Orlando Lopes
Abstract:
In this article we consider the normalized one-dimensional
three-wave interaction model
Solitary waves for this model are solutions of the form
where
and
are positive frequencies, and
,
are real-valued functions that satisfy the ODE system
For the case
, we prove existence,
uniqueness and stability of solitary waves corresponding
to positive solutions
that tend to zero as x tends to infinity.
The full model has more parameters, and the case we consider
corresponds to the exact phase matching. However, as we will see,
even in the simpler case, a formal proof of stability depends on
a nontrivial spectral analysis of the linearized operator.
This is so because the spectral analysis depends on some calculations
on a full neighborhood of the parameter
and the solution
is not known explicitly.
Submitted January 10, 2013. Published June 30, 2014.
Math Subject Classifications: 34A34.
Key Words: Dispersive equations; variational methods; stability.
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Orlando Lopes IMEUSP- Rua do Matao, 1010, Caixa postal 66281 CEP: 05315-970, Sao Paulo, SP, Brazil email: olopes@ime.usp.br |
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