Asymptotic Stability of N-solitons of the Fpu Lattices

We study stability of N-soliton solutions of the FPU lattice equation. Solitary wave solutions of FPU cannot be characterized as a critical point of conservation laws due to the lack of infinitesimal invariance in the spatial variable. In place of standard variational arguments for Hamiltonian systems, we use an exponential stability property of the linearized FPU equation in a weighted space which is biased in the direction of motion. The dispersion of the linearized FPU equation balances the potential term for low frequencies, whereas the dispersion is superior for high frequencies. We approximate the low frequency part of a solution of the linearized FPU equation by a solution to the linearized KdV equation around an N-soliton. We prove an exponential stability property of the linearized KdV equation around N-solitons by using the linearized Bäcklund transformation and use the result to analyze the linearized FPU equation.