Stabilization of uni-directional water-wave trains over an uneven bottom
arXiv:2002.12923 · doi:10.1007/s11071-020-05819-9
Abstract
We study the evolution of nonlinear surface gravity water-wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrödinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework is the theory of autoresonance in nonlinear systems and leads to a quasi-frozen state. Three main stages are identified: amplification from small sideband amplitudes, separatrix crossing, and adiabatic conversion to orbits oscillating around an elliptic fixed point. Analytical estimates on the three stages are obtained from the low-dimensional approximation and validated by NLSE simulations. Our result will contribute to understand dynamical stabilization of nonlinear wave packets and the persistence of large undulatory events in hydrodynamics and other nonlinear dispersive media.
arXiv comment11 pages, 8 figures
References in corpus (6)
- Rogue waves and analogies in optics and oceanography
- Triggering rogue waves in opposing currents
- Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions
- Efficient, Broadband and Robust Frequency Conversion by Fully Nonlinear Adiabatic Three Wave Mixing
- Nonlinear stage of Benjamin-Feir instability in forced/damped deep water waves
- Recurrence in the high-order nonlinear Schrödinger equation: a low dimensional analysis