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Existence of multi-solitons for the focusing logarithmic non-linear Schrödinger equation. (English) Zbl 1464.35319

Summary: 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 \(H^1\cap\mathcal{F}(H^1)\). 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 \(t\to\infty)\) 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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35C08 Soliton solutions
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