Universal dynamics for the defocusing logarithmic Schrödinger equation

Abstract

We consider the nonlinear Schrödinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data, and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in working in hydrodynamical variables to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker-Planck operator.

Regularized numerical methods for the logarithmic Schrödinger equation

“…4.7 Case x). This is consistent with the fact that the convergence to a universal Gaussian profile (leaving out the oscillatory aspects, which are not described in general) is very slow, as established in [17] (logarithmic convergence in time).…”

“…4.8 shows the errors of e(t) := u ε (·, t) − u(·, t) measured in different norms, which again evidence the accuracy of the STSP scheme. In addition, the rate of dispersion of the Gaussians could indeed be estimated for large time dynamics in [17].…”

“…In the case λ < 0, stationary solutions are available, called Gaussons (as noticed in [14]; see below), which turn out to be orbitally stable, but not stable in the usual sense of Lyapunov [19,22] (see also [4] for another proof, and [44,46] for other particular solutions). In the case λ > 0, every solution is (global and) dispersive with a non-standard rate (a logarithmic perturbation of the Schrödinger rate), and after a time-dependent rescaling related to this dispersion, the large time behavior of the renormalized solution exhibits a universal Gaussian profile -a phenomenon which is rather unique in the context of Hamiltonian dispersive PDEs, see [17].…”

“…In the case λ > 0, the large time behavior of r does not depend on its initial data, r(t) ∼ 2t √ λα 0 ln t as t → ∞ (see [17]). The general dynamics is rather well understood in the case λ > 0 (see [17]), but, aside from the explicit Gaussian solutions, the dynamical properties in the case λ < 0 constitute a vast open problem: is the dynamics comparable to the one, say, of the cubic nonlinear Schrödinger equation, or is it drastically different? The numerical simulations presented in this paper tend to suggest that the dynamics associated to (1.1) is quite rich, and reveals phenomena absent (or at least unknown) in the case of the cubic nonlinear Schrödinger equation.…”

Density dependent replicator-mutator models in directed evolution

“…In the context of evolutionary genetics, families of Gaussian solutions for nonlinear and nonlocal equations can be found in [6], [1,2]. In a different context involving logarithmic non-linearities, we also refer to [5], [11] for the Schrödinger equation and to [3] for the Heat equation. Proof.…”