Timeline for Does there exist always an $L^2$ threshold below (or above) which a traveling waves of a nonlinear dispersive PDE cannot exist?
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| Feb 13, 2023 at 13:23 | comment | added | Niser | Sorry, I will not mark the answer as accepted, because my initial question (which probably I didn't formulate clearly) was to know if there exists a PDE admitting traveling waves without lower and upper bound in their $L^2$ norm... | |
| Feb 13, 2023 at 13:07 | comment | added | Carlo Beenakker | yes, there is a lower bound but no upper bound. | |
| Feb 13, 2023 at 13:02 | comment | added | Niser | Thank you. I still have a question. In the case of Gross-Pitaevskii, If I am not mistaken, the $L^2$ norm of the profile $U$ cannot $<\!<1$, so there exists a lower bound>0 where the $L^2$ norm of the traveling waves of Gross-Pitaevskii cannot go below this bound. | |
| Feb 13, 2023 at 11:52 | comment | added | Carlo Beenakker | perhaps I misunderstood your question, but first of all, for a travelling wave we need a finite system for a finite norm (since by definition, $u_0(x)$ does not go to zero for large $x$). We would typically place the system on a torus, and impose periodic boundary conditions on a length $L$; then we ask whether, for a fixed $L$, the norm of $u_0$ can become arbitrarily large; this norm will be of order $r_0 L$, so for a large norm we need a large $r_0$; this will fail if $c^\ast$ vanishes for large $r_0$, but for the case of the Gross-Pitaevskii equation it will not. | |
| Feb 13, 2023 at 11:26 | comment | added | Niser | My following question might be silly: Suppose that $c^*(r_0)$ is bounded. I don't understand why if $c<c^*(r_0)$ then this implies the $L^2$-norm of $U$ cannot be arbitrarily large. | |
| Feb 13, 2023 at 9:52 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |