to study the stability of the proposed method. The efficiency and accuracy of the proposed m...

We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrödinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrödinger group are used to get global existence in the energy space on arbitrary surfaces and three-dimensional manifolds, generalizing earlier results by Bourgain on tori. On the other hand, on specific manifolds such as sph...

We establish a relation between the classical non-linear Schrödinger equation and the KP hierarchy, and we extend this relation to the quantum case by defining a quantum KP hierarchy. We also present evidence that an integrable hierarchy of equations is obtained by quantizing the first Hamiltonian structure of the KdV equation. The connection between infinite-dimensional algebras and integrable...

Abstract. We study localised attractors of the parametrically driven, damped nonlinear Schrödinger equation. Time-periodic solitons of this equation are obtained as solutions of the boundary-value problem on a two-dimensional domain. Stability and bifurcations of periodic solitons and their complexes is classified. We show that the bifurcation diagram can be reproduced using a few-mode approxim...

We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schrödinger equation in a random δ-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and usi...

We examine numerically vortex families near band edges of the Bloch wave spectrum for the Gross-Pitaevskii equation with two-dimensional periodic potentials and for the discrete nonlinear Schrödinger equation. We show that besides vortex families that terminate at a small distance from the band edges via fold bifurcations, there exist vortex families that are continued all the way to the band e...

The discretized Schrödinger equation is one of the most commonly employed methods for solving one-dimensional quantum mechanics problems on the computer, yet many of its characteristics remain poorly understood. The differences with the continuous Schrödinger equation are generally viewed as shortcomings of the discrete model and are typically described in purely mathematical terms. This is unf...

Comparison of Finite Differences and WKB Method for Approximating Tunneling Times of the One Dimensional Schrödinger Equation Student: Yael Elmatad Overview The goal of this research was to examine the quantum mechanical phenomenon of tunneling with respect to the Schrödinger Equation. A quantum mechanical wave is said to “tunnel” when it travels (propagates) through a classically forbidden reg...