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This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{o}dinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.
This article is devoted to the construction of new numerical methods for the semiclassical Schrodinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter. This allows t
In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schr{o}dinger operators $(-Delta+m^{2})^{s}$ with $sin(0
We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{o}dinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on
We analyze dynamical properties of the logarithmic Schr{o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asy
The blowup is studied for the nonlinear Schr{o}dinger equation $iu_{t}+Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $pge 1+frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)<0$ bl