Instability and noise-induced thermalization of Fermi–Pasta–Ulam recurrence in the nonlinear Schrödinger equation

We investigate the spontaneous growth of noise that accompanies the nonlinear evolution of seeded modulation instability into Fermi–Pasta–Ulam recurrence. Results from the Floquet linear stability analysis of periodic solutions of the three-wave truncation are compared with full numerical solutions of the nonlinear Schrödinger equation. The predicted initial stage of noise growth is in a good agreement with simulations, and is expected to provide further insight into the subsequent dynamics of the field evolution after recurrence breakup.     

Confined ferrofluid droplet in crossed magnetic fields

When a ferrofluid drop is trapped in a horizontal Hele-Shaw cell and subjected to a vertical magnetic field, a fingering instability results in the droplet evolving into a complex branched structure. This fingering instability depends on the magnetic field ramp rate but also depends critically on the initial state of the droplet. Small perturbations in the initial droplet can have a large influence on the resulting final pattern. By simultaneously applying a stabilizing (horizontal) azimuthal magnetic field, we gain more control over the mode selection mechanism. We perform a linear stability analysis that shows that any single mode can be selected by appropriately adjusting the strengths of the applied fields. This offers a unique and accurate mode selection mechanism for this confined magnetic fluid system. We present the results of numerical simulations that demonstrate that this mode selection mechanism is quite robust and “overpowers” any initial perturbations on the droplet. This provides a predictable way to obtain patterns with any desired number of fingers.     

Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations

In this paper, the bifurcations and chaotic motions of higher-dimensional nonlinear systems are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam. The Kelvin viscoelastic model is chosen to describe the viscoelastic property of the beam material. Firstly, the nonlinear governing equations of nonplanar motion for an axially accelerating moving viscoelastic beam are established by using the generalized Hamilton’s principle for the first time. Then, based on the Galerkin’s discretization, the governing equations of nonplanar motion are simplified to a six-degree-of-freedom nonlinear system and a three-degree-of-freedom nonlinear system with parametric excitation, respectively. At last, numerical simulations, including the Poincare map, phase portrait and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the axially accelerating moving viscoelastic beam. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity, the amplitude of velocity fluctuation and the frequency of velocity fluctuation are respectively presented when other parameters are fixed. The Lyapunov exponents are calculated to identify the existence of the chaotic motions. From the numerical results, it is indicated that the periodic, quasi-periodic and chaotic motions occur for the nonplanar nonlinear vibrations of the axially accelerating moving viscoelastic beam. Observing the in-plane nonlinear vibrations of the axially accelerating moving viscoelastic beam from the numerical results, it is found that the nonlinear responses of the six-degree-of-freedom nonlinear system are much different from that of the three-degree-of-freedom nonlinear system when all parameters are same.     

Dynamics of gas bubbles in viscoelastic fluids. II. Nonlinear viscoelasticity

The nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear viscoelastic media are examined. The constitutive equation [Upper-Convective Maxwell (UCM)] used for the fluid is suitable for study of large-amplitude excursions of the bubble, in contrast to the previous work of the authors which focused on the smaller amplitude oscillations within a linear viscoelastic fluid [J. S. Allen and R. A. Roy, J. Acoust. Soc. Am. 107, 3167-3178 (2000)]. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of viscoelastic constitutive equations into bubble dynamics equations. The numerical method used to solve the governing system of equations (one integrodifferential equation and two partial differential equations) is outlined. An energy balance relation is used to monitor the accuracy of the calculations and the formulation is compared with the previously developed linear viscoelastic model. Results are found to agree in the limit of small deformations; however, significant divergence for larger radial oscillations is noted. Furthermore, the inherent limitations of the linear viscoelastic approach are explored in light of the more complete nonlinear formulation. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that tissue viscoelasticity may be an important consideration for the risk assessment of potential cavitation bioeffects.     

Dynamics of gas bubbles in viscoelastic fluids. I. Linear viscoelasticity

The nonlinear oscillations of spherical gas bubbles in linear viscoelastic fluids are studied. A novel approach is implemented to derive a governing system of nonlinear ordinary differential equations. The linear Maxwell and Jeffreys models are chosen as the fluid constitutive equations. An advantage of this new formulation is that, when compared with previous approaches, it facilitates perturbation methods and numerical investigations. Analytical solutions are obtained using a multiple scale perturbation method and compared with the Newtonian results for various Deborah numbers. Numerical analysis of the full equations supports the perturbation analysis, and further reveals significant differences between the viscoelastic and Newtonian cases. Differences in the oscillation phase and harmonic structure characterize some of the viscoelastic effects. Subharmonic excitations at particular fluid parameters lead to a discrete group modulation of the radial excursions; this appears to be a unique, previously undiscovered phenomenon. Implications for medical ultrasound applications are discussed in light of these current findings.     

Dynamics of the plane and solitary waves in a Noguchi network:Effects of the nonlinear quadratic dispersion

We consider a modified Noguchi network and study the impact of the nonlinear quadratic dispersion on the dynamics of modulated waves. In the semi-discrete limit, we show that the dynamics of these waves are governed by a nonlinear cubic Schrodinger equation. From the graphical analysis of the coefficients of this equation, it appears that the nonlinear quadratic dispersion counterbalances the effects of the linear dispersion in the frequency domain. Moreover, we establish that this nonlinear quadratic dispersion provokes the disappearance of some regions of modulational instability in the dispersion curve compared to the results earlier obtained by Pelap et al.(Phys. Rev. E 91 022925(2015)). We also find that the nonlinear quadratic dispersion limit considerably affects the nature, stability, and characteristics of the waves which propagate through the system. Furthermore, the results of the numerical simulations performed on the exact equations describing the network are found to be in good agreement with the analytical predictions.     

Mode selection in weakly unstable two-dimensional detonations

A formulation of the reactive Euler equations in the shock-attached frame is used to study the two-dimensional instability of weakly unstable detonation through direct numerical simulation. The results are shown to agree with the predictions of linear stability analysis. Comparisons are made with linear perturbation growth rates and oscillation frequencies as a function of transverse disturbance wavelength. The perturbation eigenfunctions predicted by linear stability analysis are directly validated through numerical simulation. Three regimes of unstable behavior – linear, weakly nonlinear, and fully nonlinear – are explored and characterized in terms of the power spectrum of the normal shock velocity for a Chapman–Jouguet detonation with weak heat release.     

Multi-pulse chaotic dynamics in non-planar motion of parametrically excited viscoelastic moving belt

This paper investigates the multi-pulse global bifurcations and chaotic dynamics for the nonlinear, non-planar oscillations of the parametrically excited viscoelastic moving belt using an extended Melnikov method in the resonant case. Using the Kelvin-type viscoelastic constitutive law and Hamilton's principle, the equations of motion are derived for the viscoelastic moving belt with the external damping and parametric excitation. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:1 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics. The paper demonstrates how to employ the extended Melnikov method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear non-planar oscillations of the viscoelastic moving belt, the Shilnikov-type multi-pulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that the chaos for the Smale horseshoe sense in motion exists.     

Parameter design of a liquid-filled sloshing system

<正>The nonlinear governing equations of the liquid sloshing modals in a cylindrical storage tank are established. Through analytical analysis,the analytical expressions of the solutions of this kind of system are obtained.With different parameters,the dynamical behaviors of the solutions are different from the trivial ones.To prevent system instability,two selection principles that the stiffness equations are positive-definite and the nonlinear terms of the system are not regenerative elements are given.Meanwhile,numerical simulations are also given,which confirm the analytical results.     

Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations

To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.     

FILAMENTATION INSTABILITY OF LASER BEAMS IN NONLOCAL NONLINEAR MEDIA

The filamentation instability of laser beams propagating in nonlocal nonlinear media is investigated. It is shown that the filamentation instability can occur in weakly nonlocal self-focusing media for any degree of nonlocality, and in defocusing media for the input light intensity exceeding a threshold related to the degree of nonlocality. A linear stability analysis is used to predict the initial growth rate of the instability. It is found that the nonlocality tends to suppress filamentation instability in self-focusing media and to stimulate filamentation instability in self-defocusing media. Numerical simulations confirm the results of the linear stability analysis and disclose a recurrence phenomenon in nonlocal self-focusing media analogous to the Fermi-Pasta-Ulam problem.     

Efficient computation of the spectrum of viscoelastic flows

The understanding of viscoelastic flows in many situations requires not only the steady state solution of the governing equations, but also its sensitivity to small perturbations. Linear stability analysis leads to a generalized eigenvalue problem (GEVP), whose numerical analysis may be challenging, even for Newtonian liquids, because the incompressibility constraint creates singularities that lead to non-physical eigenvalues at infinity. For viscoelastic flows, the difficulties increase due to the presence of continuous spectrum, related to the constitutive equations.The Couette flow of upper convected Maxwell (UCM) liquids has been used as a case study of the stability of viscoelastic flows. The spectrum consists of two discrete eigenvalues and a continuous segment with real part equal to ?1/We (We is the Weissenberg number). Most of the approximations in the literature were obtained using spectral expansions. The eigenvalues close to the continuous part of the spectrum show very slow convergence.In this work, the linear stability of Couette flow of a UCM liquid is studied using a finite element method. A new procedure to eliminate the eigenvalues at infinity from the GEVP is proposed. The procedure takes advantage of the structure of the matrices involved and avoids the computational overhead of the usual mapping techniques. The GEVP is transformed into a non-degenerate GEVP of dimension five times smaller. The computed eigenfunctions related to the continuous spectrum are in good agreement with the analytic solutions obtained by Graham [M.D. Graham, Effect of axial flow on viscoelastic Taylor–Couette instability, J. Fluid Mech. 360 (1998) 341].     

Instability and evolution of nonlinearly interacting water waves

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schr?dinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.     

Dynamics of the magnetospheric cyclotron ELF/VLF maser in the backward-wave-oscillator regime. II. The influence of the magnetic-field inhomogeneity

We study the influence of the magnetic-field inhomogeneity on the nonlinear dynamics of the absolute instability of whistler-mode waves in the Earth’s magnetosphere in the presence of a step-like deformation in the distribution function of energetic electrons. Development of this instability, implying the transition of the magnetospheric cyclotron maser to the regime of a backward-wave oscillator (BWO), was proposed earlier as a generation mechanism of magnetospheric chorus emissions. We analyze the results of numerical simulations of the simplified nonlinear equations describing the magnetospheric-BWO dynamics in the case of low efficiency of wave-particle interactions. We found that the case of an inhomogeneous magnetic field where the system length is much greater than the length characterizing the linear stage of the BWO regime has important specific features compared with the case of a homogeneous medium. The main feature of the nonlinear wave dynamics in the magnetospheric BWO in an inhomogeneous magnetic field consists in the fact that for a sufficiently large excess over the generation threshold, a sequence of separate wave packets, i.e., discrete elements, is formed. The frequency within each packet varies in time, and these discrete elements are close in their properties to the chorus elements observed in the magnetosphere. The results of calculations confirm the quantitative estimates of parameters of chorus emissions, which were performed earlier on the basis of the BWO model. ^†Deceased Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 11, pp. 977–987, November 2008.     

Kinetic step bunching instability during surface growth

We study the step bunching kinetic instability in a growing crystal surface characterized by anisotropic diffusion. The instability is due to the interplay between the elastic interactions and the alternation of step parameters. This instability is predicted to occur on a vicinal semiconductor surface Si(001) or Ge(001) during epitaxial growth. The maximal growth rate of the step bunching increases like F4, where F is the deposition flux. Our results are complemented with numerical simulations which reveal a coarsening behavior in the long time evolution for the nonlinear step dynamics.     

Modulational instability of optical beams in photovoltaic and photorefractive media due to two-photon photorefractive effect under open circuit condition

The modulational instability of broad optical beams in two-photon photorefractive (PR) photovoltaic materials under open circuit conditions has been investigated. Under linear stability framework, the one dimensional modulational instability growth rate has been estimated by considering the space charge field. Gain of the instability is shown to exist only when the photovoltaic fields orients in the same direction with respect to the optical c-axis of the medium. It is found that the behavior of the gain spectrum is different in low and high power regions. We have found by numerical simulations that the evolution of the soliton induced by the modulational instability at low photovoltaic field show the dynamical behaviors similar to those of the localized beam as the initial profile. However, it has been shown that increasing photovoltaic fields produce traveling, breathing, and mutually interacting solitons.     

Modulational instability of bright solitary waves in incoherently coupled nonlinear Schrödinger equations

We present a detailed analysis of the modulational instability (MI) of ground-state bright solitary solutions of two incoherently coupled nonlinear Schr?dinger equations. Varying the relative strength of cross-phase and self-phase effects we show the existence and origin of four branches of MI of the two-wave solitary solutions. We give a physical interpretation of our results in terms of the group-velocity-dispersion- (GVD-) induced polarization dynamics of spatial solitary waves. In particular, we show that in media with normal GVD spatial symmetry breaking changes to polarization symmetry breaking when the relative strength of the cross-phase modulation exceeds a certain threshold value. The analytical and numerical stability analyses are fully supported by an extensive series of numerical simulations of the full model.     

Analysis of dynamic pattern formation in nonlinear Fabry-Perot resonators

We present a detailed analysis of transverse effects and pattern formation in bistable optical elements. The system we investigate consists of a Fabry-Perot resonator for the optical feedback element with a nematic liquid-crystal cell used as an optically nonlinear intracavity medium. On illumination with a cw-laser beam, the system causes the beam to break up into several individual spots, passing through several transitions before finally reaching a stationary state. We devise a theoretical model which is used as the basis for numerical simulations of the system. The simulation results are in good agreement with experiment. Finally, we characterize the principal instability of the system using a linear stability analysis of the theoretical model.     

Modeling polymerization of microtubules: A semi-classical nonlinear field theory approach

In this paper, for the first time, a three-dimensional treatment of microtubules’ polymerization is presented. Starting from fundamental biochemical reactions during microtubule’s assembly and disassembly processes, we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in three dimensions. We found that the dynamics of a microtubule is mathematically expressed via a cubic-quintic nonlinear Schrödinger (NLS) equation. We show that in 3D a vortex filament, a generic solution of the NLS equation, exhibits linear growth/shrinkage in time as well as temporal fluctuations about some mean value which is qualitatively similar to the dynamic instability of microtubules. By solving equations numerically, we have found spatio-temporal patterns consistent with experimental observations.     

Modulational instability on triangular dynamical lattices with long-range interactions and dispersion

The influence of long-range interactions on the stability of stationary solutions of triangular lattices described by the continuum-discrete nonlinear Schrödinger equation is analyzed. By virtue of the linear stability analysis and a variational approach we demonstrate that both soliton array and continuous-wave solutions are modulationally unstable. Analytical expressions for instability thresholds and growth rate spectra are presented and compared with the corresponding results in the approximation of a nearest neighbor interaction.