In this paper we consider a nonlinear Schrödinger equation (NLS) with random coefficients, in a regime of separation of scales corresponding to diffusion approximation. The primary goal of this paper is to propose and study an efficient numerical scheme in this framework. We use a pseudo-spectral splitting scheme and we establish the order of the global error. In particular we show that we can take an integration step larger than the smallest scale of the problem, here the correlation length of the random medium. We study the asymptotic behavior of the numerical solution in the diffusion approximation regime.

We study the convergence of the slow (or "essential") components of singularly perturbed stochastic differential systems to solutions of lower dimensional stochastic systems (the "effective", or "coarse" dynamics). We prove strong, mean-square convergence in systems where both fast and slow components are driven by noise, with full coupling between fast and slow components. We analyze a class of "projective integration" methods, which consist of a hybridization between a standard solver for the slow components, and short runs for the fast dynamics, which are used to estimate the effect that the fast components have on the slow ones. We obtain explicit bounds for the discrepancy between the results of the projective integration method and the slow components of the original system.

In this paper we discuss the existence of traveling wave solutions for a hyperbolic-elliptic system of partial differential equations. The geometric theory of singular perturbations is employed.

We consider initial-boundary value problems for a generalized time-dependent Schrödinger equation in $1D$ on the semi-axis and in $2D$ on a semi-bounded strip. For Crank-Nicolson finite-difference schemes, we suggest an alternative coupling to approximate transparent boundary conditions and present a condition ensuring unconditional stability. In the case of discrete transparent boundary conditions, we revisit the statement and the proof of stability together with the derivation of the conditions.

We discuss stochastic representations of advection diffusion equations with variable diffusivity, stochastic integrals of motion and generalized relative entropies.

The problem of a diatomic gas made up by atoms and molecules undergoing reactions of dissociation and recombination through a transition state is addressed. The gas is diffusing in a scattering and dissociating background medium. Extended Boltzmann-like kinetic equations for the relevant distribution functions and exact balance equations for mass, momentum, and energy are derived and discussed. It is shown that, in the asymptotic limit of small Knudsen numbers, a suitable scaling leads to a consistent formal derivation of hydrodynamic equations of reaction-diffusion type for number densities of atoms and molecules.

We study the Whitham equations for the fifth order KdV equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solution of the Whitham equations when the initial values are given by a step function. We classify the step-like initial data into eight different types. We construct self-similar solutions for each type.