Pohozaev identity and related things

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

$(1) \quad \label{intro-eq} \begin{cases} i\partial_t \psi_1 - \frac{r}{2} \Delta \psi_1 + \overline{\psi}_1\psi_2 = 0, \\ i\partial_t \psi_2 -\beta\psi_2 - i\vec{V}\cdot \nabla\psi_2 - \frac{\alpha}{2} \Delta \psi_2 + |\beta|\psi_1^2 = 0, \end{cases}$

where $\psi_1, \psi_2: \mathbb{R}\times\mathbb{R}^d \to \mathbb{C}$ are wave functions with the dimension $d\ge 1$, $\vec{V} \in \mathbb{R}^d$ is a constant vector, and $r, \alpha, \beta$ 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

$\begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} (t,x) = \begin{pmatrix} e^{i\omega t} Q_1\\ \\ e^{2i\omega t} Q_2 \end{pmatrix} (x) , \; \quad \omega\in \mathbb{R}^* ,$

where $Q_i(x): \mathbb{R}^d \to \mathbb{C}$, $i=1,2$. The system corresponds to this solitary wave is called the stationary state system (SSS).

Namely, we have established the following lemma.

Lemma. If $Q = (Q_1, Q_2) , Q_1, Q_2 \in H^1 (\mathbb{R}^d)$, is a weak solution of the  (SSS), then

$(2a) \quad -\omega \int |Q_1|^2$  $+ \dfrac{r}{2} \int\limits | \nabla Q_1|^2 + Re \int \overline{Q}_1^2 Q_2 = 0$

$(2b) \quad -(2\omega + \beta) \int |Q_2|^2 + Im \int \vec{V}\cdot \nabla Q_2 \overline{Q}_2 + \dfrac{\alpha}{2} \int |\nabla Q_2|^2 + |\beta| Re \int Q_1^2 \overline{Q}_2 = 0.$

The identities $(2a), (2b)$ 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 $\vec{V} = 0,\; 2\le d \le 5$ and $\lambda \ne 0$ be given, then there exists at least one (weak) solitary wave solution of (1).