Consider the non-linear Schrödinger equation (NLS) in the following version: \begin{align} u_{xx} + i u_t +2|u|^2 u = 0, \text{ where }u:\mathbb{R}\times [0,\infty) \to \mathbb{R}. \end{align} We shall assume that $u$ is twice differentiable wrt $x$ and once wrt $t$, i.e. $u$ is a 'strong' solution to the NLS-eq. Smoothness is not required. I think that weak solutions wouldn't help much but it would still be interesting to know.

Are there solutions to the non-linear Schrödinger equation, such that $\text{supp}_{x \in \mathbb{R}} u(x,t)$ is compact in $\mathbb{R}$ for an open non-empty interval in $t$? I suppose the answer to this is negative.

More generally I am looking for a solution $u(x,t)$ such that $\text{supp}_{(x,t) \in \mathbb{R}\times [0,\infty)}u(x,t)$ is open-non-empty or that at least vanishes on an open non-empty interval in $t$ or $x$.

I read about some statements about this but what I found covers only statements like $\lim u = 0$ or different versions of the NLS-eq.

Thanks in advance! (Sorry for bad tags)


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