Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation

Marchant, Timothy R.

doi:10.1016/j.wavemoti.2007.11.003

Cited by 33 publications

(54 citation statements)

References 23 publications

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“…Traveling wave solutions of the mKdV equation are available from a number of papers (see, e.g., [56,75]) and can also be obtained as reductions of corresponding solutions to the Gardner equation [54].…”

Section: Traveling Wavesmentioning

confidence: 99%

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

El¹,

Hoefer²,

Shearer³

2017

SIAM Rev.

107

150

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Abstract. We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg-de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.

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Section: Traveling Wavesmentioning

confidence: 99%

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

El¹,

Hoefer²,

Shearer³

2017

SIAM Rev.

107

150

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“…In simpler situation of unidirectional waves whose evolution is governed by the modified KdV (or Gardner) equation, this problem was solved in Ref. [21] where it was found that in addition to DSWs and rarefaction waves the arising structures can also include trigonometric and combined shocks or kinks depending on the sign of the higher order nonlinearity (see also earlier papers [22,23] where partial similar results were also obtained). In this paper we extend this method to the equation (1) describing evolution of nonlinear pulses in fibers.…”

Section: Introductionmentioning

confidence: 90%

Riemann problem for the photon fluid: Self-steepening effects

Ivanov

Камчатнов

2017

Phys. Rev. A

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We consider the Riemann problem of evolution of initial discontinuities for the photon fluid propagating in a normal dispersion fiber with account of self-steepening effects. The dynamics of light field is described by the nonlinear Schrödinger (NLS) equation with self-steepening term appearing due to retardation of the fiber material response to variations of the electromagnetic signal. It is shown that evolution dynamics in this case is much richer than that for the NLS equation. Complete classification of possible wave structures is given for all possible jump conditions at the discontinuity.

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“…The figure shows a typical bore solution of the mKdV equation. The wave amplitude varies in a quadratic manner through the bore [34], in contrast to the KdV bore, which has a linear amplitude variation. Again, the numerical amplitude of the lead wave corresponds closely with the modulation theory prediction of 2A = 1.…”

Section: Mkdv and Benjamin-ono Equationsmentioning

confidence: 99%

“…Modulation theory for the mKdV equation [34] gives the amplitude of the leading soliton generated from the initial condition (2) as A s = 2A, the same as for the KdV case. The approximate theory based on the mass and energy conservation equations for the mKdV equation (12) gives this lead soliton amplitude as…”

Section: Mkdv and Benjamin-ono Equationsmentioning

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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Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation

Cited by 33 publications

References 23 publications

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

Riemann problem for the photon fluid: Self-steepening effects

Approximate Techniques for Dispersive Shock Waves in Nonlinear Media

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