Spectrally accurate fully discrete schemes for some nonlocal and nonlinear integrable PDEs via explicit formulas
Schémas spectraux discrétisés en espace et en temps pour certaines EDPs intégrables nonlocales et nonlinéaires via des formules explicites
Résumé
We construct fully-discrete schemes for the Benjamin-Ono, Calogero-Sutherland DNLS, and cubic Szegő equations on the torus, which are exact in time with spectral accuracy in space. We prove spectral convergence for the first two equations, of order K^{-s+1} for initial data in H^s (T), with an error constant depending linearly on the final time instead of exponentially. These schemes are based on explicit formulas, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales. These schemes open doors for the understanding of the long-time dynamics of integrable equations.
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