Cnoidal Waves on Fermi-Pasta-Ulam Lattices

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 Friesecke and Pego (1999) to a periodic setting and the spectral theory of the periodic Schr\"odinger operator with KdV cnoidal wave potential.Comment: 25 pages, 3 figure

KdV soliton interactions: a tropical view

Via a "tropical limit" (Maslov dequantization), Korteweg-deVries (KdV) solitons correspond to piecewise linear graphs in two-dimensional space-time. We explore this limit.Comment: 10 pages, 4 figures, conference "Physics and Mathematics of Nonlinear Phenomena 2013

Perfectly invisible PT\mathcal{PT}-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry

We investigate a special class of the $\mathcal{PT}$-symmetric quantum models being perfectly invisible zero-gap systems with a unique bound state at the very edge of continuous spectrum of scattering states. The family includes the $\mathcal{PT}$-regularized two particle Calogero systems (conformal quantum mechanics models of de Alfaro-Fubini-Furlan) and their rational extensions whose potentials satisfy equations of the KdV hierarchy and exhibit, particularly, a behaviour typical for extreme waves. We show that the two simplest Hamiltonians from the Calogero subfamily determine the fluctuation spectra around the $\mathcal{PT}$-regularized kinks arising as traveling waves in the field-theoretical Liouville and $SU(3)$ conformal Toda systems. Peculiar properties of the quantum systems are reflected in the associated exotic nonlinear supersymmetry in the unbroken or partially broken phases. The conventional $\mathcal{N}=2$ supersymmetry is extended here to the $\mathcal{N}=4$ nonlinear supersymmetry that involves two bosonic generators composed from Lax-Novikov integrals of the subsystems, one of which is the central charge of the superalgebra. Jordan states are shown to play an essential role in the construction.Comment: 33 pages; comments and refs added, version to appear in JHE

Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials

We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order $\alpha >1$. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyze the propagation of localized waves when $\alpha$ is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic KdV equation, and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with H\"older-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When $\alpha \rightarrow 1^+$, we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed, and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile

Universal quantum Hawking evaporation of integrable two-dimensional solitons

We show that any soliton solution of an arbitrary two-dimensional integrable equation has the potential to eventually evaporate and emit the exact analogue of Hawking radiation from black holes. From the AKNS matrix formulation of integrability, we show that it is possible to associate a real spacetime metric tensor which defines a curved surface, perceived by the classical and quantum fluctuations propagating on the soliton. By defining proper scalar invariants of the associated Riemannian geometry, and introducing the conformal anomaly, we are able to determine the Hawking temperatures and entropies of the fundamental solitons of the nonlinear Schroedinger, KdV and sine-Gordon equations. The mechanism advanced here is simple, completely universal and can be applied to all integrable equations in two dimensions, and is easily applicable to a large class of black holes of any dimensionality, opening up totally new windows on the quantum mechanics of solitons and their deep connections with black hole physics