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]).


Stationary Solution High Dimension Space Dimension Localise Solution Imaginary Axis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I.Barashenkov, V.Makhankov, Phys. lett. A 128 (1988) 52–56MathSciNetADSGoogle Scholar
  2. [2]
    I.Barashenkov, A.Gocheva, V. Makhankov, I. Puzynin Phys. D 34 (1989) 240–254MathSciNetGoogle Scholar
  3. [3]
    H.Berestycki, P.L.Lions, Arch.rach.mech.anal. 82 (1983) 313–345MathSciNetMATHGoogle Scholar
  4. [4]
    H.Berestycki, T.Gallouët, O. Kavian, C. R. acad. sc PARIS 297 (1983) Serie I 307–310MATHGoogle Scholar
  5. [5]
    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

Personalised recommendations