Walid K. Abou Salem

Solitary wave dynamics in time-dependent potentials

The long time dynamics of solitary wave solutions of the nonlinear Schrodinger equation in time-dependent external potentials is rigorously studied. To set the stage, the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schrodinger equation with time-dependent nonlinearities and potential is established. Afterward, the dynamics of NLS solitary waves in time-dependent potentials is studied. It is shown that in the space-adiabatic regime where the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is described by Hamilton’s equations, plus terms due to radiation damping. Finally, two physical applications are discussed: the first is adiabatic transportation of solitons and the second is the Mathieu instability of trapped solitons due to time-periodic perturbations.

On the Quasi-Static Evolution of Nonequilibrium Steady States

The quasi-static evolution of steady states far from equilibrium is investigated from the point of view of quantum statistical mechanics. As a concrete example of a thermodynamic system, a two-level quantum dot coupled to several reservoirs of free fermions at different temperatures is considered. A novel adiabatic theorem for unbounded and nonnormal generators of evolution is proven and applied to study the quasi-static evolution of the nonequilibrium steady state (NESS) of the coupled system.Abstract.The quasi-static evolution of steady states far from equilibrium is investigated from the point of view of quantum statistical mechanics. As a concrete example of a thermodynamic system, a two-level quantum dot coupled to several reservoirs of free fermions at different temperatures is considered. A novel adiabatic theorem for unbounded and nonnormal generators of evolution is proven and applied to study the quasi-static evolution of the nonequilibrium steady state (NESS) of the coupled system.

Effective dynamics of solitons in the presence of rough nonlinear perturbations

The effective long-time dynamics of solitary wave solutions of the nonlinear Schrodinger equation in the presence of rough nonlinear perturbations is rigorously studied. It is shown that, if the initial state is close to a slowly travelling soliton of the unperturbed NLS equation (in H1 norm), then, over a long time scale, the true solution of the initial value problem will be close to a soliton whose centre of mass dynamics is approximately determined by an effective potential that corresponds to the restriction of the nonlinear perturbation to the soliton manifold.

A Remark on the Mean-Field Dynamics of Many-Body Bosonic Systems with Random Interactions and in a Random Potential

The mean-field limit for the dynamics of bosons with random two-body interactions and in the presence of a random external potential is rigorously studied, both for the Hartree dynamics and the Gross–Pitaevskii dynamics. First, it is shown that, for interactions and potentials that are almost surely bounded, the many-body quantum evolution can be replaced in the mean-field limit by a single particle nonlinear evolution that is described by the Hartree equation. This is an Egorov-type theorem for many-body quantum systems with random interactions. The analysis is then extended to derive the Gross–Pitaevskii equation with random interactions.The mean-field limit for the dynamics of bosons with random two-body interactions and in the presence of a random external potential is rigorously studied, both for the Hartree dynamics and the Gross–Pitaevskii dynamics. First, it is shown that, for interactions and potentials that are almost surely bounded, the many-body quantum evolution can be replaced in the mean-field limit by a single particle nonlinear evolution that is described by the Hartree equation. This is an Egorov-type theorem for many-body quantum systems with random interactions. The analysis is then extended to derive the Gross–Pitaevskii equation with random interactions.