Numerical finite-gap integration of the Zabusky-Kruskal problem

The Zabusky-Kruskal problem (ZKP) is well-known ever since Zabusky and Kruskal performed their famous numerical experiment. It has become a benchmark for testing numerical schemes. However, the solution of the ZKP has no analytical expression and its long-time numerical integration is notoriously difficult. Now, after decades of effort both to develop new algorithms and to improve their performance, the best existing algorithm approaches machine accuracy for moderate evolution time. Here we solve the ZKP with high accuracy using the method of numerical finite-gap integration. The windfalls of the high accuracy are abundant: Many new analytical properties are discovered from the numerical results. The algorithms presented here can be regarded as a paradigm for high-accuracy numerical integration of large-genus algebro-geometric solutions.