Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation

Choudhury, Roy S.; Ivanov, Rossen I.; Liu, Yue

doi:10.1016/j.chaos.2006.03.057

Cited by 18 publications

(11 citation statements)

References 27 publications

(34 reference statements)

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“…Note also, that there is no second Hamiltonian formulation for the Ostrovsky equation, compatible with the one given above, i.e. the equation is not bi-Hamiltonian -indeed (41) is not completely integrable for γ = 0, [7].…”

Section: Conservation Laws and Perturbed Soliton Equationsmentioning

confidence: 98%

“…The quantities R ± (k) = b(±k)/a(k) are known as reflection coefficients (to the right with superscript (+) and to the left with superscript (−) respectively). It is sufficient to know R ± (k) only on the half line k > 0, since from (7): R ± (−k) =R ± (k) and also (7) |a(k)…”

Section: Direct Scattering Transform and Scattering Datamentioning

confidence: 99%

“…where ∂ −1 is an operator such that (∂ −1 u) x ≡ u, in general not uniquely determined. [83,7,79,85].…”

Section: Perturbations Of the Equations Of The Kdv Hierarchymentioning

confidence: 99%

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Generalised Fourier transform and perturbations to soliton equations

Grahovski

Ivanov²

2009

Discrete &Amp; Continuous Dynamical Systems - B

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A brief survey of the theory of soliton perturbations is presented. The focus is on the usefulness of the so-called Generalised Fourier Transform (GFT). This is a method that involves expansions over the complete basis of "squared solutions" of the spectral problem, associated to the soliton equation. The Inverse Scattering Transform for the corresponding hierarchy of soliton equations can be viewed as a GFT where the expansions of the solutions have generalised Fourier coefficients given by the scattering data.The GFT provides a natural setting for the analysis of small perturbations to an integrable equation: starting from a purely soliton solution one can 'modify' the soliton parameters such as to incorporate the changes caused by the perturbation.As illustrative examples the perturbed equations of the KdV hierarchy, in particular the Ostrovsky equation, followed by the perturbation theory for the Camassa-Holm hierarchy are presented.

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Section: Conservation Laws and Perturbed Soliton Equationsmentioning

confidence: 98%

Section: Direct Scattering Transform and Scattering Datamentioning

confidence: 99%

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Generalised Fourier transform and perturbations to soliton equations

Grahovski

Ivanov²

2009

Discrete &Amp; Continuous Dynamical Systems - B

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“…While both the SPE the mKdV equation are integrable [1,25] so associated with them are an infinite number of conserved quantities, the arguments in [10,5] show that the α term in the RSPE destroys the integrable structure. Nevertheless there are conserved quantities associated with (1.2).…”

Section: Conserved Quantities and Hamiltonian Formulationmentioning

confidence: 99%

Solitary Waves of the Regularized Short Pulse and Ostrovsky Equations

Costanzino

Manukian

Jones³

2009

SIAM J. Math. Anal.

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We derive a model for the propagation of short pulses in nonlinear media. The model is a higher order regularization of the short pulse equation (SPE). The regularization term arises as the next term in the expansion of the susceptibility in derivation of the SPE. Without the regularization term there do not exist traveling pulses in the class of piecewise smooth functions with one discontinuity. However, when the regularization term is added we show, for a particular parameter regime, that the equation supports smooth traveling waves which have structure similar to solitary waves of the modified KdV equation. The existence of such traveling pulses is proved via the Fenichel theory for singularly perturbed systems and a Melnikov type transversality calculation. Corresponding statements for the Ostrovsky equations are also included.

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“…One of the main concerns regarding Equation was to determine the existence of solitary wave solutions and its stability. Serval contributions have been devoted to treat with this problem (see and the references therein). It is worth mentioning that Levandosky and Liu considered a generalized Ostrovsky equation