We prove that multi-soliton solutions of the Toda lattice are both linearly and nonlinearly stable. Our proof uses neither the inverse spectral method nor the Lax pair of the model but instead studies the linearization of the Ba ̈cklund transformation which links the $(m-1)$-soliton solution to the m-soliton solution. We use this to construct a conjugation between the Toda flow linearized about an m-soliton solution and the Toda flow linearized about the zero solution, whose stability properties can be determined by explicit calculation. We also show that this result can be used as a starting point to give a simple proof of the stability of solitary waves in more general FPU type lattices.

This is joint work with Nick Benes and Aaron Hoffman.