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I have obtained theoretically a solution to a nonlinear Schrödinger-type equation in dimension two. I also proved that is not constant. Now, I wonder if it depends on two variables and not only in one, that is, it is not one-dimensional. So my question is if there exist some mathematical techniques in order to stablish that?

I don't know any, so possible ways of proving it will be appreciated! Thanks in advance.

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    $\begingroup$ What do you know about the solution? That would help point you in a useful direction. For instance, this is automatically true if it lies in some $L^p$ space for $p<\infty$ (and is nontrivial). Also, does the equation have any symmetries (scaling, rotation, etc.)? $\endgroup$ Nov 15 '19 at 19:13
  • $\begingroup$ It depends a bit on what you meant by that "it is not one-dimensional". In particular, do you consider a radially symmetric solution to be one dimensional or not? $\endgroup$ Nov 15 '19 at 19:16
  • $\begingroup$ can't you say that any function $f(x,y)$ of two variables $x,y$ is also a function of a single variable $z=f(x,y)$? Perhaps you want to specify what relationship between $x$ and $y$ you would allow? $\endgroup$ Nov 15 '19 at 19:37
  • $\begingroup$ The equation is invariante under multiplication by constant of modulus one. The solution is smooth, by standard elliptic theory. What I understand for not be one dimensional is that $x_1$and $x_2$ both appears in the expression. $\endgroup$ Nov 16 '19 at 11:23
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    $\begingroup$ Try to understand the solutions that only depend on $x_1$ (this is likely easy, as the PDE becomes an ODE if we assume no dependence on $x_2$). Then once you understand them, try to see if they might be the solution you obtained. For example, if you obtained your solution as an energy minimizer, compute their energy and see if you can find any solution with smaller energy. Same for $x_2$. $\endgroup$ Nov 17 '19 at 5:01