On a class of nonlinear Schr6dinger equations
This paper concerns the existence of standing wave solutions of nonlinear Schr6dinger equations. Making a standing wave ansatz reduces the problem to that of studying the semiIinear elliptic equation: • u + b(x)u = f ( x , u), x e •". (*) The function f is assumed to be "superlinear". A special case is the power nonlinearity f ( x , z) = Izl s !z where 1 sufficient conditions for the existence of nontrivial solutions u e W m ( ~ n) are established. (Received: June 18, 1991)
We study non-equilibrium velocity fluctuations in a model for the sedimentation of non-Brownian particles experiencing long-range hydrodynamic interactions. The complex behavior of these fluctuations, the outcome of the collective dynamics of the particles, exhibits many of the features observed in sedimentation experiments. In addition, our model predicts a final relaxation to an anisotropic (hydrodynamic) diffusive state that could be observed in experiments performed over longer time ranges.