A high-order nonlinear envelope equation for gravity waves in finite-depth water

Slunyaev, Alexey

doi:10.1134/1.2149072

Cited by 132 publications

(155 citation statements)

References 13 publications

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“…Indeed, it is known that deviations with respect to NLS dynamics are expected in an experiment, when the steepness of the carrier and the breather amplitude amplification are significant [8,34,35]. In fact, higher-order evolution equations such as the modified NLS [36][37][38], also known as Dysthe equations, or others [39][40][41] provide an accurate correction and a better prediction of the amplified wave field. These discrepancies with respect to NLS dynamics have an influence in the decay dynamics of the breathers after reaching the saturation point and a recurrence, referred to as Fermi-Pasta Ulam recurrence, is expected to occur [10,31,[42][43][44].…”

Section: Discussionmentioning

confidence: 99%

The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time

Chabchoub

Grimshaw

2016

Fluids

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Abstract:The nonlinear Schrödinger equation (NLS) is a canonical evolution equation, which describes the dynamics of weakly nonlinear wave packets in time and space in a wide range of physical media, such as nonlinear optics, cold gases, plasmas and hydrodynamics. Due to its integrability, the NLS provides families of exact solutions describing the dynamics of localised structures which can be observed experimentally in applicable nonlinear and dispersive media of interest. Depending on the co-ordinate of wave propagation, it is known that the NLS can be either expressed as a space-or time-evolution equation. Here, we discuss and examine in detail the limitation of the first-order asymptotic equivalence between these forms of the water wave NLS. In particular, we show that the the equivalence fails for specific periodic solutions. We will also emphasise the impact of the studies on application in geophysics and ocean engineering. We expect the results to stimulate similar studies for higher-order weakly nonlinear evolution equations and motivate numerical as well as experimental studies in nonlinear dispersive media.

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Section: Discussionmentioning

confidence: 99%

The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time

Chabchoub

Grimshaw

2016

Fluids

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“…This fact has prompted several workers to develop higher-order nonlinear terms in the evolution equation applicable to both flat bottom (e.g. Johnson, 1977;Kakutani and Michihiro, 1983;Slunyaev, 2005) and to sloping bottom (Grimshaw and Annenkov, 2011). They suggest a rescaling for kh ≈ 1.363 in order to reestablish a leading order balance between nonlinearity and dispersion.…”

Section: Introductionmentioning

confidence: 99%

Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom

Zeng

Trulsen

2012

Nat. Hazards Earth Syst. Sci.

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Abstract.We consider the effect of slowly varying depth on the values of skewness and kurtosis of weakly nonlinear irregular waves propagating from deeper to shallower water. It is known that the equilibrium value of kurtosis decreases with decreasing depth for waves propagating on constant depth. Waves propagating over a sloping bottom must continually adjust toward a new equilibrium state. We demonstrate that weakly nonlinear waves may need a considerable horizontal propagation distance in order to adjust to a new shallower environment, therefore the kurtosis can be notably different from the equilibrium value for each corresponding depth both on top of and beyond a bottom slope. A change of depth can provoke a wake-like spatially non-uniform distribution of kurtosis on the lee side of the slope. As an application, we anticipate that the probability of freak waves on or near the edge of the continental shelf may exhibit a rather complicated spatial structure for wave fields entering from deep sea.

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“…Similar situations also occur in other contexts and, thus, corresponding versions of Eq. (2) have been derived and used, e.g., in nonlinear metamaterials [4], but also in water waves in finite depth [5][6][7]. Moreover, in the context of optics, and for relatively long propagation distances, higher-order nonlinear dissipative effects, such as the SRS effect, of strength σ R > 0, are also important [3].…”

Section: Motivation and Presentation Of The Modelmentioning

confidence: 99%

“…A prime example is the nonlinear Schrödinger (NLS) equation which constitutes one of the universal nonlinear evolution equations, with applications ranging from deep water waves to optics [2]. Remarkable phenomena are also exhibited by its higher-order variants, emerging in a diverse spectrum of applications, such as nonlinear optics [3], nonlinear metamaterials [4], and water waves in finite depth [5][6][7]. On the other hand, dissipative variants of NLS models incorporating gain and loss have also been used in optics [8], e.g., in the physics of mode-locked lasers [9,10] (see also the relevant works [11,12]) and polariton superfluids [13] -see, e.g., Ref.…”

Section: Introductionmentioning

confidence: 99%

Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

et al. 2016

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The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects, gives rise to rich dynamics: this extends from Poincaré-Bendixson-type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections, or space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that the third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (i.e., even in the absence of the other higher-order effects) for the existence of the periodic, quasi-periodic and chaotic spatiotemporal structures. Suitable lowdimensional phase space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

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A high-order nonlinear envelope equation for gravity waves in finite-depth water

Cited by 132 publications

References 13 publications

The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time

The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time

Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom

Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

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