Envelope solitons of surface waves in a plasma column

Abstract

The problem of envelope solitons of surface waves is considered on the basis of results for the nonlinear dispersion relation of the waves in a plasma column. The soliton solutions are derived as particular cases of the general solutions obtained by a universal procedure and expressed in terms of Jacobi elliptic functions. Since the two types of interactions, namely the (ω + ω) – ω and the (ω – ω) + ω interactions (where ω is the frequency of the carrier wave) included in the nonlinear dispersion relation act in opposite ways, existence both of bright and dark solitons is shown to be possible. The effect of the ponderomotive force that in our case is expressed through the contribution of the (ω – ω) + ω interaction leads to the formation of dark solitons. The effect of the losses is also considered.

Jacobi elliptic functions: A review of nonlinear oscillatory application problems

“…F φ a ; k À Á ¼ Z φ a 0 dψ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À k 2 sin 2 ψ q ; (12) and the Jacobi elliptic functions are then given by…”

“…Note also that a link can be established between Eqs. (20) and (12), with just a differently specified upper limit of the integrals.…”

“…Early descriptions of many of them date back to at least 1892, when the book by Greenhill [5] appeared, presenting a variety of such problems: a simple pendulum, catenaries, Watt's governor, Lintearia, a uniform chain that rotates steadily with a constant angular velocity about an axis to which the chain is fixed at two points, etc. Later applications include nonlinear plasma oscillations [6], Duffing oscillators [7], rigid plates satisfying the Johansen yield criterion [8], nonlinear transverse vibrations of a plate carrying a concentrated mass [9], a beam supported at an axially oscillating mount [10], doubly periodic cracks subjected to concentrated forces [11], surface waves in a plasma column [12], coupled modes of nonlinear flexural vibrations of a circular ring [13], dual-spin spacecrafts [14], spacecraft motion about slowly rotating asteroids [15], nonlinear vibration of buckled beams [16], a nonlinear wave equation [17], deep-water waves with two-dimensional surface patterns [18], oscillations of a body with an orbital tethered system [19], etc.…”

Self-focusing of surface waves in a cylindrical plasma waveguide

“…After the pioneering work by Alanakyan (1967), Boev & Prokopov (1975) and Agranovich & Chea'rnyak (1982), where nonlinear dispersion laws for surface waves in semi-infinite media were derived, many investigations have been made of self-phase modulation and the possibility of envelope soliton formation with a surface mode as a carrier wave (see Shivarova 1992 and references therein). With a view to experimental verification (Grozev, Shivarova & Tanev 1991) of the results obtained for nonlinear surface-wave behaviour in plasma waveguides, investigations of cylindrical configurations (Rasmussen 1978;Gradov & Stenflo 1983;Grozev & Shivarova 1984;Gradov, Stenflo & Sunder 1985;Shivarova & Yu 1986;Stenflo & Gradov 1986;Grozev, Shivarova & Boardman 1987;Yu & Stenflo 1991;Stenflo & Yu 1993) are of prime interest. The nonlinear plasma mechanisms involved in these studies are the effect of the ponderomotive force and the high-frequency plasma response at the secondharmonic frequency.…”

Nonlinear Surface Waves in a Strongly Inhomogeneous Plasma

“…If the amplitude of the surface waves is large the plasma will affect the wave due to different nonlinear mechanisms and give rise to slowly-varying amplitude modulations. In the present work where purely electronic motion is considered and ions are assumed to form a neutralizing background there are different types of nonlinear mechanisms to take into account; the low-frequency plasma response due to the ponderomotive force [4, 51 and the high-frequency response through second harmonic generation [6,7] and third order nonlinearities. In an informal fashion it is possible to show that a nonlinear Schrodinger equation arises as the evolution equation for these slow modulations.…”