Modulational instability and formation of a nonlinear oscillatory structure in a “focusing” medium

El, Gennady; Gurevich, A.; Khodorovskii, V.V.; Крылов, А. Л.

doi:10.1016/0375-9601(93)90015-r

Cited by 70 publications

(86 citation statements)

References 2 publications

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“…A simple wave bore solution is then not expected to exist. However, if the nonlocality is large enough the onset of modulational(MI) instability is delayed [27], so a bore-type solution exists for experimental length scales [28]. The approximate method is found to give solutions in good agreement with numerical solutions of these nonlocal equations.…”

Section: Introductionmentioning

confidence: 86%

“…As it has an inverse scattering solution [29], it has modulation equations from which a bore solution can be determined [28]. However, the NLS equation (18) is focusing, so that wavetrains show MI.…”

Section: Nls Equationmentioning

confidence: 99%

“…This is because the modulation equations for the NLS equation (18) are hyperbolic in the soliton and linear wave limits [28]. A bore-type solution can then be constructed with a soliton at the leading edge and linear waves at the trailing edge [28]. The step initial condition…”

Section: Nls Equationmentioning

confidence: 99%

“…This initial condition will generate a bore until MI takes over [28]. Modulation theory for the NLS equation [28] gives the amplitude of the lead soliton of the dispersive shock wave as A s = 2A.…”

Section: Nls Equationmentioning

confidence: 99%

“…This approximate method will now be used to find the amplitude of the lead solitary wave in dispersive shock waves for which the governing equations have no modulation equations which can be put into Riemann invariant form, or which are not hyperbolic in the dispersionless limit [28].…”

Section: Nls Equationmentioning

confidence: 99%

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Approximate Techniques for Dispersive Shock Waves in Nonlinear Media

Marchant

Smyth

2012

J. Nonlinear Optic. Phys. Mat.

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Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained. Abstract Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained.

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Section: Introductionmentioning

confidence: 86%

Section: Nls Equationmentioning

confidence: 99%

Section: Nls Equationmentioning

confidence: 99%

Section: Nls Equationmentioning

confidence: 99%

Section: Nls Equationmentioning

confidence: 99%

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Approximate Techniques for Dispersive Shock Waves in Nonlinear Media

Marchant

Smyth

2012

J. Nonlinear Optic. Phys. Mat.

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Dispersive Shock Waves in Reorientational and Other Optical Media

Marchant

2012

Nematicons

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Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

Biondini

Mantzavinos

2017

Comm Pure Appl Math

109

154

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We characterize the long‐time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity by using a variant of the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest‐descent method of Deift and Zhou for oscillatory Riemann‐Hilbert problems. First, we formulate the IST over a single sheet of the complex plane without introducing the uniformization variable that was used by Biondini and Kovačič in 2014. The solution of the focusing NLS equation with nonzero boundary conditions is thereby associated with a matrix Riemann‐Hilbert problem whose jumps grow exponentially with time for certain portions of the continuous spectrum. This growth is the signature of the well‐known modulational instability within the context of the IST. We then eliminate this growth by performing suitable deformations of the Riemann‐Hilbert problem in the complex spectral plane. The results demonstrate that the solution of the focusing NLS equation with nonzero boundary conditions remains bounded at all times. Moreover, we show that, asymptotically in time, the xt‐plane decomposes into two types of regions: a left far‐field region and a right far‐field region, where the solution equals the condition at infinity to leading order up to a phase shift, and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. Finally, we show how, in the latter region, the modulus of the leading‐order solution, initially obtained as a ratio of Jacobi theta functions, can be reduced to the well‐known elliptic solutions of the focusing NLS equation. These results provide the first characterization of the long‐time behavior of generic perturbations of a constant background in a modulationally unstable medium. © 2017 Wiley Periodicals, Inc.

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Modulational instability and formation of a nonlinear oscillatory structure in a “focusing” medium

Cited by 70 publications

References 2 publications

Approximate Techniques for Dispersive Shock Waves in Nonlinear Media

Approximate Techniques for Dispersive Shock Waves in Nonlinear Media

Dispersive Shock Waves in Reorientational and Other Optical Media

Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

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