| Abstract: |
| The study of long-time behavior in dispersive equations typically relies on energy estimates that require localization assumptions on initial data. In this talk, we explore large-time dynamics without imposing any such localization conditions. We discuss a variant of the Gross-Pitaevskii equation on the real line which is used to model superfluidity properties of Bose-Einstein condensates. This model can be viewed as intermediate between the classical Gross-Pitaevskii equation, by keeping leading order dispersion, and the complex Ginzburg-Landau equation, by adding typical balance terms. We identify a relaxation mechanism close to the wave functions $re^{i\omega t + ikx}$, which arises through the interaction between diffusive- and dispersive effects. This relaxation process allows to show nonlinear stability of the wave functions in $C_b^k$ against perturbations from $C_b^{k+1+\sigma}$ for all $\sigma>0$. In contrast, we also establish finite time blow-up solutions in $C_b^k$ whenever we include initial $C_b^{k+1}$-perturbations: this shows that our stability result is sharp concerning global existence. To the best of our knowledge, this work provides a first analysis of long-time dynamics in dispersive models without any localization assumptions on initial data, and may motivate further research directions. Specifically, we expect that the blow-up vs. stability phenomenon also appears for the Lugiato-Lefever equation. |
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