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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