2012/08/14 by Gero Friesecke, Friesecke, Gero, Alice Mikikits-Leitner +1
Mathematics · Physics and Astronomy · #35Q51 #35Q53 #70F #70H12 #82B28 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #msc:35Q51 #msc:35Q53 #msc:70F #msc:70H12 #msc:82B28 #nlin.SI
paper · pdf · doi:10.48550/arxiv.1208.2805
25 pages, 3 figures
arxiv created 2012/08/14 · arxiv updated 2012/08/15
We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion [qn = V'(qn+1-qn) - V'(qn-qn-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 Friesecke and Pego (1999) to a periodic setting and the spectral theory of the periodic Schrödinger operator with KdV cnoidal wave potential.