• Corpus ID: 235436211

On Soliton Resolution for a Lattice

@inproceedings{Hatzizisis2021OnSR,
  title={On Soliton Resolution for a Lattice},
  author={Nicholas Hatzizisis and Spyridon Kamvissis},
  year={2021}
}

• N. Hatzizisis, S. Kamvissis
• Published 15 June 2021
• Mathematics, Physics

The soliton resolution conjecture for evolution PDEs of dispersive type states (vaguely) that generic initial data of finite energy give rise asymptotically to a set of receding solitons and a decaying background radiation. In this letter, we investigate a possible extension of this conjecture to discrete lattices of the Fermi-Pasta-Ulam-Tsingou type (rather than PDEs) in two cases; the case with initial data of finite energy and a more general case with initial data that are a short range perturbation of a periodic function. In the second case, inspired by rigorous results on the Toda lattice, we suggest that the soliton resolution phenomenon is replaced by something somewhat more complicated: a short range perturbation of a periodic function actually gives rise to different phenomena in different regions. Apart from regions of (asymptotically) pure periodicity and regions of solitons in a periodic background, we also observe “modulated” regions of fast oscillations with slowly varying parameters like amplitude and phase. We have conducted some numerical calculations to investigate if this trichotomy (pure periodicity + solitons + modulated) persists for any discrete lattices of the Fermi-Pasta-Ulam-Tsingou type. For small perturbations of integrable lattices like the linear harmonic lattice, the Langmuir chain and the Toda lattice, this is true. But in general even chaotic phenomena can occur. 1. Historical Introduction and a Statement of the Soliton Resolution Conjecture in a Periodic Background A classical observation going back to the seminal discovery of Zabusky and Kruskal ([17]) states that a local (or “short range”) perturbation of the trivial stationary solution of a completely integrable soliton PDE (like KdV or NLS) or lattice (like Toda), eventually splits into a number of receeding solitons plus a (uniformly) decaying “background radiation”. The first complete description for the long time asymptotics of the KdV equation were given in [1]. Rigorous proofs can be constructed for any system solvable via the inverse scattering theory. Such proofs routinely employ the asymptotic analysis of the associated Riemann-Hilbert factorisation problems, at least in the case of one space dimension ([4], [6], [8], [9]), where the inverse scattering problem is equivalent to a Riemann-Hilbert factorisation problem in the complex plane. More recently, an even more daring conjecture has begun to take shape ([15], [2]): for any dispersive PDE of NLS or KdV type in any spatial dimension (!), generic initial data of bounded energy give rise asymptotically to a set of receeding solitons and a decaying background radiation. 2000 Mathematics Subject Classification. Primary 37K40, 37K45; Secondary 35Q15, 37K10.