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Title: On the derivation of the wave kinetic equation for NLS
Abstract A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $$T_{\mathrm {kin}} \gg 1$$ and in a limiting regime where the size L of the domain goes to infinity and the strength $$\alpha $$ of the nonlinearity goes to $$0$$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $$T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$$ and $$\alpha $$ is related to the conserved mass $$\lambda $$ of the solution via $$\alpha =\lambda ^2 L^{-d}$$ . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $$(\alpha , L)$$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $$\alpha $$ approaches $$0$$ like $$L^{-\varepsilon +}$$ or like $$L^{-1-\frac {\varepsilon }{2}+}$$ (for arbitrary small $$\varepsilon $$ ), we exhibit the wave kinetic equation up to time scales $$O(T_{\mathrm {kin}}L^{-\varepsilon })$$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $$T_*\ll T_{\mathrm {kin}}$$ and identify specific interactions that become very large for times beyond $$T_*$$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $$T_*$$ toward $$T_{\mathrm {kin}}$$ for such scaling laws seems to require new methods and ideas.  more » « less
Award ID(s):
1900251
PAR ID:
10349989
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Forum of Mathematics, Pi
Volume:
9
ISSN:
2050-5086
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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