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.

As mentioned earlier, the three conservation laws of mass, momentum, and energy work to inhibit the low-to-high frequency cascade necessary to prove the theorem.  For instance, one can easily show that any cascade involving three or fewer frequencies is morally incompatible with the three conservation laws (since three equations in three unknowns gives basically just one possible value for the amplitudes at the three frequencies).  The simplest cascade is the four-wave model discussed earlier, in which four frequencies at the corners of a rectangle transfer their mass, energy, and momentum from two diagonally opposite corners to the other two diagonally opposite corners.   One can in fact rigorously show that such four-wave cascades (plus negligible noise terms) can actually occur in (1) in the high frequency limit, by using perturbation theory to eliminate all frequencies except the four frequencies of the rectangle (thus reducing the PDE to an ODE), using symmetry to assume that diagonally opposite frequencies have the same Fourier coefficients, and then using the conservation laws to reduce the dimension of the ODE down to a scalar first-order ODE that can be solved more or less exactly.  The dynamics of the resulting solution (for the model ODE, rather than the PDE that it approximates) resembles an “Arnold whisker” in the theory of Arnold diffusion – it asymptotically converges to a “two-wave” periodic orbit based on one diagonal of the rectangle as $t \to -\infty$, and to a two-wave periodic orbit on the other diagonal as $t \to +\infty$, lying in the unstable manifold of the former orbit and stable manifold of the latter.

This four-wave solution can be constructed from arbitrarily small initial data, but it only gets us a little way towards our theorem, as it only bumps up the $H^s$ norm by a factor of about $2^{(s-1)/2}$.  To establish the theorem, we needed to chain a lot of these four-wave solutions together, in the spirit of the classic paper of Arnold that established the possibility of diffusion from one invariant torus in a certain Hamiltonian system to another.  In terms of the approximating ODE (which was now more complicated, although still manageable thanks to symmetry reductions), the main task was to concoct a solution that started out near one invariant torus, bounced towards another torus, did a ricochet off of that to the next torus, and so on until one reached a torus extremely far away from the original torus.  The best way we could describe this procedure succinctly was by citing a classic Super Bowl commercial for McDonald’s featuring Michael Jordan and Larry Bird:

(This citation, of course, appears in the bibliography to the paper, like any other reference.)

Actually showing such a solution exists turned out to require a very delicate stability and control theory analysis, in which stable, neutral, and unstable directions were treated in very different ways.  As one might imagine, the initial data has to be extremely special in order to obtain a multi-ricochet solution.

Once such a solution for the approximating ODE is constructed, one still has to embed it somehow back into the PDE (1).  Here one faces a combinatorial and number-theoretic problem of being able to place a large number of frequencies in the lattice ${\Bbb Z}^2$, such that certain specific quadruples of these frequencies form rectangles, and all other quadruples do not.  [The frequencies turn out to be organised in a series of “generations”, in which a pair of “parent” frequencies from each generation pass on their mass to a pair of “child” frequencies to the next generation, with one child inheriting virtually all of the energy, and the other child inheriting close to nothing; the children in that generation then pair up and become parents themselves, and so forth.  In the first generation, all frequencies initially have the same energy (having comparable magnitudes and amplitudes), but by the end of the process, almost all the energy in the last generation is concentrated in a single, extremely large, frequency.]  As is well known, the problem of constructing rectangles in the lattice is closely related to that of constructing Pythagorean triples $a^2+b^2=c^2$, and so our construction ultimately relies on some basic facts about those triples (basically, we need that the unit vectors $(a/c, b/c)$ arising from Pythagorean triples are dense on the unit circle).

There are still some further directions to pursue here.  Most obviously, the conjecture stated above remains open; the perturbation theory we use stops working after a long, but finite, amount of time, which has prevented us so far from resolving the conjecture, though the task may not be completely hopeless with current technology.  The other thing is that the solutions we construct for our Theorem, much like the shots in the above video, are extremely special, and do not really indicate what is going on for generic data.  The only non-trivial result we have for such data is that they do not scatter to a linear solution even after quotienting out by phase rotations, unless they are one of the explicit “one-wave” solutions $u(t,x) = A e^{i(|n|^2 t + n \cdot x - A^2 t + \theta)}$.  This is done by a simple compactness argument involving, of all things, Liouville’s theorem that harmonic functions of polynomial growth are in fact polynomial.