On the propagation of high regularity for the logarithmic Schr{\"o}dinger equation

TL;DR

This paper studies how high regularity of solutions to the 1D logarithmic Schrödinger equation behaves over time, revealing conditions for both loss and preservation of smoothness, supported by theoretical analysis and numerical simulations.

Contribution

It provides new insights into the conditions under which high regularity is lost or preserved for solutions, including the role of initial data symmetry and boundary conditions.

Findings

Solutions with broad odd initial data experience instantaneous blow-up in high Sobolev norms.

H^3 regularity is preserved under specific symmetry and boundary conditions.

Numerical simulations support the theoretical results.

Abstract

We investigate both the instantaneous loss and the persistence of high regularity for the one-dimensional logarithmic Schr{\"o}dinger equation in symmetric domains under various boundary conditions. We show that for a broad class of odd initial data, the $H^s$-norm of solutions exhibits instantaneous blow-up for all $s > 7/2 $. Conversely, we establish that $H^3$-regularity is preserved for solutions that are odd with first-order cancellation, non-vanishing behavior away from the origin and Neumann boundary conditions on symmetric bounded domains. These theoretical results are further supported and illustrated by numerical simulations.