Nobuo Yajima

Perturbation Method for a Nonlinear Wave Modulation. II

A perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein‐Gordon equation. In these systems, and probably in most physical systems, an assumed condition for a mode of l = 0 is not valid. Consequently, the direct application of the method is impossible. In the present paper, we shall illustrate by these examples how this difficulty can be overcome to allow us to use the method. As a result we shall find that, in either case, the original equation can be reduced to the nonlinear Schrodinger equation.

Interactions of Solitary Waves –A Perturbation Approach to Nonlinear Systems–

An approximate method of studying interactions between two solitary waves which propagate in opposite directions is presented. In the first approximation, the solution is described by s superposition of two solitary waves which are governed by their respective Korteveg-de Vries equation. The second order approximation gives a small correction where the two waves overlap one another. The method is extended to the system, in which there exist n “quasi-simple” waves (the simple waves under the effects of higher derivative terms, such as dispersions of dissipations). The possibility that n “quasi-simple” waves can be superposed to describe nonlinear systems is studied. Applications to ion acoustic waves in collisionless plasmas and shallow water waves are discussed.

Soliton and Nonlinear Explosion Modes in an Ion-Beam Plasma System

Nonlinear wave evolutions are studied both numerically and analytically in an ion-beam plasma system. In the linearly stable case solitons associated with linear eigen modes inherent in the system are shown to be mutually independent K-dV solitons so long as their amplitudes are smaller than a certain critical value. In the case of supercritical amplitudes, nonlinear explosion modes are found, whose analytical solution is obtained near the critical point.

A Perturbation Approach to Nonlinear Systems. II. Interaction of Nonlinear Modulated Waves

The reductive perturbation method is extended to apply to a strongly dispersive system, in which mudulated plane waves interact each other through nonlinear interactions. The interaction between two envelope solitons is examined.

Interaction of Ion-Acoustic Solitons in Multi-Dimensional Space.II

Numerical computations are made to study the collision process between two cylindrical or spherical solitons. The soliton resonance is found to play an important role in collision processes between two curved solitons as well as between two plane solitons

Scattering of Lattice Solitons from a Mass Impurity

A perturbation theory for the inverse scattering transforms applies to the Toda lattice system with a mass impurity. As an example, scattering of a soliton from an impurity is considered. When the mass of impurity is slightly different from that of host particles, the soliton nearly passes through the impurity without changing its amplitude.

Soliton Modes in an Unstable Plasma --Nonlinear Phenomena in an Electron-Beam Plasma--

A synergetic approach is applied to the study of nonlinear evolution of unstable waves in an electron beam plasma. When the beam velocity is sufficiently large compared with the thermal velocity of electrons, the system is linearly unstable against electrostatic perturba· tions of large wavelengths, but stabilized in its nonlinear stage by the effect of ponderomotive force of high·frequency fields. The results show that the amplitude of unstable plane wave behaves such as ~sech rt as a function of time t, where r is the linear growth rate, never growing up infinitely. It is also shown that the plane wave is unstable for long wavelength modulation and is stabilized by emission of a series of envelope solitons.

K-dV Soliton in Inhomogeneous System

The propagation of a soliton in an inhomogeneous medium is investigated numerically by the following K-dV type equation; u t +6 u u x + u x x x + c ( t ) u =0. The inhomogeneous coefficient c ( t ) is assumed to be zero except for the period, 0 ≤ t ≤ T . When the integral of c ( t ) from t =0 to t = T is negative, a soliton grows and generates small solitons. In the case where the integral is positive, a soliton diminishes and does not produce new soliton. In either case, the solution depends on the ratio of the distance that an initial soliton is propagated in the period 0 ≤ t ≤ T to the width of the soliton.

Effect of Bottom Irregularities on Small Amplitude Shallow Water Waves

Behavior of finite amplitude long gravity waves in a water layer with an irregular bottom surface is investigated by means of a nonlinear perturbation method. Under the assumption that the irregularities are of small size, a simple set of nonlinear equations is presented. There arise two effects in the propagation characteristic of shallow water waves, change in phase velocity and damping of amplitude. For the one-dimensional and unidirectional motion of shallow water waves the set of equations is reduced to a simpler one, the generalized Korteweg-de Vries equation. The decay rate of a solitary wave is also obtained.

Instability of Coherent Ion Acoustic Wave in a Phonon Gas

The ion-acoustic wave propagation in a field of randomly excited low-frequency waves (phonon gas) is investigated, by obtaining the linear response function of plasmas in a turbulent phonon state. The turbulent phonons have an influence on the propagation of the coherent wave, to lead a decreased phase velocity. The coherent wave becomes unstable due to the resonant coupling with the beat wave of turbulent phonons. As the result of the instability, the coherent wave with long wavelength piles up in a plasma with turbulent phonons. The results qualitatively agree with the recent experiment.