Exact Solutions to the Toda Lattice Equation

The Toda lattice equation is a nonlinear partial differential-difference equation, and it is used to model the behavior of molecules in a one-dimensional crystal lattice in solid state physics. In the Toda lattice model the neighboring molecules interact with each other through exponentially decreasing forces. The Toda lattice equation is an integrable nonlinear evolution equation and it admits exact solutions known as solitary wave solutions, or solitons for short. In an N-soliton solution to the Toda lattice equation, there are N individual solitons interacting with each other when they are close, and behaving as individual entities without much interaction when they are far apart from each other. By using a recently developed method we construct N-soliton solutions to the Toda lattice equation in terms of three constant matrices A,B,C, with respective sizes N x N, N x 1 , 1 x N, for any positive integer N. The formula for the N-soliton solution is then expressed explicitly in terms of A,B,C and an N x N matrix P satisfying the auxiliary matrix equation P-APA=BC. In our research we analyze the solvability of the auxiliary matrix equation, study the relationship between the physical properties of the individual solitons and the eigenvalues of the matrix A, and animate soliton solutions by using the software Mathematica.