Stationary solutions of the wave equation in a medium with nonlinearity saturation

Vakhitov, N. G.; Колоколов, А. А.

doi:10.1007/bf01031343

Cited by 903 publications

(754 citation statements)

References 7 publications

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“…We reach the same conclusion if we apply the Vakhitov-Kolokolov criterion [35] to the soliton solutions. The criterion states that, if ∂N s ∂λ 2 < 0, then solitons are unstable and they are stable in the opposite case.…”

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confidence: 50%

Deep-water internal solitary waves near critical density ratio

Агафонцев

Dias

Кузнецов

2007

Physica D: Nonlinear Phenomena

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Bifurcations of solitary waves propagating along the interface between two ideal fluids are considered. The study is based on a Hamiltonian approach. It concentrates on values of the density ratio close to a critical one, where the supercritical bifurcation changes to the subcritical one. As the solitary wave velocity approaches the minimum phase velocity of linear interfacial waves (the bifurcation point), the solitary wave solutions transform into envelope solitons. In order to describe their behavior and bifurcations, a generalized nonlinear Schrödinger equation describing the behavior of solitons and their bifurcations is derived. In comparison with the classical NLS equation this equation takes into account three additional nonlinear terms: the so-called Lifshitz term responsible for pulse steepening, a nonlocal term analogous to that first found by Dysthe for gravity waves and the six-wave interaction term. We study both analytically and numerically two solitary wave families of this equation for values of the density ratio ρ that are both above and below the critical density ratio ρcr. For ρ > ρcr, the soliton solution can be found explicitly at the bifurcation point. The maximum amplitude of such a soliton is proportional to √ ρ − ρcr, and at large distances the soliton amplitude decays algebraically. A stability analysis shows that solitons below the critical ratio are stable in the Lyapunov sense in the wide range of soliton parameters. Above the critical density ratio solitons are shown to be unstable with respect to finite perturbations.

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supporting

confidence: 50%

Deep-water internal solitary waves near critical density ratio

Агафонцев

Dias

Кузнецов

2007

Physica D: Nonlinear Phenomena

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“…It seems to have indeed first appeared in the paper [VK73] of Vakhitov and Kolokolov (1968), in the context of nonlinear optical waveguides. 20 Assumption 1 of [GSS87] is about the well-posedness of the Cauchy problem for (10.1) which, under our hypotheses, follows from Section 3.2.…”

Section: Stabilitymentioning

confidence: 99%

“…In the NLS context, this convexity condition takes the form of the condition (10.14) of Section 10. This stability condition seems to have first appeared in 1968 in a paper of Vakhitov and Kolokolov [VK73], where stability of trapped modes in a cylindrical nonlinear optical waveguide is discussed by formal arguments. In fact, the NLS equation is a standard model for slowly modulated waves in nonlinear media, for instance in nonlinear optics, see [SS99,Mai10].…”

Section: A Brief History Of Orbital Stabilitymentioning

confidence: 99%

Orbital Stability: Analysis Meets Geometry

Bièvre

Genoud

Nodari

2015

Nonlinear Optical and Atomic Systems

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ABSTRACT. We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schrödinger equation, for the wave equation, and for the Manakov system.

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“…To this end, we note, first of all, that the E͑ ͒ dependences displayed in Figs. 2 and 6 satisfy the Vakhitov-Kolokolov (VK) criterion, dE /d Ͻ 0, according to which solitons cannot be unstable against perturbation eigenmodes with purely real instability growth rates [41]. However, the VK criterion ignores perturbations with complex growth rates.…”

Section: Dynamical Problems: Stability Of the Solitons Bound Statesmentioning

confidence: 99%

Solitons in Bragg gratings with saturable nonlinearities

et al. 2007

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We introduce two different systems of coupled-mode equations to describe the interaction of two waves coupled by the Bragg reflection in the presence of saturable nonlinearity. The basic model assumes the ordinary linear coupling between the modes. It may be realized as a photorefractive waveguide, with a Bragg lattice permanently written in its cladding. We demonstrate the presence of a cutoff point in the system's bandgap, with gap solitons existing only on one side of it. Close to this point, the soliton's norm diverges with power −3 / 2. The soliton family between the cutoff point and the edge of the bandgap is stable. In this model, stationary bound states of two in-phase solitons are found too, but they are unstable, transforming themselves into breathers. Another model assumes a photoinduced longitudinal bulk grating, with the corresponding intermode coupling subject to saturation along with the nonlinearity. In that model, another cutoff point is found, with the soliton's norm diverging near it with power −2. Solitons are stable in this model too (while it does not give rise to twosoliton bound states). Collisions between moving solitons are always quasi-elastic, in either model.

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Stationary solutions of the wave equation in a medium with nonlinearity saturation

Cited by 903 publications

References 7 publications

Deep-water internal solitary waves near critical density ratio

Deep-water internal solitary waves near critical density ratio

Orbital Stability: Analysis Meets Geometry

Solitons in Bragg gratings with saturable nonlinearities

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