Morikazu Toda

Vibration of a Chain with Nonlinear Interaction

Vibration of a chain of particles interacting by nonlinear force is investigated. Using a transformation exact solutions to the equation of motion are aimed at. For a special type of interaction potential of the form \begin{aligned} \phi(r){=}\frac{a}{b}e^{-br}+ar+\text{const.},\ (a,b{>}0) \end{aligned} exact solutions are actually obtained in terms of the Jacobian elliptic functions. It is shown that the system has N “normal modes”. Expansion due to vibration or “thermal expansion” of the chain is also discussed.

Wave Propagation in Anharmonic Lattices

Analytic solutions to the equation of motion in anharmonic one-dimensional lattice are given. Wave-trains and solitary-waves which propagate in the lattice are studied with reference to the limiting cases of the system of hard spheres and to the continnm limit.

The Exact N-Soliton Solution of the Korteweg-de Vries Equation

The exact N -soliton solution of the Korteweg-de Vries equation is obtained through the procedure suggested by Gardner, Greene, Kruskal and Miura. From this solution, it is shown that solutions are stable and behave like particles. The collisions are well described by the phase shifts. Explicit calculation of the phase shifts assures the conservation of the total phase shift. This fact turns out to be a special expression for the constant motion of he center of mass.

A Soliton and Two Solitons in an Exponential Lattice and Related Equations

Relations between a nonlinear (exponential) lattice, the Boussinesq equation and the Korteweg-de Vries equation are clarified and therefrom the exact solutions for the two-soliton state are given in each case for both the head-on and the overtaking collisions.

Bäcklund Transformation for the Exponential Lattice

A Backlund transformation associated with the equation of motion for an exponential lattice is found. It is shown that recursive application of the transformation provides an algebraic recursion formula for the solutions. Using the recursion formula, two-soliton solution is obtained and a method for constructing N -soliton solution is presented. It is also shown that the fundamental equations of inverse method and conservation laws can be derived from the transformation.

A Canonical Transformation for the Exponential Lattice

A canonical transformation which gives the relation between two solutions of an exponential lattice is presented. Using this relation a new solution can be obtained from a known solution. It is thus a discrete version of the Backlund transformation.

Interaction of Soliton with an Impurity in Nonlinear Lattice

Interaction of a soliton with a mass impurity in the exponential lattice has been studied by means of a computer simulation. We particularly investigate the decrease of the soliton amplitude due to an impurity. The present computation shows that the decrease of amplitude is proportional to ( A IN D ) 2 in the region of \(A_{\text{IN}}|D| \lesssim 1\), where A IN denotes the incident soliton amplitude and the mass defect D is defined by 1- Q , if Q denotes the mass ratio of the impurity to the host particle. The relation holds whether the impurity is lighter ( Q 1) than the host particle. The result agrees qualitatively with the perturbation theory based on the inverse method. In the case of a light impurity, there remains a localized mode near the impurity site, after a soliton has passed through the impurity. The frequency of the localized mode decreases as the amplitude of the mode increases, in a qualitative agreement with a simple mode-coupling theory.

The Classical Specific Heat of the Exponential Lattice

The classical specific heat of the exponential lattice at constant length is derived and its asymptotic behaviours are studied.

On the Theory of the Brownian Motion

The Brownian motion of an oscillator in a thermostat is considered assuming simple forms of interaction between them. No further assumption is made except that the thermostat is always in thermal equilibrium in itself. Solving the Liouville equation or its counterpart in quantum mechanics the long time evolution of the system is clarified. Thus the theory connects automatically the irreversible and the equilibrium behaviors of the system without any ad hoc assumption as in conventional theories. The results include, as a special case, the equation derived by Kramers and Chandrasekhar using the theory of stochastic processes. It is shown that the oscillating part of the distribution function or the density matrix plays an important role to which the peculiar way of damping of the oscillator is to be attributed.

Experiment on Soliton-Impurity Interaction in Nonlinear Lattice Using LC Circuit

Interaction of a soliton with a mass impurity has been experimentally investigated in a nonlinear electric circuit which is equivalent to an infinite or a semi-infinite exponential lattice. After a soliton collided with a light mass impurity, there remains a localized mode in the vicinity of the impurity. The frequency of the localized mode is above the cut-off frequency of the system, and increases as the mass ratio is decreased. In an infinite system, the soliton amplitude is decreased by a light mass impurity in proportion to the square of mass defect and of the amplitude. In the case of a heavy impurity, the transmitted wave breaks into multiple solitons. In a semi-inifinite system with a light impurity on a surface, the reflected signal is explained by a collisionless shock model.