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We consider the logarithmic Schr\"odinger equation in the focusing regime.For this equation, Gaussian initial data remains Gaussian. In particular, theGausson-a time-independent Gaussian function-is an orbitally stable solution.In the general case in dimension $d = 1$, the solution with Gaussian initialdata is periodic, and we compute some approximations of the period in the caseof small and large oscillations, showing that the period can be as large aswanted for the latter. The main result of this article is a principle ofnonlinear superposition: starting from an initial data made of the sum ofseveral standing Gaussian functions far from each other, the solution remainsclose (in $L^2$) to the sum of the corresponding Gaussian solutions for a longtime, in square of the distance between the Gaussian functions.

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