No abstract

We consider the logarithmic Schrödinger equation (logNLS) in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson -a time-independent Gaussian function -is an orbitally stable solution. In this paper, we construct multi-solitons (or multi-Gaussons) for logNLS, with estimates in H 1 ∩ F (H 1 ). We also construct solutions to logNLS behaving (in L 2 ) like a sum of N Gaussian solutions with different speeds (which we call multi-gaussian). In both cases, the convergence (as t → ∞) is faster than exponential. We also prove a rigidity result on these multi-gaussians and multi-solitons, showing that they are the only ones with such a convergence.

No abstract

We analyze dynamical properties of the logarithmic Schrödinger 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 asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.

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.