The Spectrum and Construction of Solutions

Abstract

In the preceding section we have seen that the lattice with exponential interaction admits many particular solutions, and the cyclic lattice has as many conserved quantities as the number of particles in the lattice. Thus we have obtained fundamental knowledge of nonlinear wave propagation. In this chapter, the equations of motion are written in matrix formalism, conserved quantities are derived from them, and it is shown that the initial value problem can be solved exactly for a infinite lattice. Further, we consider equations of motion dL/dt = BL — LB with generalized matrices B , and relation of the system discussed by Kac and Moerbeke to the lattice with exponential interaction. It is then shown that we can derive certain other solutions from a known solution (the Bäcklund transformation) by the use of a generating function for canonical transformation. Further, the methods, including that by Moser , of integration for a lattice of finite number of particles with exponential interaction are discussed. Finally, we study continuum approximation for the above subjects, deriving formulas for the Korteweg-de Vries equation related to a nonlinear continuous medium.