Shingo Ishiwata

Higher Order Solution of Nonlinear Waves.: II. Shock Wave Described by Burgers Equation

A method of renormalization has been applied to a nonlinear differential equation which describes, in the continuum approximation, a weak shock wave propagating in a dissipative nonlinear LC circuit. In the lowest order of approximation, a well-known Burgers equation is derived. In the next order, a higher order equation which brings about correction to the Burgers equation is obtained. This equation contains secular terms and a term whose particular solution does not satisfy a boundary condition. A method of renormalization is introduced to eliminate such secular terms and an improper particular solution violating a boundary condition. It is shown that the renormalized solution can be described by the Burgers shock wave with the modifications in the width and the velocity.

Soliton in a Random System

Soliton propagation in a random medium has been numerically studied. As a model of the system, we consider a one-dimensional Toda lattice where two different mass particles are distributed randomly but the interaction potential is uniform. It is shown that the amplitude of a soliton decreases as n -1 ∼ n -1.2 ( n denotes the lattice point) in the course of propagation and that the damping is enhanced with soliton amplitude. The possibility of soliton damping with hysteresis is suggested.

Generation of a Vortex Ring with High Reynolds Number by an Exploding Wire in Water

A method is presented for generating a vortex ring in water by an exploding wire installed in a metal pipe. As a vapor bubble accompanied by wire explosion grows, water starts to flow with a high speed resulting in a formation of a vortex ring at the outlet of the pipe. The growth speed of a vapor bubble by the exploding wire in water is measured as 10∼20 m/s, in consistent with measurements of the translational velocity, 4∼5 m/s, of vortex ring and the rotational velocity, about 10 m/s, of fluid at the edge of vortex core. The Reynolds number based on the diameter of orifice, the velocity of vortex ring and the kinematic viscosity reaches the order of 10 5 . The magnitude of vorticity and the circulation are in the order of 10 3 /s and 10 -1 m 2 /s. These values are larger by about 10 times than those reported previously.

LOCALIZED STATE OF HARD CORE CHAIN AND CYCLOTOMIC POLYNOMIAL : HARD CORE LIMIT OF DIATOMIC TODA LATTICE

Abstract We consider a one-dimensional classical hard core chain with different alternating masses m and M . For a certain mass ratio M m , there exists a localized state which consists of three adjacent particles and propagates. Then its mass ratio is given by a polynomial with integer coefficients, which turns out to be the cyclotomic polynomial. We can derived the complete series of such mass ratios.

Shock Wave in a Nonlinear LC Circuit

A shock wave in a nonlinear LC circuit with dissipation has been investigated theoretically and experimentally. A shock wave solution of a stationary waveform traveling with a constant velocity and a higher-order correction have been obtained theoretically. In the experiment, it has been demonstrated that a shock wave with a constant amplitude propagates stably in a nonlinear LC circuit with dissipation. The velocity and width of the shock wave have been observed to be consistent with the theoretical prediction, provided that the amplitude is small enough to be consistent with the theoretical approximation.

Experiment on Solitary Wave in a Symmetric Electric Circuit

A new type of nonlinear LC circuit has been introduced in order to investigate experimentally the propagation and interaction of solitary waves which is symmetric with respect to the inversion of voltage. A nonlinear capacitor in the shunt branch is composed of two diodes connected in series but inversely with each other. Both positive and negative solitary waves are stably propagated in the circuit. The velocity and width of a solitary wave deviate from the theoretical prediction, as the amplitude increases. This discrepancy comes from the fact that the dynamic response of nonlinear diodes is different from the static one. The phase shift resulting from a head-on collision of two solitary waves has also been observed.

Nonlinear waves in mass-spring systems with velocity-dependent friction

Abstract The role of velocity-dependent friction in stick–slip model is numerically examined by introducing a modified model. Contrary to monotonous motion in linear systems without sliding friction or with constant sliding friction, a motion full of variety appears in a nonlinear system with velocity-dependent friction. Linear and nonlinear analyses of stick–slip motion are given theoretically.

Formation of a large-scale structure in a nonlinear dissipative system

Dynamic evolution of a small-scale structure to a large-scale structure is studied by a nonlinear dissipative LC ladder circuit. The circuit is described by a Burgers equation in the lowest order. We confirmed that the observed propagation of a shock wave and the confluence of shock waves are well described by the theory. A single hump wave is used to model a small-scale structure. When two single-hump waves with different amplitudes are applied to the circuit, they evolve into a large-scale structure in the course of propagation. Such a formation of a large-scale structure is unique in a nonlinear dissipative system, while the nonlinear dispersive system is known to preserve the nature of small-scale structures even after their interaction as known for solitons.

Experiment on Shock Wave in a Random LC Circuit

Shock wave in a random system has been investigated theoretically and experimentally. A nonlinear LC ladder circuit with random inhomogeneity of linear inductor in series branch has been studied. The circuit is equivalent to a random Toda lattice. It is shown theoretically that wave propagation in the system is described by the Burgers equation in the case of strong randomness with weak nonlinearity. The transition from an oscillatory shock wave to a shock wave has been observed on a random LC ladder circuit experimentally. The velocity and width of shock wave agree with the theoretical prediction from the Burgers equation in a region of strong randomness.

Self-modulation and self-focusing of ion wave in two-electron-temperature plasma

Self-modulation and self-focusing of ion wave in a two-electron-temperature plasma composed of cold and hot electrons and positive ion have been theoretically investigated based on the nonlinear Schrodinger equation describing the wave. For the ion wave of large wavenumber, the modulational instability occurs irrespective of temperature and density ratios of the hot electron to the cold electron. If the temperature ratio is larger than ∼10, the ion wave of any wavenumber turns out to be unstable modulationally at a critical density ratio. The self-focusing of ion wave, on the other hand, scarcely occurs. Between two critical densities, however, the ion wave of small wavenumber becomes unstable with respect to transverse modulation if the temperature ratio is superior to ∼10.