The Backlund Transform, first developed in the context of differential geometry, has been classically used to obtain multi-soliton states in completely integrable infinite dimensional dynamical systems. It has recently been used to study the stability of these special solutions. We offer here a dynamical perspective on the Backlund Transform, prove an abstract orbital stability theorem, and demonstrate its utility by applying it to the sine-Gordon equation and the Toda lattice.
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 Backlund} 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.
By combining results of Mizumachi on the stability of solitons for the Toda lattice with a simple rescaling and a careful control of the KdV limit we give a simple proof that small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam (FPU) model are linearly stable and hence by the results of Friesecke and Pego that they are also nonlinearly, asymptotically stable.
We prove the existence of asymptotic two-soliton states in the Fermi-Pasta-Ulam model with general interaction potential. That is, we exhibit solutions whose difference in $ell^2$ from the linear superposition of two solitary waves goes to zero as time goes to infinity.
We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intemediate values of time these solutions describe the interaction of two counterpropagating pulses. These solutions are stable with respect to perturbations in $ell^2$ and asymptotically stable with respect to perturbations which decay exponentially at spatial $pm infty$.}
