Tsunehiko Kakutani

Solitary Waves on a Two-Layer Fluid

This paper deals with weakly nonlinear long gravity waves on a stably stratified two-layer fluid. By using the reductive perturbation method, it is found that the fast mode is always governed by a Korteweg-de Vries (K-dV) equation whose coefficients depend on the thickness ratio and the density ratio. On the other hand, the slow mode is also governed, in general, by another K-dV equation except near and at the critical thickness ratio. At the critical thickness ratio, however, the slow mode is shown to be governed by a modified K-dV equation with cubic nonlinearity and near the critical thickness ratio it is governed by an equation of a combined form of the K-dV and modified K-dV equation. Steady solitary wave solutions to these equations are investigated in detail. A special solution representing a dispersive bore or hydraulic jump is also found.

Reductive Perturbation Method in Nonlinear Wave Propagation II. Application to Hydromagnetic Waves in Cold Plasma

Silicon diodes and silicon controlled rectifiers are provided with reduced minority carrier lifetimes without a significant increase in forward voltage drop or reverse leakage current by means of a gadolinium dopant which is diffused into the semiconductor wafers after first being applied on one face thereof by means of a conventional sputtering process. Gadolinium is sputtered onto the base side of the rectifiers from a foil of the pure metal in a vacuum. The wafers are then placed in a furnace at a temperature above 820 DEG C where the diffusion process takes place.

Krylov‐Bogoliubov‐Mitropolsky method for nonlinear wave modulation

The Krylov‐Bogoliubov‐Mitropolsky perturbation method is applied to systems of nonlinear dispersive waves including plasma waves such as ion‐acoustic, magneto‐acoustic, and electron plasma waves. It is found that long time slow modulation of the complex wave amplitude can be described by the nonlinear Schrodinger equation for a very wide class of nonlinear dispersive waves.

Effect of an Uneven Bottom on Gravity Waves

Effect of an uneven bottom on the long gravity waves is investigated by using a nonlinear perturbation method. The following simple nonlinear equation is obtained, to the lowest order of perturbation, for the free-surface elevation h (1) : \(\frac{\partial h^{(1)}}{\partial\eta}+c_{1}h^{(1)}\frac{\partial h^{(1)}}{\partial\xi}+c_{2}\frac{\partial^{3}h^{(1)}}{\partial\xi^{3}}-c_{3}\frac{\mathrm{d}B}{\mathrm{d}\eta}h^{(1)}{=}0\), where ξ and η are stretched space-time coordinates, and the coefficients c 1 , c 2 and c 3 are all functions of the bottom surface B (η). If the bottom is flat (d B /dη=0), the equation reduces to the well-known Korteweg-de Vries equation. It is interesting to note that, due to the presence of the last term, damping or instability may occur depending upon the sign of the local slope, d B /dη, of the bottom. A comment is given on the relationship between the conventional shallow-water (or long-wave) theory and the present analysis.

Effect of Viscosity on Long Gravity Waves

The effect of viscosity is examined on long gravity waves of small but finite amplitude. The reductive perturbation method combined with the usual boundary layer theory reveals that the inviscid Korteweg-de Vries equation is not affected by the viscosity if O(α -5 )< R , where R is the Reynolds number and α(≪1) the wavenumber. For O(α -1 )< R ≤O(α -5 ), the effect of viscosity modifies the Korteweg-de Vries equation and yields new types of equation. On the other hand, for R <O(α -1 ), the complex phase velocity becomes purely imaginary and the free surface is found to be governed by a nonlinear diffusion equation which was first obtained by Nakaya.

Nonlinear Capillary Waves on the Surface of Liquid Column

Weakly nonlinear capillary waves on a liquid column of circular cross-section are investigated on the basis of the derivative expansion method. It is found that the complex amplitude of a quasi-monochro-matic travelling wave can be described by a nonlinear Schrodinger equation in a frame of reference moving with the group velocity. The known property of this equation reveals that wave trains of constant amplitude are modulationally unstable, which suggests a new possibility of the break-up of column. On the other hand, the complex amplitude of a quasi-monochromatic standing wave near the cut-off is governed by another type of nonlinear Schrodinger equation in which the role of the time and that of the space are interchanged. This equation makes it possible to estimate the nonlinear effect on the cut-off wave-number.

Note on Higher Order Terms in Reductive Perturbation Theory

It is remarked that the second order correction given by Ichikawa et al. for a single ion acoustic K-dV (Korteweg-de Vries) soliton contains a secular term. To eliminate the secularity, the method of multiple scales combined with the reductive perturbation method is proposed. It is then found that not only the second order correction is modified so as to be secular-free but also the phase factor of the lowest K-dV soliton suffers a modification proportional to its amplitude.

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.

Hydromagnetic Flow due to a Rotating Disk

The steady flow of an incompressible, viscous and electrically conducting fluid due to a rotating disk of infinite radius is investigated in the presence of a transverse magnetic field. This is an extension of the well-known Karmans problem to the hydromagnetic case. It is assumed that the magnetic Prandtl number of the fluid is so small that the electric currents flowing in the fluid do not affect the magnetic field. It is found that the magnitudes of the radial and the axial components of the velocity decrease rapidly as the hydromagnetic interaction parameter N increases, while the velocity gradient of the circumferential component at the disk increases. Therefore, the torque due to viscous friction acting on the disk increases with increasing N . It is also found that both of the displacement-thickness and the angle of yaw at large distances from the disk decrease with increasing N .

Marginal State of Modulational Instability —Note on Benjamin-Feir Instability—

This paper deals with the marginal state of modulational instability of a nonlinear wave which is governed by a cubic nonlinear Schrodinger (shortly NS) equation far from the marginal state. Considering the gravity water wave as a typical example for which the coefficient of the nonlinear term of the NS equation does vanish at the marginal state, we reduce a new governing equation near the marginal state instead of the NS equation. Based on this new equation, the Benjamin-Feir instability of the Stokes wave is found to be enhanced by the higher order nonlinearity.