Takuji Kawahara

Oscillatory Solitary Waves in Dispersive Media

Steady solutions of a generalized Korteweg-de Vries equation which has an additional fifth order derivative term are investigated on the basis of a numerical calculation. It is found that either compressive or rarefactive solitary waves are possible to exist according as the dispersion is negative or positive and that the solitary waves take the oscillatory structure when the coefficient of the fifth order derivative term dominates over that of the third order one.

Approximate Equations for Long Nonlinear Waves on a Viscous Fluid

A systematic perturbation method is applied to three-dimensional long waves on a viscous liquid film, and the nonlinear evolution equation incorporating the effects of dissipation and dispersion is derived. It is shown that both the fourth-order derivative term as well as the three-dimensionality have stabilizing effects.

Interactions of traveling fronts: An exact solution of a nonlinear diffusion equation

Abstract An exact solution which describes the coalescence of two traveling fronts of the same sense into a front connecting two stable constant states is found in terms of the direct method for a simple nonlinear diffusion equation. Head-on collisions of two fronts of opposite sense are also examined numerically.

Pulse interactions in an unstable dissipative‐dispersive nonlinear system

An attempt is made to understand several features of the wave evolutions in an unstable dissipative‐dispersive nonlinear system in terms of the interactions of localized solitonlike pulses. It is found that the wave evolutions can be qualitatively well described by weak interactions of pulses, each of which is the steady solution to the original evolution equation. The oscillatory structure of a tail of the pulse for weakly dispersive cases is responsible for the existence of bound states of pulses, which explains the numerical result that the interpulse distances in the initial value problem take certain fixed values or values in the definite regions. In cases of monotone tails for strongly dispersive cases, the effects of pulse interactions become repulsive, which explains the result that the pulses asymptotically tend to be arranged periodically, adjusting to the periodic boundary conditions in the numerical simulation.

The Derivative-Expansion Method and Nonlinear Dispersive Waves

The method of derivative-expansion with multiple scales is applied in a generalized from to the analysis of weak nonlinear dispersive waves. It is shown that in a certain case a nonlinear Schrodinger equation, which describes a nonlinear slow amplitude modulation of wave trains, can be derived from the condition that the perturbation expansion be free from secular terms. A reduction to the Korteweg-de Vries equation for long waves is also discussed.

Nonlinear Self-Modulation of Capillary-Gravity Waves on Liquid Layer

The derivative expansion method is applied to an investigation of the weakly nonlinear self-interactions of capillary-gravity waves on a liquid layer of uniform depth. The stability characteristics of a wave train are examined on the basis of the nonlinear Schrodinger equation. It is shown that the effect of capillarity is of critical importance to the modulational instability.

CHANGES IN AMPLITUDE OF THE EEG INDUCED BY A PHOTIC STIMULUS

Occipital EEG responses to a single photic flash were studied by using a complex demodulation technique. Individual EEGs were first digitally filtered by making use of the Ormsby band-pass filter. Envelopes of the alpha (8-12 c/sec) and of the low frequency (3-7 c/sec) activities were obtained in terms of the demodulation calculation. Both individual and averaged data of raw, filtered, and demodulated EEGs were utilized for the present study. Two typical kinds of EEG responses were observed. One showed clearly the blocking as well as the enhancement of the alpha activity. The alpha activity first exhibited a reduction in amplitude in the range 50-200 msec after the stimulus (the minimum amplitude at about 150 msec). Then an evocation, rather than merely a recovery, of the alpha was observed in the range 500-800 msec (maximum amplitude at about 650 msec). The other typical example showed the low frequency evoked potential prominently. This response appeared almost locked to the stimulus and showed roughly a bi-phasic pattern in the time range where the alpha blocking took place (30-200 msec). Phases of the evoked low frequency component were almost locked to the stimulus but no distinct phase coherence was observed for the enhanced alpha wavetrains. Detection by demodulation of the phase change of the alpha was critical for a single EEG. It is pointed out that changes of the alpha must be studied on individual EEGs.

Cylindrical quasi-solitons of the Zakharov-Kuznetsov equation

Abstract Evolutions and interactions of two-dimensional solitary waves of the Zakharov-Kuznetsov equation are investigated numerically. Formations of cylindrical bell-shaped pulses are observed in the initial value problems. A single bell-shaped pulse propagates stably without any deformation like a soliton. Two similar pulses exchange their amplitudes without merging and two dissimilar ones undergo overtaking collision. After a collision of two pulses, the strong pulse becomes somewhat stronger and the weak one becomes weaker with radiation of ripples, so that the collision process is slightly inelastic. Generated ripples are very small for center-to-center collision of similar pulses. The cylindrically symmetric pulses of the Zakharov-Kuznetov equation are thus found to behave approximately soliton-like. Some properties of collision process are interpreted in terms of the conservation laws.

Nonlinear hydromagnetic solitary waves in a collision-free plasma with isothermal electron pressure

Nonlinear hydromagnetic solitary waves propagating in a collision-free plasma of cold ions and isothermal electrons are investigated on the basis of hydrodynamical transport equations. Both cases of wave propagation along and across a magnetic field are studied in detail. In the parallel case, two types of solitary wave are found. The first type is an ordinary one which reduces to the zero-pressure wave in the cold limit and the other is a particular one which has no counterpart in the cold plasma. In the transverse case, only one type of solitary wave similar to that in the cold plasma is found. The effect of electron pressure shortens the relative widths of the waves in the parallel case (for the first type) while widens those in the transverse case. Magneto-ion-dynamics is formulated for the plasma under consideration, where the relation between the two-fluid model and the one-fluid magneto-hydrodynamics is established.

Coupled Van der Pol oscillators -- A model of excitatory and inhibitory neural interactions

A system of mutually coupled Van der Pol equations is derived from an extended version of the Wilson and Cowan model for the dynamics of a number of excitatory and inhibitory neural subsets. In the lowest order of approximation, interactions between excitatory and inhibitory subsets appear as linear elastic coupling, while those within and between excitatory and excitatory subsets appear as nonlinear frictional coupling. The case of two coupled oscillators is investigated by the method of averaging and the stability conditions for two mode oscillations are obtained. Internal resonance is also discussed briefly in the case of identical oscillators.