Self-similar Solutions of the Cubic Wave Equation

Details Self-similar Solutions of the Cubic Wave Equation Self-similar Solutions of the Cubic Wave Equation

2009-05-23

arxiv.org/abs/0905.3834v1

Nonlinearity 23:225-236,2010

Mathematics

Analysis of PDEs

14 pages, 1 figure

Scientific paper

10.1088/0951-7715/23/2/002

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 labeled by an integer index $n$ which counts the number of oscillations of the solution. The linearized operator around the $n$-th 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.

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