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From the three-body problem to KAM

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I believe that many physicists may have come across the KAM theory in the physics literature. For me, I have read about KAM many times in papers discussing the discrete regime of wave turbulence at low nonlinearity. Nevertheless, I never had a deep understanding of this concept except knowing that the system behaves "like a linear one" or "like an integrable one". Recently, in a project with one of my students, we found a breather solution in a derivative NLS, for which we suspected that the oscillation part is supported on a KAM tori. This motivated me to seek further understanding about the KAM theory. However, most of the mathematical materials that I found online are not immediately accessible, as many of them are written in terms of symplectic manifold which requires further efforts to understand. This is until finally I found a book " The KAM story: a friendly introduction to the content, history, and significance of classical Kolmogorov-Arnold-Moser Theo...
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Useful inequalities in estimation

Here I would like to collect a bunch of useful inequalities in estimation. This list is by no means complete, but I will keep adding new things when appropriate. (1) Cauchy-Schwarz inequality : Let $(X,\langle\cdot,\cdot\rangle)$ be an inner product space, and $||x||=\sqrt{\langle x,x \rangle}$ (think of $L^2$ norm), then for all $x,y\in X$, $|\langle x,y \rangle| \leq ||x||\cdot||y||$. This inequality has an obvious geometric interpretation for $\mathbb{R}^2$ (or $\mathbb{R}^3$) vector space, but let's prove it for the general case. Proof: If $x=0$, the general inequality obviously holds. If $x\neq 0$, let $\hat{x}\equiv \frac{x}{||x||}$, $y_{\parallel}\equiv \langle \hat{x},y \rangle \hat{x}$, $y_{\perp}\equiv y-y_{\parallel}$, then $0\leq ||y_{\perp}||^2 = ||y-y_{\parallel}||^2 = ||y-\langle \hat{x},y \rangle\hat{x}||^2 = \langle  y-\langle \hat{x},y \rangle\hat{x}, y-\langle \hat{x},y \rangle\hat{x} \rangle = \langle  y-\langle \hat{x},y \rangle\hat{x}, y \rangle - \langle...
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Local well-posedness problem

In working with my collaborators to justify the diffusion equation resulted from wave turbulence systems dominated by non-local interactions, we need to prove the local well-posedness of some wave kinetic equation. The system under our consideration is a three-dimensional Majda-McLaughlin-Tabak  ( MMT) equation, with its wave kinetic equation given in the form of      $\frac{\partial n_k}{\partial t} = 4\pi {\large{\int}}_{\mathbb{R}^9} |k|^{2\beta} |k_1|^{2\beta} |k_2|^{2\beta} |k_3|^{2\beta} (n_1n_2n_3 + n_2n_3n_k - n_2n_1n_k - n_3n_1n_k)$     $\delta(k+k_1-k_2-k_3) \delta(\omega+\omega_1-\omega_2-\omega_3) dk_1dk_2dk_3$, with $k\in \mathbb{R}^3$ and $\omega=|k|^\alpha$. This equation describes the evolution of wave action spectrum as a result of four-wave interactions. I will not detail more about the physical meaning of the equation and notations used therein, because that will distract readers too much from the main theme of the post. Anyway, one ...
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The Start

This is Yulin Pan, an assistant professor in the department of Naval Architecture and Marine Engineering at the University of Michigan. Throughout my education I was in engineering departments, receiving all my degrees in engineering. I do have a minor degree in mathematics from MIT, but to achieve that I only needed to take three courses from the math department there. And for me, I took all three courses from the very applied side of mathematics. In summary, my college education of mathematics is highly limited to engineering mathematics, numerical methods and some applied asymptotic analysis to nonlinear problems.  In spite of the lack of training in rigorous mathematics, I do keep a keen interest in mathematics over my research career. During my Ph.D., I got interested in the field of wave turbulence, based on which I later developed my Ph.D. thesis. The field of wave turbulence is quite technical in terms of the derivation, which is usually considered as part of theoretical cl...
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