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1. Large-Data Equicontinuity for the Derivative NLS

   https://doi.org/10.1093/imrn/rnab374  

   Harrop-Griffiths, Benjamin; Killip, Rowan; Vişan, Monica (January 2022, International Mathematics Research Notices)

   Abstract We consider the derivative nonlinear Schrödinger equation in one spatial dimension, which is known to be completely integrable. We prove that the orbits of $L^2$ bounded and equicontinuous sets of initial data remain bounded and equicontinuous, not only under this flow, but also under the entire hierarchy. This allows us to remove the small-data restriction from prior conservation laws and global well-posedness results. 

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2. The unconditional uniqueness for the energy-supercritical NLS

   https://doi.org/10.1007/s40818-022-00130-9  

   Chen, Xuwen; Shen, Shunlin; Zhang, Zhifei (December 2022, Annals of PDE)
3. On the kinetic wave turbulence description for NLS

   https://doi.org/10.1090/qam/1554  

   Buckmaster, T.; Germain, P.; Hani, Z.; Shatah, J. (April 2020, Quarterly of Applied Mathematics)
4. On the derivation of the wave kinetic equation for NLS

   https://doi.org/10.1017/fmp.2021.6  

   Deng, Yu; Hani, Zaher (January 2021, Forum of Mathematics, Pi)

   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. 

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5. Scattering for the Cubic-Quintic NLS: Crossing the Virial Threshold

   https://doi.org/10.1137/20M1381824  

   Killip, Rowan; Murphy, Jason; Visan, Monica (January 2021, SIAM Journal on Mathematical Analysis)