Abstract
An extensive account of the exact theory of wave propagation in the one-dimensional nonlinear lattice with exponential interaction between nearest neighbor particles is given. A brief review of the development, useful particular solutions, the general method of solving the equations of motion and the relation between the discrete lattice and the con tinuous Korteweg-de Vries system are given, with some future aspect of the problems of nonlinear lattices. In this article the author wishes to present the problems related to wave propagation in nonlinear lattices with special emphasis on the one-dimensional lattice of particles with the nearest neighbor interaction of the exponential type (the exp-lattice or the Toda lattice). The development of the theory of the nonlinear lattice will be briefly reviewed in §§l and 2, and the charac teristic features of waves will be presented in §3 by showing particular solutions to the equations of motion for the lattice. In §§ 4 and 5, general theory of the exp-lattice will be followed by the general method for solving the equations of motion. If one takes the continuum limit under certain restrictions, one sees that the time evolution of the wave can be approximated by a partial differential equation which was found by Korteweg and de Vries to describe shallow water waves (KdV equation). In §6 the relation between the exp-lattice and the KdV equation will be discussed. In the final section, some remarks on further problems will be presented. 1-l. Nonlz'near lattice The equations of motion for the one-dimensional lattice of particles with nearest neighbor interaction can be written, when no external force is present, as
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