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The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition
Discrete and Continuous Dynamical Systems ( IF 1.1 ) Pub Date : 2020-07-27 , DOI: 10.3934/dcds.2020277
Guillaume Ferriere ,

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.

中文翻译:

聚焦对数Schrödinger方程:呼吸和非线性叠加分析

我们考虑聚焦系统中的对数Schrödinger方程。对于该方程式,高斯初始数据仍为高斯。特别是,与时间无关的高斯函数-高斯(Gausson)是轨道稳定的解决方案。在维数为$ d = 1 $的一般情况下,具有高斯初始数据的解是周期性的,并且在小振荡和大振荡的情况下,我们计算出周期的一些近似值,这表明周期可以与所需的大小一样大。后者。本文的主要结果是非线性叠加的原理:从初始数据开始,该数据由多个相互远离的站立式高斯函数的和构成,其解保持接近(以L ^ 2为单位)。长期以来,高斯函数以高斯函数之间的距离的平方为单位。
更新日期:2020-07-27
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