Extended criterion for the modulation instability

The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity ffalse(ufalse)=12u2 of the Korteweg–de Vries equation and the full linear dispersion relation Ωfalse(kfalse)=ktanhk of unidirectional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux function f(u) and a general linear dispersion relation Ω(k). Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit, simple criteria in terms of general f(u) and Ω(k) establishing the strict hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill–Whitham criterion for modulational instability.

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“…Note that the NLS equation () is valid when

n false(\bar{u}, k false) {normalΩ}^{''} false(k false) \neq 0

. At an inflection point of the dispersion, a higher‐order NLS approximation is required where an alternative criteria for MI can be obtained 39 …”

Section: Applications To MImentioning

confidence: 99%

“…At an inflection point of the dispersion, a higher-order NLS approximation is required where an alternative criteria for MI can be obtained. 39…”

Section: Application ( ) ( )mentioning

confidence: 99%

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

et al. 2021

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“…• The above scenario of a rogue wave formation as a result of focusing dispersion packets and a small number of solitons (one or two) is apparently typical for the equations in which the interaction of solitons leads to their repulsion in space (as in the classical Korteweg-de Vries equation). It is closely related to the phenomenon of modulation instability, [7,12,[29][30][31], or rather its absence in systems with repulsive solitons, [20,34]. Such equations are called defocusing, and for them, in our opinion, the dispersive focusing mechanism is fundamental for the occurrence of rogue waves.…”

Section: Discussionmentioning

confidence: 99%

“…Let us now summarize the role of the obtained solutions in the context of rogue waves. By inverting solution given by the equation ( 23) with the amplitude (30), one can guarantee the dispersion focusing of these packets into a rogue wave of a large Gaussian shape. Differences in the dispersion laws for different media appear in the shape of focusing wave packets.…”

Section: Bell-shaped Rogue Wavementioning

confidence: 99%

Dispersive focusing in fractional Korteweg–de Vries-type equations

Tobisch

Pelinovsky

2020

J. Phys. A: Math. Theor.

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In this paper we study dispersive enhancement of a wave train in systems described by the fractional Korteweg–de Vries-type equations of the form u t + α n u n u x + β m ( D m { u } ) x = 0 , D m { u } = − | k | m u ( k ) where the operator D m {u} is written in the Fourier space, α n , β m are arbitrary constants and n, m being rational numbers (positive or negative). Using both approximate and exact solutions of these wave equations we describe constructively the process of dispersive focusing. It is based on a time-reversing approach with the expected rogue wave chosen as the initial condition for a solution of these equations. We demonstrate the qualitative difference in the shape of the focused wavetrains for various n and m. Our results can be used for prediction of the rogue wave appearance arising in many types of weakly nonlinear and weakly dispersive wave systems in physical context.

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“…Note that the NLS equation ( 50) is valid when n(u, k)Ω (k) = 0. At an inflection point of the dispersion, a higher order NLS approximation is required where an alternative criteria for MI can be obtained [3].…”

Section: Nonlinear Schrödinger Equation Approximationmentioning

confidence: 99%

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

Binswanger,

Hoefer,

Ilan

et al. 2020

Preprint

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The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity f (u) = 1 2 u 2 of the Korteweg-de Vries equation and the full linear dispersion relation Ω(k) = √ k tanh k of uni-directional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux function f (u) and a general linear dispersion relation Ω(k). Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit, simple criteria in terms of general f (u) and Ω(k) establishing the strict hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill-Whitham criterion for modulational instability.

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Extended criterion for the modulation instability

Cited by 10 publications

References 54 publications

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

Dispersive focusing in fractional Korteweg–de Vries-type equations

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

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