The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition

Guillaume Ferriere

doi:10.3934/dcds.2020277

Article Contents

2020, Volume 40, Issue 11: 6247-6274. Doi: 10.3934/dcds.2020277

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The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition

Guillaume Ferriere

IMAG, Univ. Montpellier, CNRS, Montpellier, France

Received: November 2019

Revised: May 2020

Published online: November 2020

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Abstract

We consider the logarithmic Schrödinger equation 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 the general case in dimension $ d = 1 $, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the sum of several standing Gaussian functions far from each other, the solution remains close (in $ L^2 $) to the sum of the corresponding Gaussian solutions for a long time, in square of the distance between the Gaussian functions.

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Mathematics Subject Classification: Primary: 35Q55, 35B10, 35B15, 35B40; Secondary: 34C25, 34C27.

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Figure 1. Phase portraits. At the left-hand side, the arrows give the direction of the flow; the green and magenta lines are the horizontal and vertical isoclines respectively; the red lines are some trajectories, which fits the level set of $ H(p, q) $. At the right-hand side, a part of some trajectories are drawn along with the isoclines again

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Figure 2. Plot of some trajectories of $ \tau_\gamma $ in the phase space for 5 real $ \gamma $ between $ 10 $ and $ 50 $

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Figure 3. Plot of the breather ($ \lambda = 0.5 $) with initial data $ \exp \Bigl( \frac{1}{2} - \delta x^2 \Bigr) $ with $ \delta^{-1} = 2 \times 35^2 $. The first "peak" is at $ t = 43.86 $

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