Fermi-Pasta-Ulam Problem Research Articles

Embedded exponential Runge–Kutta–Nyström methods for highly oscillatory Hamiltonian systems

Exponential/extended Runge–Kutta–Nyström (ERKN) methods have been validated by numerous theoretical and numerical results to be more efficient and suitable than classical Runge–Kutta–Nyström (RKN) methods in dealing with highly oscillatory Hamiltonian systems. Once numerical solutions are required to achieve a prescribed local error tolerance for practical problems, the embedded ERKN pairs with adaptive stepsize are needed. Given the higher efficiency of high-order methods than low-order methods, this paper focuses on high-order embedded ERKN pairs. Using the particular mapping from RKN methods into ERKN methods, we establish an approach to constructing high-order embedded ERKN pairs. By diagonalizing the frequency matrix, an implementation algorithm with less calculation amount than the direct calculation procedure is proposed for high-dimensional problems. In addition, the dispersion and dissipation of the ERKN pairs ERKN4(3) and ERKN8(6) are analyzed. Furthermore, the application of ERKN pairs to an important highly oscillatory problem, i.e., the wave equation prescribed with different boundary conditions is discussed. For the periodic boundary condition, a special implementation algorithm incorporated with FFT techniques is presented, which decreases the calculation amount in one time step from O(N2) to O(Nlog⁡N). Finally, numerical results with the FPU problem, the Klein–Gordon equation in the nonrelativistic limit regime, and a two-dimensional wave equation demonstrate the superiority of ERKN pairs over classical RKN pairs.

When discrete fronts and pulses form a single family: FPU chain with hardening-softening springs

We consider a version of the classical Hamiltonian FPU (Fermi–Pasta–Ulam) problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink–antikink bundle. Direct numerical simulations and Floquet analysis of linear stability suggest that all of the obtained solutions are effectively stable. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem both qualitatively and quantitatively.

Feedback resonance in Fermi–Pasta–Ulam chain

A controlled version of the celebrated Fermi–Pasta–Ulam problem is introduced. The feedback control algorithm based on Speed-gradient approach is proposed and analyzed by extensive computer simulation. It is demonstrated that the feedback controlled system tends to approximate equipartition state much faster than it happens in the open loop (classical) system. Apparently a substantial change in system behavior is achieved with a small intensity control: less than 0.5% of the total system energy. Such a behavior has been observed and analyzed previously for oscillators with one degree-of-freedom (DOF) under the name of feedback resonance (Fradkov A.L., Physica D, 128(1999), 159-168. The results of this paper suggest that feedback resonance phenomenon may be observed in multi-DOF oscillator chains, particularly in FPU chain.

Dual-wavelength pumped latticed Fermi–Pasta–Ulam recurrences in nonlinear Schrödinger equation

We show that the nonlinear stage of the dual-wavelength pumped modulation instability (MI) in nonlinear Schrödinger equation (NLSE) can be effectively analyzed by mode truncation methods. The resulting complicated heteroclinic structure of instability unveils all possible dynamic trajectories of nonlinear waves. Significantly, the latticed-Fermi–Pasta–Ulam recurrences on the modulated-wave background in NLSE are also investigated and their dynamic trajectories run along the Hamiltonian contours of the heteroclinic structure. It is demonstrated that there has much richer dynamic behavior, in contrast to the nonlinear waves reported before. This novel nonlinear wave promises to inject new vitality into the study of MI.

Numerical simulation of modified nonlinear Schrodinger equation and turbulence generation

This article presents a numerical model to study wave turbulence in fluids. The model equation is derived by incorporating energy conservation (along with the usual fluid equations of a compressible flow), and the source of nonlinearity is the rise in temperature due to the acoustic wave's high amplitude. The nonlinear Schrödinger (NLS) and modified nonlinear Schrödinger (MNLS) equations have been derived and then solved numerically. A numerical simulation of the MNLS equation is used for investigating the turbulence generation and a semi-analytical method to understand the physics of localized structures of nonlinear waves. The numerical simulation is based on a pseudo-spectral approach to resolve spatial regimes, a finite difference method for temporal evolution. The results show a periodic pattern viz. Fermi–Pasta–Ulam (FPU) recurrence for NLS, while turbulence generation breaks down the FPU recurrence in MNLS. The turbulent power spectrum in the inertial sub-range approximately follows the Kolmogorov–Zakharov scaling (∼k−1.2).

Nonlinear energy localisation in a model of plane metamaterial

Applying the concepts of nonlinear normal modes and limiting phase trajectories introduced by Manevitch in Manevitch (Arch Appl Mech 77:301–312, 2007) to a two-dimensional mass–spring system, the authors propose a generalised method to tune a plane metamaterial and get the desirable resonant behaviour at short wavelengths. Indeed, the account of nonlinear coupling between the oscillators enables the localisation of energy leading the origin of a bandgap at short wavelengths regardless the existence of external disturbances. Moreover, further restrictions on the modes amplitude allow the observation of Fermi–Pasta–Ulam–Tsingou recurrence and super-recurrence in the two-dimensional metamaterial. These findings can open the way to further research in order to improve efficiency and performance of resonant metamaterials.

Phase space topology of four-wave mixing reconstructed by a neural network.

The dynamics of ideal four-wave mixing in optical fiber is reconstructed by taking advantage of the combination of experimental measurements together with supervised machine learning strategies. The training data consist of power-dependent spectral phase and amplitude recorded at the output of a short fiber segment. The neural network is shown to be able to accurately predict the nonlinear dynamics over tens of kilometers, and to retrieve the main features of the phase space topology including multiple Fermi-Pasta-Ulam recurrence cycles and the system separatrix boundary.

Fermi–Pasta–Ulam–Tsingou recurrence in two-core optical fibers

The Fermi–Pasta–Ulam–Tsingou recurrence (FPUT) refers to the property of a multi-mode nonlinear system to return to the initial states after complex stages of evolution. FPUT in two-core optical fibers (TCFs) is studied by taking symmetric continuous waves as initial conditions. Analytically the dynamics is governed by linearly coupled nonlinear Schrödinger equations. A ‘cascading mechanism’ is investigated theoretically to elucidate the FPUT dynamics. Higher-order modes grow in ‘locked step’ with the fundamental one until a breather emerges, which results in a triangular spectrum, in consistency with observations in many experiments. In terms of optical physics, FPUT with in-phase perturbations is identical to that in single-core fibers (SCFs). For quadrature-phase perturbations (those with π/2 phase shifts between the two cores), symmetry is broken and the FPUT dynamics is rich. For normal dispersion, FPUT otherwise absent in SCFs arises for TCFs. The optical fields change from a phase-shifted FPUT pattern to an ‘in-phase’ pattern. The number of FPUT cycles eventually saturates. For anomalous dispersion, in-phase FPUT and ‘repulsion’ between wave trains in the two cores are observed. Preservation of the phase difference between waves in the two cores depends on the dispersion regime. An estimate based on modulation instability provides a formula for predicting the first occurrence of FPUT. These results enhance the understanding of the physics of TCFs, and help the evaluation of the performance of long-distance communication based on multi-core fibers.

Transition fronts and their universality classes.

Steadily moving transition (switching) fronts, associated with local transformation, symmetry breaking, or collapse, are among the most important dynamic coherent structures. The nonlinear mechanical waves of this type play a major role in many modern applications involving the transmission of mechanical information in systems ranging from crystal lattices and metamaterials to macroscopic civil engineering structures. While many different classes of such dynamic fronts are known, the interrelation between them remains obscure. Here we consider a minimal prototypical mechanical system, the Fermi-Pasta-Ulam (FPU) chain with piecewise linear nonlinearity, and show that there are exactly three distinct classes of switching fronts, which differ fundamentally in how (and whether) they produce and transport oscillations. The fact that all three types of fronts could be obtained as explicit Wiener-Hopf solutions of the same discrete FPU problem allows one to identify the exact mathematical origin of the particular features of each class. To make the underlying Hamiltonian dynamics analytically transparent, we construct a minimal quasicontinuum approximation of the FPU model that captures all three classes of the fronts and reveals interrelation between them. This approximation is of higher order than conventional ones (KdV, Boussinesq) and involves mixed space-time derivatives. The proposed framework unifies previous attempts to classify the mechanical transition fronts as radiative, dispersive, topological, or compressive and categorizes them instead as irreducible types of dynamic lattice defects.

The Fermi–Pasta–Ulam–Tsingou recurrence for discrete systems: Cascading mechanism and machine learning for the Ablowitz–Ladik equation

The Fermi–Pasta–Ulam–Tsingou recurrence phenomenon for the Ablowitz–Ladik equation is studied analytically and computationally. Wave profiles periodic in the discrete coordinate may return to the initial states after complex stages of evolution. Theoretically this dynamics is interpreted through a cascading mechanism where higher order harmonics exponentially small initially grow at a faster rate than the fundamental mode. A breather is formed when all modes attain roughly the same magnitude. Numerically a fourth-order Runge–Kutta method is implemented to reproduce this recurring pattern. In another illuminating perspective, we employ data driven and machine learning techniques, e.g. back propagation, hidden physics and physics-informed neural networks. Using data from a fixed time as a learning basis, doubly periodic solutions in both the defocusing and focusing regimes are obtained. The predictions by neural networks are in excellent agreement with those from numerical simulations and analytical solutions.

Idealized four-wave mixing dynamics in a nonlinear Schrödinger equation fiber system

The observation of ideal four-wave mixing dynamics is notoriously difficult to implement experimentally due to the generation of higher-order sidebands and optical loss, which limit the potential interaction distance. Here, we overcome this problem with an experimental technique that uses programmable phase and amplitude shaping to iterate the wave mixing initial conditions injected into an optical fiber. This extends the effective propagation distance by two orders of magnitude, allowing idealized Kerr-driven dynamics to be seen over 50 km of fiber using only one short fiber segment of 500 m. Our experiments reveal the full phase space topology, showing characteristic features of multiple Fermi–Pasta–Ulam recurrence, stationary wave existence, and the system separatrix representing the boundary between two distinct regimes of spatiotemporal evolution. Experiments are in excellent quantitative agreement with numerical solutions of the differential equation system describing the wave evolution. This experimental approach can be readily adapted to study other wave mixing and nonlinear propagation phenomena in optics.

Energy-Preserving Continuous-Stage Exponential Runge--Kutta Integrators for Efficiently Solving Hamiltonian Systems

As one of the most important properties, energy preservation is a natural requirement for numerical integrators of Hamiltonian systems. Considering the limited second-order accuracy of the existing exponential average vector filed (or discrete gradient) method, this paper is devoted to developing high-order energy-preserving exponential integrators for general semilinear Hamiltonian systems. To this end, we first formulate and analyze the continuous-stage exponential Runge--Kutta (ERK) method. After deriving the order conditions, energy-preserving conditions, and symmetric conditions for the continuous-stage ERK method, we construct a novel class of fourth-order symmetric energy-preserving continuous-stage ERK methods with a free parameter based on the established conditions and present a convergence theorem for the fixed-point iteration associated with the continuous-stage ERK method. Furthermore, we extend the proposed continuous-stage ERK method to noncanonical Hamiltonian systems and discuss implementation issues in detail. Finally, numerical results from the FPU problem, the charged-particle dynamics, and the Klein--Gordon equation demonstrate the high efficiency, good energy-preserving behavior, and applicability of large time stepsizes of the proposed fourth-order energy-preserving continuous-stage ERK method.

Nonlinear dynamics of “déjà vu” phenomenon in nonautonomous cubic–quintic NLSE models: Complete suppression of rogue waves generation

The main objective of our study consists in revealing the role of the cubic–quintic nonlinearity in the dynamics of induced modulational instability in nonautonomous nonlinear optical systems with dispersion and nonlinearity varying along the propagation distance. Analytical methods are used to obtain the criterion of modulational instability in the cubic–quintic nonautonomous systems. Nonlinear stage of the modulational instability is studied by direct computational modeling. The main results of our computer experiments clearly demonstrate the imperfect spectral energy restoration leading to variations of the Fermi–Pasta–Ulam recurrence (“déjà vu”) periods. The significance of the results consists in the important possibility of suppressing the appearance of rogue waves due to the compensation mechanisms for both the quintic nonlinearity and the changing dispersion.

Heteroclinic-structure transition of the pure quartic modulation instability

We show that, in the pure-quartic systems, modulation instability (MI) undergoes heteroclinic-structure transitions (HSTs) at two critical frequencies of {\omega} c1 and {\omega} c2 ( {\omega} c2 > {\omega} c1 ), which indicates that there are significantchanges of the spatiotemporal behavior in the system. The complicated heteroclinic structure of instability obtained by the mode truncation method reveals all possible dynamic trajectories of nonlinear waves, which allows us to discover the various types of Fermi-Pasta-Ulam (FPU) recurrences and Akhmediev breathers (ABs). When the modulational frequency satisfies {\omega} < {\omega} c2 , the heteroclinic structure encompasses two separatrixes corresponding to the ABs and the nonlinear wave with a modulated final state, which individually separate FPU recurrences into three different regions. Remarkably, crossing critical frequency {\omega} c1 , both the staggered FPU recurrences and ABs essentially switch their patterns. These HST behaviors will give vitality to the study of MI.

New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem

For the spatially-distributed Fermi–Pasta–Ulam (FPU) equation, irregular solutions are studied that contain components rapidly oscillating in the spatial variable, with different asymptotically large modes. The main result of this paper is the construction of families of special nonlinear systems of the Schrödinger type—quasinormal forms—whose nonlocal dynamics determines the local behavior of solutions to the original problem, as t→∞. On their basis, results are obtained on the asymptotics in the main solution of the FPU equation and on the interaction of waves moving in opposite directions. The problem of “perturbing” the number of N elements of a chain is considered. In this case, instead of the differential operator, with respect to one spatial variable, a special differential operator, with respect to two spatial variables appears. This leads to a complication of the structure of an irregular solution.

Thermal fluctuations in a realistic ionic-crystal model

We investigate the thermal fluctuations of the ionic motions in a Born model of ionic crystals, namely, a model in which the electrons are eliminated, being replaced by suitable effective potentials among the ions. The model is studied in its classical version, computing the Newtonian trajectories of the ions. The general motivation is that, although being an essential ingredient within Green–Kubo linear response theory, thermal fluctuations apparently were not studied systematically by molecular dynamics methods, as was done instead for the approach to equilibrium in the Fermi–Pasta–Ulam problem. The time evolution of the fluctuations is studied in terms of the time-changes of the mode-energies of the system. The stages of the “regression” of the fluctuations are described, from a first stage of strong time-correlations up to a final decorrelation, and a comparison with the process of approach to equilibrium is performed. Finally, the dependence on specific energy is investigated.

Induced modulational instability in the sign-reversal dispersion traps: Imperfect Fermi–Pasta–Ulam recurrence and partial “déjà vu” phenomenon

Our direct computer experiments reveal the main features of the induced modulational instability in the sign-reversal dispersion traps and offer a clearer view of how nonlinear analogies to the “déjà vu” phenomena have arisen. We demonstrate the crucial role of the higher-order dispersion in the origin of the imperfect Fermi–Pasta–Ulam recurrence and partial “déjà vu” phenomena. Induced modulational instability can be used to generate the steady-state soliton trains in the sign-reversal parabolic dispersion traps with exponentially decreasing dispersion.

Extreme spectral asymmetry of Akhmediev breathers and Fermi-Pasta-Ulam recurrence in a Manakov system.

The dynamics of Fermi-Pasta-Ulam (FPU) recurrence in a Manakov system is studied analytically. Exact Akhmediev breather (AB) solutions for this system are found that cannot be reduced to the ABs of a single-component nonlinear Schrödinger equation. Expansion-contraction cycle of the corresponding spectra with an infinite number of sidebands is calculated analytically using a residue theorem. A distinctive feature of these spectra is the asymmetry between positive and negative spectral modes. A practically important consequence of the spectral asymmetry is a nearly complete energy transfer from the central mode to one of the lowest-order (left or right) sidebands. Numerical simulations started with modulation instability of plane waves confirm the findings based on the exact solutions. It is also shown that the full growth-decay cycle of the AB leads to the nonlinear phase shift between the initial and final states in both components of the Manakov system. This finding shows that the final state of the FPU recurrence described by the vector ABs is not quite the same as the initial state. Our results are applicable and can be observed in a wide range of two-component physical systems such as two-component waves in optical fibers, two-directional waves in crossing seas, and two-component Bose-Einstein condensates.

Origin of “déjà vu” phenomenon in the framework of the Fermi–Pasta–Ulam problem in nonautonomous systems

The induced modulational instability is studied in the framework of the nonautonomous nonlinear Schrödinger equation (NLSE) model with varying dispersion and nonlinearity. The Hasegawa algorithm of the analysis of the modulational instability (MI) is generalized taking into account both varying dispersion and nonlinearity. Main features of the induced MI are revealed using direct computational experiments. In particular, the induced modulational instability is considered from a point of view of dynamical memory behavior, and the main restrictions leading to imperfect restoration of the information signal are found. Our direct computer experiments offer a clearer view of how nonlinear analogies to the “déjà vu” phenomena have arisen. It is shown that in the framework of the nonautonomous NLSE model, the “déjà vu” phenomena must be accompanied by the imperfect spectral energy restoration into its fundamental Fourier mode, the variation of the Fermi–Pasta–Ulam recurrence period along the propagation distance, the excitation of the higher-order modulation instability, and the rogue waves formation.

Four-wave mixing and coherently coupled Schrödinger equations: Cascading processes and Fermi-Pasta-Ulam-Tsingou recurrence.

Modulation instability, breather formation, and the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena are studied in this article. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers where the carrier envelope depends on the transverse coordinate of the Schrödinger equations. A "cascading mechanism" is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. Unequal initial conditions for the two waveguides produce symmetry breaking, with "eye-shaped" breathers in one waveguide and "four-petal" modes in the other. An analytical formula for the time or distance of breather formation for a two-waveguide system is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed. For applications to temporal optical waveguides, the present formulation can predict the distance along a weakly birefringent fiber needed to observe FPUT.