Modulational instability in nonlinear nonlocal equations of regularized long wave type

Hur, Vera Mikyoung; Pandey, Ashish Kumar

doi:10.1016/j.physd.2016.03.005

Cited by 20 publications

(33 citation statements)

References 13 publications

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“…Here, the proof follows along the same line as the arguments in Refs. and but with suitable modifications to accommodate product spaces. Throughout the subsection, κ and b 1 , b 2 are fixed and suppressed for simplicity of notation.…”

Section: Sufficiently Small and Periodic Traveling Wavesmentioning

confidence: 99%

“…However, for

ξ > 0

and small for λ in the vicinity of the origin in

double-struckC

, we may take a spectral perturbation approach in Refs. and , for instance, to address it analytically. This is the subject of investigation here.…”

Section: Modulational Instabilitymentioning

confidence: 99%

“…For instance, i 1 , i 2 , and i 3 appear in an index formula for and , which shares the dispersion relation in common with ‐; see Ref. for details. Moreover, i 2 appears in Ref.…”

Section: Modulational Instabilitymentioning

confidence: 99%

“…For , for instance, i 4 is replaced by

2 i_{3} + c_{ww}^{2} false(2 κ false) i_{2}

; see Ref. for details. To compare, a modulational instability index for the Whitham equation (see Refs.…”

Section: Modulational Instabilitymentioning

confidence: 99%

“…The proof follows along the same line as those in Refs. and , making a lengthy and complicated, but explicit, spectral perturbation for the associated linearized operator. For the zero Floquet exponent, we distinguish four eigenvalues at the origin and calculate small‐amplitude expansions of the associated eigenfunctions.…”

Section: Introductionmentioning

confidence: 99%

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Modulational instability in a full‐dispersion shallow water model

Hur

Pandey

2018

Stud Appl Math

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We propose a shallow water model that combines the dispersion relation of water waves and Boussinesq equations, and that extends the Whitham equation to permit bidirectional propagation. We show that its sufficiently small and periodic traveling wave is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value, like the Benjamin‐Feir instability of a Stokes wave. We verify that the associated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability to the leading order in the amplitude parameter. We discuss the effects of surface tension. The results agree with those from a formal asymptotic expansion and a numerical computation for the physical problem.

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Section: Sufficiently Small and Periodic Traveling Wavesmentioning

confidence: 99%

“…However, for

ξ > 0

and small for λ in the vicinity of the origin in

double-struckC

, we may take a spectral perturbation approach in Refs. and , for instance, to address it analytically. This is the subject of investigation here.…”

Section: Modulational Instabilitymentioning

confidence: 99%

“…For instance, i 1 , i 2 , and i 3 appear in an index formula for and , which shares the dispersion relation in common with ‐; see Ref. for details. Moreover, i 2 appears in Ref.…”

Section: Modulational Instabilitymentioning

confidence: 99%

“…For , for instance, i 4 is replaced by

2 i_{3} + c_{ww}^{2} false(2 κ false) i_{2}

; see Ref. for details. To compare, a modulational instability index for the Whitham equation (see Refs.…”

Section: Modulational Instabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modulational instability in a full‐dispersion shallow water model

Hur

Pandey

2018

Stud Appl Math

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Proof of Modulational Instability of Stokes Waves in Deep Water

Nguyen

Strauss

2022

Comm Pure Appl Math

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It is proven that small-amplitude steady periodic water waves with infinite depth are unstable with respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. It has been proven rigorously only in the case of finite depth. We provide a completely different and self-contained approach to prove the spectral modulational instability for water waves in both the finite and infinite depth cases.

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Nonlinear Modulational Instability of Dispersive PDE Models

Jin

Liao

Lin

2018

Arch Rational Mech Anal

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We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schrödinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs; Second, for KDV type equations the loss of derivative in the nonlinear term is overcome in two complementary cases: (1) for smooth nonlinear terms and general dispersive operators, we construct higher order approximation solutions and then use energy type estimates; (2) for nonlinear terms of low regularity, with some additional assumption on the dispersive operator, we use a bootstrap argument to overcome the loss of derivative.Proof. By assumption, there exists λ (k) with Re λ (k) > 0 such thatwhere J k , L k are defined in (1.8). It is easy to see that J k is a skew-adjoint operator and L k is a self-adjoint operator. Taking the real part of the L 2 inner product of (2.1) with L k v k , we get the following "conservation law":Re λv k , L k v k = Re J k L k v k , L k v k = 0.

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Modulational instability in nonlinear nonlocal equations of regularized long wave type

Cited by 20 publications

References 13 publications

Modulational instability in a full‐dispersion shallow water model

Modulational instability in a full‐dispersion shallow water model

Proof of Modulational Instability of Stokes Waves in Deep Water

Nonlinear Modulational Instability of Dispersive PDE Models

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