Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation

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 $H^1\cap F\left(H^1\right)$. 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 $t\to \infty$) is faster than exponential. We also prove a rigidity result on these constructed multi-gaussians and multi-solitons, showing that they are the only ones with such a convergence.