Asymptotic stability of the Toda m-soliton

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 B\"acklund} 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 $m$-solition solution and the Toda flow linearized about the zero solution, whose stability properties can be determined by explicit calculation.