Counter-propagating two-soliton solutions in the Fermi–Pasta–Ulam lattice

Abstract

We study the interaction of small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbour interaction potential. We establish global-in-time existence and stability of counterpropagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intermediate values of time these solutions describe the interaction of two counter-propagating pulses. These solutions are stable with respect to perturbations in 2 and asymptotically stable with respect to perturbations which decay exponentially at spatial ±∞.

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

“…We note that this improves on the estimate of k 3.5 which appears e.g. in [47,52]. [A number of details towards making this argument rigorous are presented in the Appendix].…”

“…It is known, both formally [46] and rigorously [47] (on long but finite time scales) that KdV approximates FPU α-type lattices for smallamplitude, long-wave, low-energy initial data. This fact has been used in the mathematical literature to determine the shape [48] and dynamical stability [49][50][51] of solitary waves and even of their interactions [52]. We remark that the above referenced remarks in the mathematical literature are valid "for ǫ sufficiently small", where ǫ is a parameter characterizing the amplitude and inverse width, as well as speed of the waves above the medium's sound speed.…”

Solitary waves propagation described by Korteweg-de Vries equation in the granular chain with initial prestress

“…[35][36][37] The KdV equation describes approximately FPU-type lattices for small-amplitude, long-wave and low-energy initial data, which has been used in the mathematical literature to determine the shape and dynamical stability of solitary waves and even their interactions. [37][38][39][40][41] Following the Nesterenko's works, 2,10,23 we further study the solitary wave when the strong external force acts on the granular chain. We also study the problem under the long-wavelength approximation.…”

N-Soliton States of the Fermi–Pasta–Ulam Lattices

“…See also [21] and [17] for results of nonlinear Schrödinger equations. Using the method of [5,6,7] as well as a decomposition argument of [22], Hoffman and Wayne [10] have studied a head-on collision of two solitary waves that propagate to the opposite directions. They give a rigorous proof of Schneider and Wayne [25] and show that the shapes of the two solitary waves are stable to perturbations.…”

“…They also prove the existence of solutions that converge to a sum of counterpropagating solitary waves (see [11]). If solitary waves move in the same direction, the interaction through their tails is effective for a longer period, and we cannot rely on the compactness arguments of [10,11].…”