Numerical study on Fermi–Pasta–Ulam–Tsingou problem for 1D shallow-water waves

•We simulated long time evolutions of gravity waves with the Boussinesq model.•Evident recurrence is observed with appropriate initial conditions.•The transition from regular to chaotic motions is observed.•We conjecture this related with the ratio between dispersion and nonlinearity.

Beginning with the first mode as the initial condition, long-term evolutions of gravity waves in shallow water are simulated based on the full nonlinear Boussinesq model. Evident recurrence is observed in long basins with appropriate initial amplitudes. Equipartition can be obtained in the case of a long basin, large initial amplitude or a long evolution time. Well-defined solitary waves are present during the recurrence stage and completely lost at the equipartition stage. The transition from regular to chaotic motion is conjectured to be related to the ratio of the dispersion and nonlinearity of the initial condition. For short basins with small initial amplitudes, nonlinearity is much smaller than dispersion, energy transfer is weak, and no recurrence can be observed. If dispersion and nonlinearity are chosen to be the same order in the initial condition, recurrence clearly emerges. However, if nonlinearity is chosen to be larger than dispersion, recurrence is absent and the system reaches equipartition rapidly.