On the Instability of the Static Soliton-like “Bubbles”

  • A. de Bouard
Conference paper
Part of the Springer Series in Nonlinear Dynamics book series (SSNONLINEAR)


The following ψ 3-ψ 5 nonlinear Schrödinger equation:
$$i \frac{{\partial \varphi }}{{\partial t}} + \Delta \varphi - {\alpha _1}\varphi + {\alpha _3}|\varphi {|^2}\varphi - {\alpha _5}|\varphi {|^4}\varphi = 0 $$
possesses, in some domain of the parameters α i , some localised solutions propagating with velocity ν which can be interpreted as rarefaction bubbles in a Bose condensate (see [1]). These localised solutions satisfy the ” boundary conditions”:
$$ \varphi \left( {x,t} \right) \to {\sqrt {{\rho _0}e} ^{i\left( {\omega t + \mu \left( {\frac{x}{{|x|}}} \right)} \right)}}when|x| \to + \infty $$
where ρ 0 is a positive constant, μ depends on the velocity ν and is equal to zero in the particular case of stationary solutions, i.e when ν = 0. Such solutions of equation (1) where found explicitly in space dimension one, and numerically in higher dimension (see [1]).




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    A. de Bouard, PhD thesis (1991), Universite Paris-Sud, Otsay France. and article in preparationGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • A. de Bouard
    • 1
  1. 1.Laboratoire d’Analyse NumériqueUniversité Paris SudF-Orsay CedexFrance

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