Self-similar solutions of the cubic wave equation

We prove that the focusing cubic wave equation in three spatial dimensions has a countable family of self-similar solutions which are smooth inside the past light cone of the singularity. These solutions are labelled by an integer index n which counts the number of oscillations of the solution. The...

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Bibliographic Details
Published in:Nonlinearity 2010-02, Vol.23 (2), p.225-236
Main Authors: Bizoń, P, Breitenlohner, P, Maison, D, Wasserman, A
Format: Article
Language:English
Subjects:
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematical analysis
Mathematical methods in physics
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Other topics in mathematical methods in physics
Partial differential equations
Physics
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:We prove that the focusing cubic wave equation in three spatial dimensions has a countable family of self-similar solutions which are smooth inside the past light cone of the singularity. These solutions are labelled by an integer index n which counts the number of oscillations of the solution. The linearized operator around the nth solution is shown to have n + 1 negative eigenvalues (one of which corresponds to the gauge mode) which implies that all n > 0 solutions are unstable. It is also shown that all n > 0 solutions have a singularity outside the past light cone which casts doubt on whether these solutions may participate in the Cauchy evolution, even for non-generic initial data.
ISSN:0951-7715
1361-6544
DOI:10.1088/0951-7715/23/2/002