Yoshi H. Ichikawa

New Integrable Nonlinear Evolution Equations

A new series of integrable nonlinear evolution equations is presented. The equations are novel in the sense that the nonlinear terms have saturation effects. It is shown that the equations have an infinite number of conservation laws and can be expressed in the Hamiltonian form.

Contribution of Higher Order Terms in the Reductive Perturbation Theory. I. A Case of Weakly Dispersive Wave

Contribution of higher order terms in the reductive perturbation theory has been investigated for nonlinear propagation of strongly dispersive ion plasma wave. The basic set of fluid equation is reduced to a coupled set of the nonlinear Schrodinger equation for the first order perturbed potential and a linear inhomogeneous equation for the second order perturbed potential. A steady state solution of the coupled set of equations has been solved analytically for the case of nonlinear amplitude modulation of the ion acoustic wave, which is nothing but the small wave number limit of the strongly dispersive ion plasma wave.

Spiky Soliton in Circular Polarized Alfvén Wave

A new type of nonlinear evolution equation for the Alfven waves, propagating parallel to the magnetic field, is now registered to the completely integrable family of nonlinear evolution equations. In spite of the extensive studies of Kaup and Newell, and of Kawata and Inoue, these analysis have been dealing with solutions for restricted boundary conditions. The present paper presents full account of stationary solitary wave solutions for the plane wave boundary condition. The obtained results exhibit peculiar structure of spiky modulation of amplitude and phase, which arises from the derivative nonlinear coupling term. A nonlinear equation for complex amplitude associated with the carrier wave is shown to be a mixed type of nonlinear Schrodinger equation, having an ordinary cubic nonlinear Schrodinger equation, having an ordinary cubic nonlinear term and the derivative of cubic nonlinear term.

A Modified Korteweg de Vries Equation for Ion Acoustic Waves

Three wave mode coupling effects on ion acoustic waves are examined by applying the Fourier transformation method of separation of the nonlinear slow processes for a two component plasma described by the Vlasov equation. Contribution of the three mode coupling terms gives rise to a modified Korteweg de Vries equation with a nonlinear term of the form of { B φ+( C /2)φ 2 }φ x , where B and C are constants determined by the unperturbed state of plasmas. It is found that velocity of a stationary solitary wave depends sensitively on the electronion temperature ratio, and that the equation obtained here enables us to account for the amplitude dependence of the solitary wave velocity observed by Ikezi, Baker and Taylor.

On the Nonlinear Schrodinger Equation for Langmuir Waves

A direct derivation of the nonlinear schrodinger equation for Langmuir waves is presented, based upon the nonlinear wave packet ansatz of Karpman and Krushkal. Both fluid and Vlasov equation formulations are used. The results obtained are essentially equivalent to those found earlier by Taniuti, et al. using reductive perturbation theory, including the importance of wave particle resonances at the group velocity for the long time behavior of the amplitude of modulated waves. Separating the wave packet considerations from the calculation of the nonlinear frequency shift makes it possible to attack the latter with whatever method facilitates the analysis of that part of the problem. In addition, certain ambiguities concerning singularities in velocity integrations are resolved, and the connection with a well-posed initial value problem is made somewhat clearer. This method can be used equally well for other waves, and may be of help particularly in situations where it is not clear, a priori, what scaling to...

Propagation of Ion Acoustic Cnoidal Wave

Higher order contributions in the reductive perturbation theory have been reexamined for the nonlinear ion acoustic wave propagation under the periodic boundary condition. The boundary condition gives rise to renormalization terms which enable us to eliminate secular contribution in the higher order terms. As a result of analysis carried up to the second order, the ion flux associated with propagation of nonlinear ion acoustic cnoidal wave is examined for various values of excitation frequencies. It has been shown explicitly that the dependence of the ion flux on the wave amplitude becomes much weaker than that of quasi-linear theory at lower frequency. This illustrates that the averaged properties such as the averaged flux in genuine nonlinear systems depend on their nonlinear behavior in very crucial way. Range of validity of so-called quasi-linear approximation appears to be restricted to very small amplitude of disturbance.

Dynamical Processes of the Dressed Ion Acoustic Solitons

Dynamical processes of the dressed ion acoustic solitons have been numerically analyzed by solving the coupled set of the first order Korteweg-de Vries equation and the second order equation, which has been derived on a basis of the reductive perturbation theory. A single soliton dressed by clouds of thc second order perturbation potential exhausts ion acoustic wave behind when it travels down free space. On the other hand, in collision processes of two dressed solitons, the clouds around the Korteweg-da Vries soliton core redistribute themselves in such a way to equalize the heights of colliding pairs. Structure of the clouds suggests that effective attractive interaction acts between the colliding pairs of the dressed solitons.

Theory of Resonance Probe

A theoretical analysis is made of the resonance phenomena in the radio-frequency probe experiments of Takayama et al. The Boltzmann-Vlasov equation is solved under the action of an external rf electric field. The solution gives the resonance peak of the de component of the electron current to the probe at the plasma frequency. For a partially ionized plasma, the peak-height IJj and the half-width ilah/2 are given by the following formulae.

Modulation Instability of Electron Plasma Wave

A modified nonlinear Schrodinger equation obtained on the basis of the reductive perturbation theory is compared with one obtained by Dewar applying the Lagrangian method. It is shown that the both results agree each other for a special case of the one dimensional electrostatic modes in a collisionless plasma. In a case of the electron plasma wave, the nonlocal nonlinear term of the modified nonlinear Schrodinger equation leads to a modulational instability, which is reduced to the nonlinear Landau damping in the limit of small amplitude.

Contribution of the Second Order Terms to the Nonlinear Shallow Water Waves

Contribution of the second order terms in the reductive perturbation theory has been investigated for the nonlinear shallow water waves. The fundamental equations are reduced to a coupled set of the Korteweg-de Vries equation for the first order horizontal velocity and a linear inhomogeneous equation for the second order arbitrary function. Structure of the coupled set of equations turns out to be the same as in the case of nonlinear ion acoustic wave. A steady state solution of the coupled set of equations has been examined in comparison with Laitons analysis of the second order contribution of the Friedrichs expansion for the nonlinear shallow water waves.