A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water

Trulsen, Karsten; Dysthe, Kristian B.

doi:10.1016/s0165-2125(96)00020-0

Cited by 279 publications

(252 citation statements)

References 15 publications

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“…There is no bound on either U or V if δ = 0. This is consistent with the work of Dysthe (1979) and Trulsen & Dysthe (1996) in which the plane-wave solutions of MNLS and BMNLS are shown to be linearly unstable.…”

Section: Discussionsupporting

confidence: 89%

“…In each plot, the regions between the curves correspond to Λ's that have positive real part. The t = 0 (zero dissipation) case of these (or equivalent) plots were obtained by Dysthe (1979), Trulsen & Dysthe (1996) and Trulsen et al (2000).…”

Section: Approximate Regions Of Growthmentioning

confidence: 99%

“…Zakharov (1968) derived the onedimensional cubic nonlinear Schrödinger equation and established that all of its plane-wave solutions are linearly unstable. Dysthe (1979) and Trulsen & Dysthe (1996), respectively, established that all plane-wave solutions of the MNLS and BMNLS equations are linearly unstable.…”

Section: Introductionmentioning

confidence: 99%

“…In order to create a model that accurately approximates (1) for a wider bandwidth of modulations, Trulsen & Dysthe (1996) generalized the asymptotic analysis used in the derivation of (2) and (3). In their analysis, they allowed for modulations of O( 1/2 ), δk/k = O( 1/2 ); and deep water, (kh) −1 = O( 1/2 ).…”

Section: Introductionmentioning

confidence: 99%

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Stability of plane waves on deep water with dissipation

Canney

Carter

2007

Mathematics and Computers in Simulation

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The Benjamin-Feir modulational instability effects the evolution of perturbed planewave solutions of the cubic nonlinear Schrödinger equation (NLS), the modified NLS, and the band-modified NLS. Recent work demonstrates that the BenjaminFeir instability in NLS is "stabilized" when a linear term representing dissipation is added. In this paper, we add a linear term representing dissipation to the modified NLS and band-modified NLS equations and establish that the plane-wave solutions of these equations are linearly stable. Although the plane-wave solutions are stable, some perturbations grow for a finite period of time. We analytically bound this growth and present approximate time-dependent regions of wave-number space that correspond to perturbations that have increasing amplitudes.

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Section: Discussionsupporting

confidence: 89%

Section: Approximate Regions Of Growthmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability of plane waves on deep water with dissipation

Canney

Carter

2007

Mathematics and Computers in Simulation

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“…As expected, the agreement becomes worse when nonlinearity is significantly increased during the breather compression process. An experimentally observed asymmetry can be explained in terms of higher-order NLS equations with odd terms such as the modified NLSE [51] or similar [52,53] equations that account for the formation of asymmetric wave packets.…”

Section: Appendix B: Water Wave Tankmentioning

confidence: 99%

Superregular Breathers in Optics and Hydrodynamics: Omnipresent Modulation Instability beyond Simple Periodicity

et al. 2015

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Since the 1960s, the Benjamin-Feir (or modulation) instability (MI) has been considered as the selfmodulation of the continuous "envelope waves" with respect to small periodic perturbations that precedes the emergence of highly localized wave structures. Nowadays, the universal nature of MI is established through numerous observations in physics. However, even now, 50 years later, more practical but complex forms of this old physical phenomenon at the frontier of nonlinear wave theory have still not been revealed (i.e., when perturbations beyond simple harmonic are involved). Here, we report the evidence of the broadest class of creation and annihilation dynamics of MI, also called superregular breathers. Observations are done in two different branches of wave physics, namely, in optics and hydrodynamics. Based on the common framework of the nonlinear Schrödinger equation, this multidisciplinary approach proves universality and reversibility of nonlinear wave formations from localized perturbations for drastically different spatial and temporal scales.

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Observations of the shape and group dynamics of rogue waves

Gemmrich

Thomson

2017

Geophysical Research Letters

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Surface elevation records from two locations in the northeast Pacific are used to examine rogue waves and the relationship to wave groups. Three hundred individual rogue waves with heights greater than 2.2 times the significant wave height are found in analyzing >2 × 106 wave groups. In contrast to recent nonlinear modeling results, we do not find that rogue waves occur at the front of wave groups. There is a tendency for steep waves to occur at the front of a group, but these are not the largest waves of the group and do not meet the rogue wave criterion. Rogue waves are most commonly located in the center of the group, but their height ratio to the neighboring crest is greater than in the average wave group. Assessing group dynamics in terms of spectral width suggests that random superposition of nonlinear waves is sufficient to explain the observations of individual rogue waves.

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A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water

Cited by 279 publications

References 15 publications

Stability of plane waves on deep water with dissipation

Stability of plane waves on deep water with dissipation

Superregular Breathers in Optics and Hydrodynamics: Omnipresent Modulation Instability beyond Simple Periodicity

Observations of the shape and group dynamics of rogue waves

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