TY - JOUR
T1 - Long-time dynamics of the modulational instability of deep water waves
AU - Ablowitz, M. J.
AU - Hammack, J.
AU - Henderson, D.
AU - Schober, C. M.
N1 - Funding Information:
This work was partially supported by the AFOSR USAF, Grant No. F49620-00-1-0031 and the NSF, Grant Nos. DMS-0070772, DMS-9803567 and DMS-9972210.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2001/5/15
Y1 - 2001/5/15
N2 - In this paper, we experimentally and theoretically examine the long-time evolution of modulated periodic 1D Stokes waves which are described, to leading-order, by the nonlinear Schrödinger (NLS) equation. The laboratory and numerical experiments indicate that under suitable conditions modulated periodic wave trains evolve chaotically. A Floquet spectral decomposition of the laboratory data at sampled times shows that the waveform exhibits bifurcations across standing wave states to left- and right-going modulated traveling waves. Numerical experiments using a higher-order nonlinear Schrödinger equation (HONLS) are consistent with the laboratory experiments and support the conjecture that for periodic boundary conditions the long-time evolution of modulated wave trains is chaotic. Further, the numerical experiments indicate that the macroscopic features of the evolution can be described by the HONLS equation. Ultimately, these laboratory experiments provide a physical realization of the chaotic behavior previously established analytically for perturbed NLS systems.
AB - In this paper, we experimentally and theoretically examine the long-time evolution of modulated periodic 1D Stokes waves which are described, to leading-order, by the nonlinear Schrödinger (NLS) equation. The laboratory and numerical experiments indicate that under suitable conditions modulated periodic wave trains evolve chaotically. A Floquet spectral decomposition of the laboratory data at sampled times shows that the waveform exhibits bifurcations across standing wave states to left- and right-going modulated traveling waves. Numerical experiments using a higher-order nonlinear Schrödinger equation (HONLS) are consistent with the laboratory experiments and support the conjecture that for periodic boundary conditions the long-time evolution of modulated wave trains is chaotic. Further, the numerical experiments indicate that the macroscopic features of the evolution can be described by the HONLS equation. Ultimately, these laboratory experiments provide a physical realization of the chaotic behavior previously established analytically for perturbed NLS systems.
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U2 - 10.1016/S0167-2789(01)00183-X
DO - 10.1016/S0167-2789(01)00183-X
M3 - Article
AN - SCOPUS:18244424907
VL - 152-153
SP - 416
EP - 433
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
ER -