We consider the numerical computation of finite-genus solutions of the Korteweg-de Vries equation when the genus is large. Our method applies both to the initial-value problem when spectral data can be computed and to dressing scenarios when spectral data is specified arbitrarily. In order to compute large genus solutions, we employ a weighted Chebyshev basis to solve an associated singular integral equation. We also extend previous work to compute period matrices and the Abel map when the genus is large, maintaining numerical stability. We demonstrate our method on four different classes of solutions. Specifically, we demonstrate dispersive quantization for “box” initial data and demonstrate how a large genus limit can be taken to produce a new class of potentials.
Time evolution initial data obtained by taking a large-$g$ limit of $g$ gaps placed in the inteval $[0,2]$ under some distribution. Observe the rogue character.
Time evolution initial data obtained by taking a large-$g$ limit of $g$ gaps placed in the inteval $[0,2]$ under some distribution.
$u(x, 1.03 \pi)$ solving KdV equation in the form $u_{t}-$ $u u_{x}+u_{x x x}=0$ with box initial data. This plot shows dispersive quantization. The solution appears to be piecewise smooth at rational-times- $\pi$ times and fractal otherwise. For the KdV equation, this was first observed by Chen and Olver. These plots are produced using a genus $g=300$ approximation. More details concerning the computation of this solution can be found in Section 5.4.
![Time evolution initial data obtained by taking a large-$g$ limit of $g$ gaps placed in the inteval $[0,2]$ under some distribution. Observe the rogue character.](/~bilman/publication/kdv-periodic-nist/rogue_hi.gif)
![Time evolution initial data obtained by taking a large-$g$ limit of $g$ gaps placed in the inteval $[0,2]$ under some distribution.](/~bilman/publication/kdv-periodic-nist/contour-g-30_huf230fc3a3efaac4af565f9027c4237c8_598952_44a6dfdbc26bafcd081b058b8da42a70.webp)
