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Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, and I have just uploaded to the arXiv the paper “Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation“, which we have submitted to Inventiones Mathematicae. This paper concerns the numerically observed phenomenon of weak turbulence for the periodic defocusing cubic non-linear Schrödinger equation

$-i u_t + \Delta u = |u|^2 u$ (1)

in two spatial dimensions, thus u is a function from ${\Bbb R} \times {\Bbb T}^2$ to ${\Bbb C}$.  This equation has three important conserved quantities: the mass

$M(u) = M(u(t)) := \int_{{\Bbb T}^2} |u(t,x)|^2\ dx$

the momentum

$\vec p(u) = \vec p(u(t)) = \int_{{\Bbb T}^2} \hbox{Im}( \nabla u(t,x) \overline{u(t,x)} )\ dx$

and the energy

$E(u) = E(u(t)) := \int_{{\Bbb T}^2} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{4} |u(t,x)|^4\ dx$.

(These conservation laws, incidentally, are related to the basic symmetries of phase rotation, spatial translation, and time translation, via Noether’s theorem.) Using these conservation laws and some standard PDE technology (specifically, some Strichartz estimates for the periodic Schrödinger equation), one can establish global wellposedness for the initial value problem for this equation in (say) the smooth category; thus for every smooth $u_0: {\Bbb T}^2 \to {\Bbb C}$ there is a unique global smooth solution $u: {\Bbb R} \times {\Bbb T}^2 \to {\Bbb C}$ to (1) with initial data $u(0,x) = u_0(x)$, whose mass, momentum, and energy remain constant for all time.

However, the mass, momentum, and energy only control three of the infinitely many degrees of freedom available to a function on the torus, and so the above result does not fully describe the dynamics of solutions over time.  In particular, the three conserved quantities inhibit, but do not fully prevent the possibility of a low-to-high frequency cascade, in which the mass, momentum, and energy of the solution remain conserved, but shift to increasingly higher frequencies (or equivalently, to finer spatial scales) as time goes to infinity.  This phenomenon has been observed numerically, and is sometimes referred to as weak turbulence (in contrast to strong turbulence, which is similar but happens within a finite time span rather than asymptotically).

To illustrate how this can happen, let us normalise the torus as ${\Bbb T}^2 = ({\Bbb R}/2\pi {\Bbb Z})^2$.  A simple example of a frequency cascade would be a scenario in which solution $u(t,x) = u(t,x_1,x_2)$ starts off at a low frequency at time zero, e.g. $u(0,x) = A e^{i x_1}$ for some constant amplitude A, and ends up at a high frequency at a later time T, e.g. $u(T,x) = A e^{i N x_1}$ for some large frequency N. This scenario is consistent with conservation of mass, but not conservation of energy or momentum and thus does not actually occur for solutions to (1).  A more complicated example would be a solution supported on two low frequencies at time zero, e.g. $u(0,x) = A e^{ix_1} + A e^{-ix_1}$, and ends up at two high frequencies later, e.g. $u(T,x) = A e^{iNx_1} + A e^{-iNx_1}$.  This scenario is consistent with conservation of mass and momentum, but not energy.  Finally, consider the scenario which starts off at $u(0,x) = A e^{i Nx_1} + A e^{iNx_2}$ and ends up at $u(T,x) = A + A e^{i(N x_1 + N x_2)}$.  This scenario is consistent with all three conservation laws, and exhibits a mild example of a low-to-high frequency cascade, in which the solution starts off at frequency N and ends up with half of its mass at the slightly higher frequency $\sqrt{2} N$, with the other half of its mass at the zero frequency.  More generally, given four frequencies $n_1, n_2, n_3, n_4 \in {\Bbb Z}^2$ which form the four vertices of a rectangle in order, one can concoct a similar scenario, compatible with all conservation laws, in which the solution starts off at frequencies $n_1, n_3$ and propagates to frequencies $n_2, n_4$.

One way to measure a frequency cascade quantitatively is to use the Sobolev norms $H^s({\Bbb T}^2)$ for $s > 1$; roughly speaking, a low-to-high frequency cascade occurs precisely when these Sobolev norms get large.  (Note that mass and energy conservation ensure that the $H^s({\Bbb T}^2)$ norms stay bounded for $0 \leq s \leq 1$.)  For instance, in the cascade from $u(0,x) = A e^{i Nx_1} + A e^{iNx_2}$ to $u(T,x) = A + A e^{i(N x_1 + N x_2)}$, the $H^s({\Bbb T}^2)$ norm is roughly $2^{1/2} A N^s$ at time zero and $2^{s/2} A N^s$ at time T, leading to a slight increase in that norm for $s > 1$.  Numerical evidence then suggests the following

Conjecture. (Weak turbulence) There exist smooth solutions $u(t,x)$ to (1) such that $\|u(t)\|_{H^s({\Bbb T}^2)}$ goes to infinity as $t \to \infty$ for any $s > 1$.

We were not able to establish this conjecture, but we have the following partial result (“weak weak turbulence”, if you will):

Theorem. Given any $\varepsilon > 0, K > 0, s > 1$, there exists a smooth solution $u(t,x)$ to (1) such that $\|u(0)\|_{H^s({\Bbb T}^2)} \leq \epsilon$ and $\|u(T)\|_{H^s({\Bbb T}^2)} > K$ for some time T.

This is in marked contrast to (1) in one spatial dimension ${\Bbb T}$, which is completely integrable and has an infinite number of conservation laws beyond the mass, energy, and momentum which serve to keep all $H^s({\Bbb T}^2)$ norms bounded in time.  It is also in contrast to the linear Schrödinger equation, in which all Sobolev norms are preserved, and to the non-periodic analogue of (1), which is conjectured to disperse to a linear solution (i.e. to scatter) from any finite mass data (see this earlier post for the current status of that conjecture).  Thus our theorem can be viewed as evidence that the 2D periodic cubic NLS does not behave at all like a completely integrable system or a linear solution, even for small data.  (An earlier result of Kuksin gives (in our notation) the weaker result that the ratio $\|u(T)\|_{H^s({\Bbb T}^2)} / \|u(0)\|_{H^s({\Bbb T}^2)}$ can be made arbitrarily large when $s > 1$, thus showing that large initial data can exhibit movement to higher frequencies; the point of our paper is that we can achieve the same for arbitrarily small data.) Intuitively, the problem is that the torus is compact and so there is no place for the solution to disperse its mass; instead, it must continually interact nonlinearly with itself, which is what eventually causes the weak turbulence.

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