WKB analysis of the logarithmic nonlinear Schrödinger equation in an analytic framework

Abstract

We are interested in a WKB analysis of the logarithmic nonlinear Schrödinger equation with “Riemann-like” variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed. In particular, our framework is not only compatible with the Gross–Pitaevskii equation with logarithmic nonlinearity, but also allows initial data (and solutions) which can converge to [Formula: see text] at infinity.

Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation

“…In the second part of [8], the authors have shown the existence of solutions to the isothermal Euler-Korteweg system (which corresponds to the case µ = 0 in the system (1.1)), but again their proof is strongly based on the link with the logarithmic Schrödinger equation, which does not seem to be useful in our case due to the ill-posed nonlinear Langevin potential 1 2i log (ψ/ψ * ). In a more regular framework, note that in the recent work [15], the author shows the local existence in time of solutions to the Euler-Korteweg system satisfying some analytic regularity, which may be extended to the Euler-Langevin-Korteweg equations.…”