A perturbation theory for the Korteweg-De Vries equation

Karpman, V. I.; Maslov, Eugene M.

doi:10.1016/0375-9601(77)90107-4

Cited by 112 publications

(61 citation statements)

References 5 publications

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“…The solution thus constructed has been used extensively to develop a direct perturbation theory for the KdV equation [22,23] The above results coincide with those derived by IST [14].…”

Section: Detailed Description Of the Phase Shiftsupporting

confidence: 72%

Multisoliton perturbation theory for the Benjamin-Ono equation and its application to real physical systems

Matsuno

1995

Phys. Rev. E

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A direct perturbation theory is developed to study the effects of small perturbations on the interaction process of algebraic solitons of the Benjamin-Ono (BO) equation. Using the method of multiple scales, the modulation equations for the amplitude and the phase of each soliton are derived in the lowest approximation. As practical applications of the theory, the interaction of two solitons is investigated for the two different types of perturbations that appear in real physical systems. One is a dissipative perturbation (BO -Burgers equation) and the other is a dispersive perturbation (higher-order BO equation). In both cases, the changes of the soliton parameters due to small perturbation are calculated by numerical integrations and their characteristics are elucidated in detail. Among them, the phase shift caused by the dispersive perturbation is a remarkable feature that has never been observed in the collision process of algebraic solitons.

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“…The solution thus constructed has been used extensively to develop a direct perturbation theory for the KdV equation [22,23] The above results coincide with those derived by IST [14].…”

Section: Detailed Description Of the Phase Shiftsupporting

confidence: 72%

Multisoliton perturbation theory for the Benjamin-Ono equation and its application to real physical systems

Matsuno

1995

Phys. Rev. E

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“…There is also a monotonically increasing positive phase shift induced by the dissipation. The modulated amplitude, translation speed and phase shift are natural consequences of the perturbed evolution of the scattering data in the inverse-scattering (1ST) formulation (21,23). Alternatively, it is possible to obtain the modulation in a direct singular perturbation theory as the consequence of the perturbed solitary wave attempting to satisfy leadingand secondary-order energy balances for the dissipative system (25,40).…”

Section: Problem Formulation and Derivation Of The Korteweg-de Vries-mentioning

confidence: 99%

“…Here, we focus on determining the evolution of rj(x) and %o(x)-The dependence of r) and £ 0 on the stretched coordinate % c 811 be obtained as solvability conditions for the perturbation expansion (3.2) (25,42), as a result of averaged conservation laws (40), or directly from the evolution of the scattering data in the perturbed 1ST formalism (21,23). Again, all these approaches are, of course, equivalent.…”

Section: Problem Formulation and Derivation Of The Korteweg-de Vries-mentioning

confidence: 99%

“…We may briefly summarize our principal results as follows. The main pulse evolves adiabatically so as to satisfy a leading-order energy balance (21,23,25) which will result in an algebraic decay in the sou'ton amplitude and translation speed relative to the Korteweg-Moens phase speed. (The soliton always travels faster than the Korteweg-Moens phase speed due to the action of nonlinearity.)…”

Section: Introductionmentioning

confidence: 99%

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Viscoelastic Modulation of Solitary Pressure Pulses in Nonlinear Fluid-Filled Distensible Tubes

Swaters

Sawatzky

1989

Q J Mechanics Appl Math

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SUMMARYThe propagation of small-but-finite-amplitude weakly-dispersive pressure pulses in a viscoelastic fluid-filled tube is shown to be governed by a Korteweg-de Vries-Burgers (KdVB) equation. We present a direct singular perturbation theory to describe the viscoelastic modulation of solitary pressure pulses for small retardation times. The main pulse evolves so as to satisfy a leading-order energy balance resulting in an algebraic decay in the amplitude and translation speed of the solution. Higher-order energy balances imply a monotonically increasing positive phase shift.A complete description of the first-order perturbation pressure is given. A pressure shelf of finite extent is continuously excited in the lee of the propagating pulse. The shelf extends from the current location of the solitary pressure pulse to the Korteweg-Moens phase position. The transition of the pressure shelf back to a zero background state at the Korteweg-Moens phase position is accomplished through a series of viscoelastically modified spatially-decaying high-wavenumber oscillations. Ahead of the main pulse a uniformly valid perturbation pressure field is obtained via a combination WKB-power-series and similarity-solution procedure.

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“…A similar behavior can be found (numerically and experimentally) for the KdV equation with dissipation on an infinite interval. However, in this case, some problems arise in the derivation of an analytical approximation since the decay of the mass functional M (u) = u then has to be taken into account; see [18]- [20].…”

Section: Introduction Consider the Uniformly Damped One-dimensional mentioning

confidence: 99%

Dissipation in Hamiltonian Systems: Decaying Cnoidal Waves

Derks¹,

Groesen²

1996

SIAM J. Math. Anal.

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Abstract. The uniformly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions can be viewed as a Hamiltonian system with dissipation added. The KdV equation is the Hamiltonian part and it has a two-dimensional family of relative equilibria. These relative equilibria are space-periodic soliton-like waves, known as cnoidal waves.Solutions of the dissipative system, starting near a cnoidal wave, are approximated with a long curve on the family of cnoidal waves. This approximation curve consists of a quasi-static succession of cnoidal waves. The approximation process is sharp in the sense that as a solution tends to zero as t → ∞, the difference between the solution and the approximation tends to zero in a norm that sharply picks out their difference in shape. More explicitly, the difference in shape between a solution and a quasi-static cnoidal-wave approximation is of the order of the damping rate times the norm of the cnoidal-wave at each instant.

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A perturbation theory for the Korteweg-De Vries equation

Cited by 112 publications

References 5 publications

Multisoliton perturbation theory for the Benjamin-Ono equation and its application to real physical systems

Multisoliton perturbation theory for the Benjamin-Ono equation and its application to real physical systems

Viscoelastic Modulation of Solitary Pressure Pulses in Nonlinear Fluid-Filled Distensible Tubes

Dissipation in Hamiltonian Systems: Decaying Cnoidal Waves

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