The focusing logarithmic Schr\"odinger equation: analysis of breathers and nonlinear superposition

Abstract

We consider the logarithmic Schr\"odinger equation 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 the general case in dimension $d = 1$, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the sum of several standing Gaussian functions far from each other, the solution remains close (in $L^2$) to the sum of the corresponding Gaussian solutions for a long time, in square of the distance between the Gaussian functions.