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

We consider the logarithmic Schr{"o}dinger 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 $H^1\cap \mathcal{F}(H^1)$. 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 $t\to \mathrm{\infty }$) 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.