Nonlinear Modulational Instability Research Articles

Nonlinear Evolution of the Kelvin–Helmholtz Instability of Supersonic Tangential Velocity Discontinuities

A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for the instability of two layers of immiscible, inviscid, arbitrarily compressible fluids in relative motion. Such configurations are of relevance in a variety of astrophysical and space configurations. For modes ofallwavenumbers on, or in the stable neighborhood of, the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein–Gordon equation. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interacting vortices observed empirically, or are disrupted via nonlinear modulation instability.

Frequency stability enhancement from carrier-envelope resonance in a surface acoustic wave delay line oscillator

We have built a new injection-locking set-up which allows the study of low frequency noise in an oscillator. The interaction between a driving 100 MHz signal and a surface acoustic wave delay line oscillator undergoing a nonlinear modulational instability is investigated. Driving amplitude is used to control the synchronization range and therefore the frequency stability of a newly discovered carrier-envelope resonance process. In the soliton-locked regime, fundamental and subharmonic steps attached to a Lorentzian shape frequency-amplitude characteristic are found. They are explained thanks to a driven nonlinear Schrödinger model and the joined synchronization map.

Fibre nonlinearity in dispersion-compensated conventional fibre transmission

The authors investigate the fibre non-linear effect, self-phase modulation (SPM) and modulation instability (MI), in long distance (>1000 km) dispersion-compensated (DC) conventional optical fibre transmission using numerical simulation and experiments. They compare the results with those in the dispersion-shifted fibre (DSF) transmissions and clarify the remarkable characteristics of DC transmissions i.e. the characteristic behaviour of the waveform deformation and the absence of the MI. The optimum amount of the dispersion compensation is also discussed, and it is shown that the dispersion window for a transmission thousands of kilometres long is obtained in the average positive dispersion region, and the negative transmitter α-parameter is advantageous.

Effect of modulation instability on phase modulation–amplitude modulation conversion in optical fibers

Phase modulation-amplitude modulation conversion in dispersive fibers is affected by nonlinear phase modulation and modulation instability. A differential equation is derived under adiabatic approximations and is solved analytically. The results agree well with experimental values. A simple measurement of the nonlinear coeff icient is thus feasible.

Analysis of Spatial Structure in a Predator-Prey Model with Delay II: Nonlinear Theory

A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for a predator-prey model including spatial diffusion and Volterra-type distributed delays in the interspecies interaction terms. For delays modeled by the “weak” generic kernel, the slow evolution of the amplitude of the spatially nonuniform states predicted by the linear analysis is shown to be governed by a complicated Ginzburg–Landau/Newell-Whitehead equation. Both the spatially-dependent and space-independent versions of this equation are analyzed to obtain the regimes of the physical parameter space where the linear nonuniform solutions either asymptote to a fixed amplitude wave pattern with an amplitude dependent frequency modulation, evolve to other permanent spatially-dependent wave solutions or patterns via nonlinear modulational instability, or decay to zero.

Nonlinear evolution of Langmuir and electromagnetic pulses in a warm, unmagnetized plasma: modulational instability, integrability and self-focusing in 2 + 1 dimensions

The nonlinear dynamics of wave envelopes modulated in 2 + 1 dimensions is considered for two systems in plasma physics: (i) Langmuir pulses and (ii) intense (but weakly relativistic) electromagnetic (EM) pulses. Using singular perturbation techniques applied to an envelope approximation, both problems are reduced to the two-dimensional nonlinear Schrödinger (2DNLS) system, which describes the dynamics of two coupled slowly varying potentials. The general 2DNLS system exhibits a rich variety of phenomena, including enhanced (compared with ‘longitudinal’ propagation) modulational stability and (1D) soliton formation; decay of 1D solitons over long time scales; self- focusing regimes (determined by a virial-type condition); as well as integrability and 2D solitons. Applying our recent results on the 2DNLS system, we determine which of these phenomena can actually occur here and compute the parameter regimes. (i) The 2DNLS system for the Zakharov equations is modulationally unstable for all parameter values. It also has an integrable sector and a self-focusing regime. (ii) The 2DNLS system describes coupled ‘longitudinal’ and ‘transverse’ modulations of linearly or circularly polarized EM pulses propagating through a warm unmagnetized two-component neutral plasma with arbitrary masses (i.e. electron—positron or electron—ion). The pulse can accelerate particles to weakly (but not fully) relativistic velocities; relativistic, ponderomotive and harmonic effects all contribute to the nonlinear terms. The resulting 2DNLS system does not admit a self-focusing regime. Parameter values leading to an integrable case (the so-called ‘Davey—Stewartson I’ equations, which admit 2D soliton solutions) are computed; however, the required values would not be attainable in a laboratory or astrophysical setting. None the less, the existence of new nonlinear modulational instabilities associated with the second spatial degree of freedom already represents an important potential limitation on any (1 + 1)-dimensional approach to nonlinear evolution and modulational instability of plasma EM waves.

Nonlinear Wave Number Shift and Modulational Instability for Large Amplitude Waves in a Relativistic Magnetised Plasma

Properties of large amplitude waves in a relativistic magnetised plasma are studied using the method of reductive perturbation. The plasma under consideration consists of warm adiabatic ions and isothermal warm electrons, under the influence of a magnetic field. A onsideration of large amplitude waves demands study of the relativistic situation. In the present case we consider both the electrons and ions to be relativistic. A KdV equation is derived from which a nonlinear Schrodinger equation is deduced by further scaling. Lastly we derive an expression for nonlinear wave number shift, critical angle of propagation and the condition for modulational instability. Our analysis is applicable to both laboratory and space plasmas.

Bifurcations and three-wave-mixing instabilities in nonlinear propagation in birefringent dispersive media

Polarization modulationalal instability in a nonlinear dispersive birefringent medium refers to the exponential growth of polarized sidebands due to mixing with an orthogonally polarized pump. We show that this instability extends to the nonlinear regime of strong coupling between the waves. By reducing the coupled nonlinear Schr\odinger equations that govern the interaction to a finite-dimensional integrable Hamiltonian system, we show that nonlinear modulational instability originates from bifurcations of spatial eigensolutions of the three-wave interaction. Spatial recurrence of the field evolutions and period doubling related to the existence of spatially unstable eigensolutions are relevant for applications which make use of polarization modulational instability (or birefringence-matched third-order three-wave mixing) in the strong-depletion regime. In contrast to previous results obtained by means of linearized equations, we find that efficient frequency conversion, when strong coupling is accounted for, occurs for initially wave-vector-mismatched waves.

Dynamics of the nonlinear modulational instability in optical fibers

We study the nonlinear evolution in optical fibers of a modulationally unstable continuous wave. We find that a good description of the propagation can be analytically obtained by a simple truncated three-wave model. The dynamic viewpoint may give an important physical insight in different processes, such as frequency conversion of modulated signals, all-optical generation of temporal codes, and the onset of spatiotemporal chaos.

Variational approach to Langmuir waves described by the Zakharov equations

Nonlinear modulational instability and the evolution of a pulse that is initially non-solitonic for Langmuir waves described by the Zakharov equations are considered. The average Lagrangian method has been used to derive the nonlinear dispersion relation for Stokes waves. It is found that the region of instability increases with the amplitude of the perturbed Langmuir field. The propagation of the pulse is studied with the help of the Rayleigh-Ritz optimization method. It is noted that the width of the pulse oscillates rapidly for high initial pulse velocity.

Nonlinear break up of an infinite cylinder in the presence of a uniform axial magnetic field

The effect of a uniform axial magenetic field on the nonlinear instability of a self-gravitating infinite cylinder is examined. Using the method of multiple scales, it is found that while the nonlinear (modulational) instability cannot be completely suppressed, the presence of a magnetic field does increase the range of stable wave numbers. The evolution of the amplitude is governed by a non-linear Schrodinger equation which gives the criterion for modulational instability.

Nonlinear capillary-gravity waves in magnetic fluids

The nonlinear wave propagation of capillary-gravity waves on the surface of a ferrofluid of finite depth is investigated by employing the method of multiple scales. The stability analysis reveals the existence of different regions of instability. We show that the nonlinear modulational instability cannot be suppressed by the application of a strong magnetic field, however strong it may be. The influence of the magnetic field is not only quite signficant but its effect is also different for different regions of stability.

Wave Propagation in a Solid Ice Pack

The analysis presented in this paper was inspired by the report that the R/V Polarstern has encountered surface waves of large amplitude hundreds of kilometers inside the ice pack in the Weddell Sea. This paper presents analysis of processes that affect waves in an ice pack, namely the refraction of waves at the pack edge, the effects of pack compression on wave propagation, wave train stability and buckling stability in the ice pack. Sources of pack compression and interaction between wave momentum and pack compression are discussed. Viscous damping of propagating waves are also studied. Significant results include the conditions for total reflection of waves at the pack edge, the strong effect of pack compressive stress on wave group speed, with the concomitant possibility of extreme local concentration of wave energy. The result that compressive stress in the pack leads to very rapid development of wave packets, through changes in the parameters for weakly nonlinear modulational instability of the wave field is also notable. The analysis suggests an explanation for the change in wave dispersion observed from the ship between the time of first arrival of the waves and after the pack was partially broken up by the first waves.

Study of Quasiperiodic Solutions of the Nonlinear Schrödinger Equation and the Nonlinear Modulational Instability

A simple and direct method is described for constructing exact quasiperiodic solutions of the focusing nonlinear Schr\"odinger equation. An algebraic constraint is imposed on the initial values of a set of auxiliary variables. The solution, expressed in terms of these auxiliary variables, is quasiperiodic in general, and represents a wider class of solutions than those obtained from the inverse-scattering transform. As an example of our method, we present solutions corresponding to a modulated plane wave.

Modulational instability of a magneto-hydrodynamic jet

The effect of a magnetic field on the nonlinear capillary instability of a fluid jet is examined using the method of multiple scales. It is a well-known result that a sufficiently strong magnetic field can, in the limit of zero viscosity and resistivity, completely suppress the linear capillary instability. It turns out that while the nonlinear (modulational) instability cannot be completely suppressed, the presence of a magnetic field does greatly increase the range of stable wave numbers.

Nonlinear Propagation and Modulational Instability of Electrostatic Ion Cyclotron Wave

Three-dimensional nonlinear Schrödinger equation for an electrostatic ion cyclotron wave coupled to an ion acoustic wave is derived with the use of the reductive perturbation method. Electrons are assumed to obey the Boltzmann law under the influence of the electrostatic potential. Cold fluid equations are used for the ions. Instability of a plane electrostatic ion cyclotron wave against a two-dimensional modulation is investigated and the unstable regions are graphically plotted in the wavenumber space.