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

Abstract : 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 $t \rightarrow \infty$) 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.
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https://hal.archives-ouvertes.fr/hal-02498540
Contributor : Guillaume Ferriere <>
Submitted on : Wednesday, March 4, 2020 - 3:19:25 PM
Last modification on : Thursday, March 5, 2020 - 7:01:13 PM
Long-term archiving on: : Friday, June 5, 2020 - 2:54:16 PM

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  • HAL Id : hal-02498540, version 1
  • ARXIV : 2003.02571

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Guillaume Ferriere. Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrodinger Equation. 2020. ⟨hal-02498540⟩

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