Loss of stability of a solitary wave through exciting a cnoidal wave on a Fermi-Pasta-Ulam ring (original) (raw)

Solitary Wave Interactions with Continuous Waves

International Journal of Bifurcation and Chaos, 2006

Solitary wave propagation under interaction with continuous waves is studied in the context of the Nonlinear Schrödinger Equation. An analytical approach, based on the conserved quantities of the wave evolution, is used to study transverse velocity variations for the case of nonzero transverse wavenumber difference between the solitary and continuous waves. The method is applicable for any number of transverse dimensions and any kind of nonlinearity. Moreover, the presence of a coherent continuous background is shown to be responsible for the creation of solitary rings and spirals, under interaction with solitary structures with nonzero topological charge. Numerical simulations for specific cases were used to confirm the analytical results.

Polychromatic Solitary Waves in a Periodic and Nonlinear Maxwell System

SIAM Journal on Applied Dynamical Systems, 2012

We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite timescales , the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations, the so-called extended nonlinear coupled mode equations, or xNLCME. Numerical studies support the existence of long-lived spatially localized coherent structures, featuring a slowly varying envelope and a train of carrier shocks. In this paper we explore, by analytical, asymptotic, and numerical methods, the existence and properties of spatially localized structures of the xNLCME system for the case where the refractive index profile consists of a periodic array of Dirac delta functions. We consider, in particular, the limit of small amplitude solutions with frequencies near a spectral band edge. In this case, stationary xNLCME is well approximated by an infinite system of coupled, stationary, nonlinear Schrödinger (NLS) equations, the extended nonlinear Schrödinger system, xNLS. We embed xNLS in a one-parameter family of equations, xNLS , which interpolates between infinitely many decoupled NLS equations (= 0) and xNLS (= 1). Using bifurcation methods we show existence of solutions for a range of ∈ (− 0, 0) and, by a numerical continuation method, establish the continuation of certain branches all the way to = 1. Finally, we perform time-dependent simulations of a truncated xNLCME and find the small-amplitude near-band edge gap solitons to be robust to both numerical errors and the NLS approximation.

Propagation dynamics on the Fermi-Pasta-Ulam lattices

Frontiers of Physics, 2013

The spatiotemporal propagation of a momentum excitation on the finite Fermi-Pasta-Ulam lattices is investigated. The competition between the solitary wave and phonons gives rise to interesting propagation behaviors. For a moderate nonlinearity, the initially excited pulse may propagate coherently along the lattice for a long time in a solitary wave manner accompanied by phonon tails. The lifetime of the long-transient propagation state exhibits a sensitivity to the nonlinear parameter. The solitary wave decays exponentially during the final loss of stability, and the decay rate varying with the nonlinear parameter exhibits two different scaling laws. This decay is found to be related to the largest Lyapunov exponent of the corresponding Hamiltonian system, which manifests a transition from weak to strong chaos. The mean-free-path of the solitary waves is estimated in the strong chaos regime, which may be helpful to understand the origin of anomalous conductivity in the Fermi-Pasta-Ulam lattice.

Nonlinear Waves: Classical and Quantum Aspects

NATO Science Series II: Mathematics, Physics and Chemistry, 2005

Towards algebro-geometric integration of the Gross-Pitaevskii equation V.Z. Enolskii 1 Introduction 2 Utilization of the Schrödinger equation 3 Solutions in terms of hyperelliptic functions 4 Two component Gross-Pitaevskii equation and the Manakov system 9 On modeling adiabatic N-soliton interactions V.S. Gerdjikov 1 Introduction 2 N-soliton trains of the NLS and HNLS equations 3 N-soliton trains of the MNLS equation 4 The importance of the CTC model 18 5 Dynamical regimes of the HNLS soliton trains 6 The perturbed NLS and perturbed CTC 6.1 Second order dispersion and nonlinear gain 6.2 Quadratic and periodic potentials 7 Analysis of the Perturbed CTC 8 Discussion Dynamical stabilization of nonlinear waves F. Abdullaev 1 Introduction 2 Dynamics of solitons in BEC with rapidly oscillating trap 3 Stable two dimensional bright soliton under Feschbach resonance management 4 Stable two dimensional dispersion-managed soliton 5 Conclusions v vi

Generation and intrinsic dynamics of ring dark solitary waves

Applied Physics B: Lasers and Optics, 1997

We present experimental results on the evolution of ring dark solitary waves generated by computersynthesized holograms. The data obtained and the detailed comparative numerical simulations show that this approach ensures reproduction of the correct intensity and phase portrait of the dark wave at the entrance of the nonlinear medium. The transverse dynamics at both even and odd initial conditions is studied and compared with the theory and simulations.

Secondary solitary wave formation in systems with generalized Hertz interactions

Physical Review E, 2002

We consider a chain of monodisperse elastic grains of radius R where the grains are barely in contact. The grains repel upon contact via the Hertz-type potential, Vϰ␦ n , nϾ2, where ␦у0, is the grain-grain overlap, ␦ϵ2RϪ(u iϩ1 Ϫu i), where u i denotes the displacement of grain i from its original equilibrium position. This being a computational study, we consider n to be arbitrary. Our dynamical simulations build on several earlier studies by Nesterenko, Coste, and Sen and co-workers that have shown that an impulse propagates as a solitary wave of fixed spatial extent, ϱϽL(n)Ͻ1, through a chain of grains. Here, we develop on a recent study by Manciu, Sen, and Hurd ͓Phys. Rev. E 63, 016614 ͑2001͔͒ that shows that colliding solitary waves in the chains of interest spawn a well-defined hierarchy of multiple secondary solitary waves ͑SSWs͒ that carry ϳ0.5% or less of the energy of the original solitary waves. We show that the emergence of SSWs is a complex process where nonlinear forces and the discreteness of the grains lead to the partitioning of the available energy into hierarchies of SSWs. The process of formation of SSWs involves length scales and time scales that are controlled by the strength of the nonlinearity in the system. To the best of our knowledge, there is no formal theory that describes the dynamics associated with the formation of SSWs. Calculations for cases where the Hertz-type potential can be symmetric in the overlap parameter ␦, i.e., where ␦ can be both positive and negative, suggest that the formation of secondary solitary waves may be a fundamental property of certain discrete, nonlinear systems.

Dissipative cnoidal waves (Turing rolls) and the soliton limit in microring resonators

Optica, 2019

Single solitons are a special limit of more general waveforms commonly referred to as cnoidal waves or Turing rolls. We theoretically and computationally investigate the stability and accessibility of cnoidal waves in microresonators. We show that they are robust and, in contrast to single solitons, can be easily and deterministically accessed in most cases. Their bandwidth can be comparable to single solitons, in which limit they are effectively a periodic train of solitons and correspond to a frequency comb with increased power. We comprehensively explore the three-dimensional parameter space that consists of detuning, pump amplitude, and mode circumference in order to determine where stable solutions exist. To carry out this task, we use a unique set of computational tools based on dynamical system theory that allow us to rapidly and accurately determine the stable region for each cnoidal wave periodicity and to find the instability mechanisms and their time scales. Finally, we focus on the soliton limit, and we show that the stable region for single solitons almost completely overlaps the stable region for both continuous waves and several higher-periodicity cnoidal waves that are effectively multiple soliton trains. This result explains in part why it is difficult to access single solitons deterministically.

Origin of the persistent oscillations of solitary waves in nonlinear quadratic media

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1996

We reveal the origin of the persistent oscillations of solitary waves in a quadratically nonlinear medium. It is found that the oscillations are closely correlated to a nontrivial discrete eigenmode of the corresponding linear eigenvalue problem. In addition to this discrete eigenmode a quasibound mode, weakly coupled to the continuous spectrum, is identified. This gives rise to a long-living beating of the solitary wave amplitude.

Solitary waves in excitable systems with cross-diusion

2000

We consider a FitzHugh-Nagumo system of equations where the traditional diffusion terms are replaced with linear cross-diffusion of components. This system describes solitary waves that have unusual form and are capable of quasi-soliton interaction. This is different from the classical FitzHugh-Nagumo system with self-diffusion, but similar to a predator-prey model with taxis of populations on each other's gradient which we considered earlier. We study these waves by numerical simulations and also present an analytical theory, based on the asymptotic behaviour which arises when the local dynamics of the inhibitor field are much slower than those of the activator field.