Reflection Of Solitons Research Articles

Reflection and Transmission of a Shallow-Water Soliton over a Barrir

This paper deals with a shallow-water soliton incident upon a submerged barrier. Introducing an `edge-layer' near the barrier, the matched-asymptotic expansion method is applied to derive uniformly valid solutions. Assuming Boussinesq equations are valid for the shallow-water regions, two `reduced' boundary conditions are derived from the matching conditions. This boundary-value problem is further reduced to an `initial value' problem for K-dV equations, which is solved to investigate reflection and transmission of an incident soliton. For a moderate height of the barrier, the soliton can pass over the barrier as if it were almost transparent. As the barrier becomes high, the transmitted soliton is suppressed, while the reflected wave is enhanced and evolves into one soliton with slowly decaying ripples behind.

Perturbation analysis of a parametrically changed sine-Gordon equation.

A long Josephson junction with a spatially varying inductance is a physical manifestation of a modified sine-Gordon equation with parametric perturbation. Soliton propagation in such Josephson junctions is discussed. First, for an adiabatic model where the inductance changes smoothly compared with soliton size, transmission or reflection of the soliton is described using a simple energy analysis. Next, the soliton propagation is solved on the basis of a perturbation theory constructed by McLaughlin and Scott. Radiation as well as soliton trajectories are presented numerically. Agreement between such solutions and the results of direct numerical integration by means of a finite-difference method is excellent.

Reflection of a shallow-water soliton. Part 2. Numerical evaluation.

Reflection of a shallow-water soliton at a plane beach is investigated numerically based on the edge-layer theory developed in Part 1 of this series. The offshore behaviour in the shallow-water region is first obtained by solving the Boussinesq equation under the ‘reduced’ boundary condition. The spatial and temporal variations of the surface elevation are displayed for two typical values of the inclination angle of the beach. Using these solutions, the nearshore behaviour is then evaluated to obtain the surface elevation and the velocity distribution in the edge layer. Both offshore and nearshore behaviours furnish a full knowledge of the reflection problem of a shallow-water soliton. To check the applicability of the edge-layer theory, a ‘computational experiment’ is carried out based on the boundary-element method, in which the Laplace equation is solved numerically under the full nonlinear boundary conditions at the free surface without introducing the edge-layer concept. Both results show a fairly good agreement for the overall reflection behaviour of a shallow-water soliton except for the surging movement at the shoreline.

Reflection and transmission of a shallow-water soliton

Reflection and transmission of a shallow-water soliton

Nonlinear reflection and refraction of planar ion-acoustic plasma solitons.

Experimental observations on the reflection and refraction of a planar ion-acoustic soliton from a metallic mesh electrode are performed in a uniform double-plasma device. Reflection and refraction angles are observed to depend on the incident wave amplitude, showing a nonlinear Snell's law.

Solution of quantum Gel'fand-Levitan-Marchenko equations for the sine-gordon model in the soliton sector for ?=?/gn

The solutions of the quantum Gel'fand-Levitan-Marchenko equations for the sine-Gordon model are studied in the sector containing solitons. The case ..gamma.. = ..pi../v (v integer), when there is no soliton reflection, is considered. A recursive procedure for constructing solutions is obtained, and examples in sectors containing a small number of solitons are considered.

On a mechanism of soliton bunching in Josephson junctions

The interaction of solitons in long Josephson junctions governed by the perturbed sine-Gordon equation is investigated. The theory of soliton bunching presented by Karpman et al. is refined, generalized, and supported by new numerical material. It is demonstrated that soliton reflections from the junction ends may play a significant role in stabilization of the distance between the solitons in a bunched state.

Reflection of a shallow-water soliton. Part 1. Edge layer for shallow-water waves

To investigate reflection of a shallow-water soliton at a sloping beach, the edge-layer theory is developed to obtain a ‘reduced’ boundary condition relevant to the simplified shallow-water equation describing the weakly dispersive waves of small but finite amplitude. An edge layer is introduced to take account of the essentially two-dimensional motion that appears in the narrow region adjacent to the beach. By using the matched-asymptotic-expansion method, the edge-layer theory is formulated to cope with the shallow-water theory in the offshore region and the boundary condition at the beach. The ‘reduced’ boundary condition is derived as a result of the matching condition between the two regions. An explicit edge-layer solution is obtained on assuming a plane beach.

Comparison between experiment and perturbation theory for solitons in Josephson junctions

Experiments have been made on long inline and overlap Josephson junctions at various temperatures and current densities. The junctions had parameters such that the recently developed perturbation theory for soliton motion according to the modified sine-Gordon equation should be applicable. A comparison showed that this is the case, and the damping constant was derived as a function of the temperature. In addition, results were obtained for the soliton-antisoliton annihilation process. A fine structure in the zero-field steps at low temperatures is interpreted as being due to plasma oscillations in connection with soliton reflections at the boundaries.

Reflection of an ion-acoustic soliton by a bipolar potential wall structure

Abstract A potential wall structure consisting of a negatively biased grid and a positively biased plate is inserted in a DP-machine plasma. Ion-acoustic solitons are seen to be reflected at this obstacle as soon as the bias of the plate is sufficiently positive to reflect ions.

Reflection of ion acoustic soliton from the sheath around a negatively biased grid

Experimental observations on the partial reflection of ion acoustic solitary waves from sharp density gradients, produced by a negatively biased grid immersed in the plasma, are presented. Reflection occurs where the density gradient scale size is of the same order as the width of the soliton. The reflected wave packet apppears to be highly dispersive and travels with subsonic velocity.

Reflection and Transmission of Lattice Solitons

The scattering of lattice solitons by a mass interface is studied through the use of reductive perturbation method. In the lowest order approximation, the incident soliton and the reflected and transmitted waves propagate independently of each other and each of them is governed by the generalized Korteweg-de Vries equation. The boundary condition at the mass interface is the same as that for linear waves. Reflection and transmission of solitons through the interface are qualitatively discussed. The results are compared with those of the recent numerical experiments. · § I. Introduction Since Todan found the existence of pulse-like waves passing through one another without losing their identities in an anharr~J.Onic one-dimensional lattice system with the nearest neighbor interaction of the exponential type, much work on the lattice solitons has been made.J Recently, an interesting numerical experi­ ment was carried out by Yoshida, Nakayama and Sakuma. 3J They investigated reflection of solitons in an anharmonic lattice system with the cubic and quartic interaction potential. It is seen from their results that the solitons are reflected as stable entities from a fixed boundary, preserving their identities. However, the reflection of solitons from a free boundary is quite different; for the lattice system with only quartic anharmonicity the incident soliton reflects at the boundary transforming its type, for example, from compressive type into rarefactive one, and vice versa, whilst, for a system in which there exists cubic nonlinearity the incident soliton decays with time into a train of ripples after the reflection. They further studied the scattering of lattice solitons from a mass interface. The soliton disintegrates into reflected and transmitted waves, in a way such that their ampli­ tudes are determined by the reflection coefficient calculated by a linear wave theory. This implies that a soliton, when passing through the mass interface, behaves like a linear wave. With regard to the reflection problem of lattice solitons, Toda has made the following conjecture :4l Reflection of a soliton at a fixed boundary can be generally treated by the method of image because the fixed boundary condition can be replaced by an image soliton beyond the boundary. However, reflection at a free boundary provides a quite new problem to be worked out. On the analogy of a harmonic lattice, it is anticipated that a compressive (rarefactive) __ soliton will be reflected

Chopping Phenomenon of a Nonlinear System

It is shown by numerical experiments that a system with nonlinear interaction may exhibit chopping phenomenon induced by solitons. Reflection of solitons at a free end is also studied.