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 the dynamical stabilization of nonlinear wave packets and the persistence of large undulatory events in hydrodynamics and other nonlinear dispersive media.
Highlights
Modulational instability (MI) is an ubiquitous phenomenon for wave packets propagating in a weakly nonlinear medium [1]
If the envelope of the wave-packet is narrowbanded, the nonlinear stage of the evolution can be modeled by means of the universal nonlinear Schrödinger equation (NLSE)
We study the nonlinear stage of evolution of modulational instability in surface water waves over a water body of gradually increasing depth
Summary
Modulational instability (MI) is an ubiquitous phenomenon for wave packets propagating in a weakly nonlinear medium [1]. The family of Akhmediev breather (AB) is the prototype of the nonlinear evolution of MI: in the time-like NLSE, an initially slightly modulated time-periodic train of pulses reaches its peak value at a given point in space, as a result of the exponential sideband growth, as is followed by the recovery of the initial state known as Fermi-Pasta-Ulam recurrence [8]. Because of this characteristic feature, i.e., extreme waves appearing from nowhere and suddenly disappearing [9], it is a candidate solution for the explanation of rogue waves and other nonlinear systems.
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