Differential Difference Evolution Equations. Part 2: Darbouse Transformation for the Toda Lattice

摘要

The Toda lattice equation is represented in the form of the condition of compatibility of the system of linear equations corresponding to a nonhermitian Lax representation. The Darboux invariance of this linear system is defined and proved. Results are used to construct formulas for the solutions of the Toda lattice equation. These formulas involving determinants are applicable to an arbitrary initial solution of the Toda equation, for example, to a solution growing at infinity.