Cnoidal waves on Fermi-Pasta-Ulam lattices. (English) Zbl 1382.37080
Summary: We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion
\[
\ddot{q}_n=V'(q_{n+1}-q_n)-V'(q_n-q_{n-1})
\]
with generic nearest-neighbour potential \(V\). We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength\(^{-3}\) suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of G. Friesecke and R. L. Pego[ Nonlinearity 12, No. 6, 1601–1627 (1999; Zbl 0962.82015)] to a periodic setting and the spectral theory of the periodic Schrödinger operator with KdV cnoidal wave potential.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 35C08 | Soliton solutions |
| 35Q51 | Soliton equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 70H12 | Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics |
| 82B28 | Renormalization group methods in equilibrium statistical mechanics |
Citations:
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