In this post, we are going to discuss some issues related to a nice identity, Pohozaev identity, and give some results I did in my master thesis.
Consider the system of nonlinear Schrodinger equations
where are wave functions with the dimension , is a constant vector, and are given real parameters.
Systems of this type arise in some physical models, such as nonlinear optics (e.g, see [Menyuk, Schiek and Torner (1994); Solitary waves due to χ(2) : χ(2) cascading, J. Opt. Soc. Am. B, 11(12), 2434–2443]).
We study the solitary wave, i.e, waves which are localized pulses that propagate without change of shape, of this system. This wave is actually a particular solution of (1) of the form
where , . The system corresponds to this solitary wave is called the stationary state system (SSS).
Namely, we have established the following lemma.
Lemma. If , is a weak solution of the (SSS), then
The identities are called Pohozaev identities. By these identities, we could find some non-existence results of solitary waves of (1).
For the existence, by applying variational method and some compact embeddings of Strauss-Lions type, we obtain the following result.
Theorem. Assume that and be given, then there exists at least one (weak) solitary wave solution of (1).