Shelves and the Korteweg-de Vries equation

Knickerbocker, C. J.; Newell, Alan C.

doi:10.1017/s0022112080000407

Cited by 95 publications

(64 citation statements)

References 9 publications

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“…Thus, our problem is now broken into two regions: the region that matches imposed non-decaying boundary condition behaviour at infinity that is unaffected by the soliton, and the region in which the O(e) correction term is valid and the solution is quasi-stationary. We introduce a boundary layer in which there is a transition from the non-zero value in the perturbation term to the boundary conditions at infinity (see also Knickerbocker & Newell 1980;Kodama & Ablowitz 1981). Note in this section we will consider the more general case when u ∞ is a function of Z = ez.…”

Section: Boundary Layermentioning

confidence: 99%

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Perturbations of dark solitons

et al. 2011

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A direct perturbation method for approximating dark soliton solutions of the nonlinear Schrödinger (NLS) equation under the influence of perturbations is presented. The problem is broken into an inner region, where the core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton that propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated, including linear and nonlinear damping type perturbations. Keywords: perturbation theory; solitons; opticsPerturbation theory as applied to solitons that decay at infinity, i.e. so-called bright solitons, has been developed over many years (cf. Karpman & Maslov 1977;Kodama & Ablowitz 1981;Herman 1990). The analytical work employs a diverse set of methods including perturbations of the inverse scattering transform (IST), multi-scale perturbation analysis, perturbations of conserved quantities, etc.; the analysis applies to a wide range of physical problems. In optics, a central equation that describes the envelope of a quasi-monochromatic wave train is the nonlinear Schrödinger (NLS) equation, which in normalized form readswhere D, n are constants. In this paper, we consider the NLS equation in a typical nonlinear optics context, where D corresponds to the group-velocity dispersion (GVD), n > 0 is related to the nonlinear index of refraction, z is the

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Section: Boundary Layermentioning

confidence: 99%

“…They were needed to effectively understand the KdV equation under perturbation (cf. Knickerbocker & Newell 1980;Ablowitz & Segur 1981). In the KdV equation, there is a small shelf produced in the wake of the soliton.…”

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confidence: 99%

Perturbations of dark solitons

et al. 2011

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“…In the region 0 < 0 < 6 c (x) the amplitude of the shelf is small (that is, O(n)) and varies slowly with respect to the variables T and 6 (that is, the z and 6 derivatives are O(p)) (24,25,26). Thus, in this slowly-varying small-amplitude shelf region the evolution of the tail will be described by an asymptotic expansion of the form…”

Section: Shelf Emergence and Subsequent Evolutionmentioning

confidence: 99%

“…The shelf represents a small-amplitude (on the order of the non-dimensional retardation time) dilation of the tube. The dilation extends from the decaying main pulse to the current position associated with a hypothetical Korteweg-Moens pulse (24,26). Beginning at the Korteweg-Moens phase position, the shelf undergoes a series of viscoelastically modified high-wavenumber oscillations (that is, a dissipative and dispersive wavetrain) in the transition to a zero-amplitude background state.…”

Section: Introductionmentioning

confidence: 99%

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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“…The propagation of plane solitary waves over variable depth is now well understood ( [3,14,10,11,7]). In this problem it is shown that, starting from an initial solitary-wave solution of the KdV equation ( [12]), the propagation over a region of varying depth introduces a shelf directly behind the primary wave, a left-going (reflected) 'shelf' and another right-going 'shelf' (re-reflection).…”

Section: Introductionmentioning

confidence: 99%

Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth

Killen

Johnson

2001

Mathematics and Computers in Simulation

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A problem in nonlinear water-wave propagation on the surface of an inviscid, stationary fluid is presented.The primary surface wave, suitably initiated at some radius, is taken to be a slowly evolving nonlinear cylindrical wave (governed by an appropriate Korteweg-de Vries equation); the depth is assumed to be varying in a purely radial direction.We consider a sech 2 profile at an initial radius (which is, following our scalings, rather large), and we describe the evolution as it propagates radially outwards. This initial profile was chosen because its evolution over constant depth is understood both analytically and numerically, even though it is not an exact solitary-wave solution of the cylindrical KdV equation. The propagation process will introduce reflected and re-reflected components which will also be described. The precise nature of these reflections is fixed by the requirements of mass conservation.The asymptotic results presented describe the evolution of the primary wave, the development of an outward shelf and also an inward (reflected) shelf. These results make use of specific depth variations (which were chosen to simplify the solution of the relevant equations), and mirror those obtained for the problem of 1-D plane waves over variable depth, although the details here are more complex due to the axi-symmetry.

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Shelves and the Korteweg-de Vries equation

Cited by 95 publications

References 9 publications

Perturbations of dark solitons

Perturbations of dark solitons

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

Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth

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