Fractal solutions of dispersive partial differential equations on the torus

Burak Erdoğan and I just put “Fractal solutions of dispersive partial differential equations on the torus” on the arXiv. We study cancellation in exponential sums and how this leads to bounds for the fractal dimension of solutions to certain PDE, the ultimate “square root cancellation” implying exact knowledge of the dimension.

In this post, we consider the case of Schrödinger’s equation with initial data {\chi := \chi_{[0 , \pi]}} for simplicity. Here,

\displaystyle if_t + f_{xx} = 0 , \ \ t \in \mathbb{R}, \ \ x \in \mathbb{R} / 2 \pi \mathbb{Z} \ \ , \ \ f(0,x) = \chi(x) ,

and the solution is given by

\displaystyle f(t,x) = \sum_{n \in \mathbb{Z}} \widehat{\chi}(n) e^{i t n^2 + i n x}.

Note the solution is periodic in both {x} and {t}.

For a line {\mathcal{L} \subset \left( \mathbb{R} / 2 \pi \mathbb{Z} \right)^2}, we are interested in the fractal dimension of the real and imaginary parts of the graph of {f |_{\mathcal{L}} (t,x)}. This dimension must lie in the interval {[1,2]}.

An important case to consider is {t = a/q}, which leads to so–called quantization and is related to the Talbot effect. In this case, it was shown by Berry and Klein that

\displaystyle f(t,x) = \sum_{j=0}^{q-1} c_j \chi_{[0 , 1/q] + j/q} (x), \ \ \ \ \ (1)

for some {c_j \in \mathbb{C}} that are Gauss sums. Thus {f} is a linear combination of at most {q} intervals and has fractal dimension 1. See page 4 of this paper of Chen and Olver for some pictures of this phenomenon.

The story is entirely different for irrational {t}. To see why, observe for a sequence {\frac{a_n}{q_n} \rightarrow t}, the functions {f(\frac{a_n}{q_n} , x)} as given in (1) increase in complexity as {q_n} increases.

To make this precise, we use the theory of Besov spaces and show that bounds for the {L^p} norm of

\displaystyle H_N(t,x) : = \sum_{N \leq n < 2N} e^{itn^2 + i nx}, \ \ (t,x) \in \mathcal{L},

imply fractal dimension bounds for the real and imaginary parts of {f_{\mathcal{L}}(t,x)}.

Remark: The starting point of the theory of of Besov spaces that we use can be illustrated by the following basic facts from analysis:

  • The graph of a real valued {C^{\gamma}} function has fractal dimension {\leq 2 - \gamma},
  • For {g \in C^{\gamma}}, one has {\widehat{g}(n) \ll n^{-\gamma}}.

Thus we see that both fractal dimension and Fourier decay are related to the smoothness of {g}. In our situation, we have more information than that of the conclusion of the second bullet, that is we know that

\displaystyle \sum_{N \leq n < 2N} \widehat{g}(n),

exhibits cancellation and so we are able to use more involved theory to roughly reverse the implication in the second bullet as well as provide lower bounds. ♠

Let us first consider the case of horizontal lines in space–time, that is {t} fixed. Then by orthogonality, Weyl’s inequality, and the divisor bound, one has for almost every {t},

\displaystyle ||H_N||_{L_x^2} = N^{1/2} , \ ||H_N||_{L_x^{\infty}} \ll N^{1/2 + \epsilon}, \ ||H_N||_{L_x^4} \ll N^{1/2 + \epsilon}.

By a previous blog post, this implies

\displaystyle ||H_N||_{L_x^1} \gg N^{1/2 - \epsilon}.

One can use this to show that the fractal dimension of {f_{\mathcal{L}}(t,x)} is equal to {3/2} for almost every horizontal line, recovering a theorem of Rodnianski. In one part of our paper, we adapt the above strategy to show the following.

Theorem (oblique): The fractal dimension of the real and imaginary parts of the solution to (1) restricted to almost every

\displaystyle \mathcal{L} = \{(t,x) : a t + bx = c\},

is in the interval

\displaystyle [7/4 , 19/10]. ♠

Note that the lower bound 7/4 is bigger than the 3/2 for horizontal lines. To show the upper bound in Theorem (oblique), we study exponential sums such as

\displaystyle \sup_x |\sum_{n \leq N} e^{i(c+x)n^2 + i nx}| , \ \ c \in \mathbb{R} / 2 \pi \mathbb{Z}.

In this case, square root cancellation would imply the fractal dimension is exactly {7/4}.

In our paper, we study a variety of dispersive PDE, for instance the Airy, KDV, non–linear Schrödinger, free water boundary, Boussinesq, and gravity–capillary wave equation. To handle non–linear equations, we use smoothing estimates, which roughly state that the solution to a nonlinear PDE is the linear part plus a smoother perturbation.

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