What is the connection between these two PDEs?

Asked Feb 28 '13 at 10:12

Active Mar 04 '13 at 07:37

Viewed 131 times

Does anyone see a or know of some connection between solutions to the two Gross-Pitaevskii-Equations

$i \partial_t \phi = \Delta \phi +\phi(1-|\phi|^2) ~~~~~~~~~~~ (1)$

and

$i \partial_t \phi = \Delta \phi -\phi(1-|\phi|^2) ~~~~~~~~~~~ (2)$

Are there any known results as "If $\phi$ is a solution to (1) then ... is a solution to (2)". I can't seem to find any.

Our viewpoint is $\phi : \mathbb{R} \times \mathbb{R}^N \to \mathbb{C}$, $(t,x) \mapsto \phi(t,x)$.

edited Mar 04 '13 at 07:37

asked Feb 28 '13 at 10:12

mjb

3

Consider a simplified situation: stationary solution ($\partial_t\phi = 0$), small $\phi$ (that is, $|\phi|^3$ is negligible). Then the equations are $\Delta \phi+ \phi = 0$ and $\Delta \phi - \phi = 0$, which are very different animals. The latter satisfies the maximum principle and has unique solution for any Dirichlet data. The former does neither. – Mar 06 '13 at 20:18
Care to post this as an answer? – mjb Aug 22 '13 at 09:38

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