Kuznetsov–Ma waves train generation in a left-handed material

We analyze the behavior of an electromagnetic wave which propagates in a left-handed material. Second-order dispersion and cubic–quintic nonlinearities are considered. This behavior of an electromagnetic wave is modeled by a nonlinear Schrödinger equation which is solved by collective coordinates theory in order to characterize the light pulse intensity profile. More so, a specific frequency range has been outlined where electromagnetic wave behavior will be investigated. The perfect combination of second-order dispersion and cubic nonlinearity leads to a robust soliton. When the quintic nonlinearity comes into play, it provokes strong and long internal perturbations which lead to Benjamin–Feir instability. This phenomenon, also called modulational instability, induces appearance of a Kuznetsov–Ma waves train. We numerically verify the validity of Kuznetsov–Ma theory by presenting physical conditions which lead to Kuznetsov–Ma waves train generation. Thereafter, some properties of such waves train are also verified.     

Airy–Bessel modulated self-similar rogue waves in a nonlinear Schrödinger equation model

We construct explicit rogue wave solutions of the generalized nonlinear Schrödinger equation under Airy and Bessel-modulated nonlinearity and an external potential, by employing a multivariate self-similarity transformation. A set of conditions in the form of analytical relations among dispersion, external potential and nonlinearity have been found for rogue wave generation. Different dimensions of controllability of the rogue waves are also being explored by varying parameters associated with the nonlinearity coefficients by judiciously choosing the form of the parametric functions.     

Nonautonomous solitons,breathers and rogue waves for the cubic-quintic nonlinear Schrödinger equation in an inhomogeneous optical fibre

Under investigation in this paper is a cubic-quintic nonlinear Schrödinger equation which can describe the propagation of ultrashort pulses in an inhomogeneous optical fibre. Lax pair and conservation laws are constructed from which the integrability of the equation can be verified. Through a gauge transformation, spectral problem for the equation is converted to an Ablowitz–Kaup–Newell–Segur spectral problem. Nonautonomous solitons and breathers are derived based on the Darboux transformation (DT), while nonautonomous rogue waves are obtained via the generalized DT. Influence of the group-velocity dispersion, gain-or-loss coefficient and group-velocity coefficient on the propagation and interaction of the nonautonomous solitons, breathers and rogue waves is also discussed. The gain-or-loss coefficient influences the amplitude of nonautonomous soliton. The characteristic line and velocity for the nonautonomous soliton are dependent on the group-velocity dispersion and group-velocity coefficient, but independent of the gain-or-loss coefficient. For the second-order nonautonomous soliton, if the two spectral parameters have the same real part, the bound-state structure can be formed, and the intensity becomes amplified in the attractive procedure. There exist two types of the nonautonomous breathers. Quasi-periods for the nonautonomous breathers are given, and effects of those coefficients on the quasi-periods are also discussed: The group-velocity dispersion can influence the trajectories of the nonautonomous breathers and rogue waves, while the gain-or-loss coefficient affects the backgrounds for the nonautonomous breathers and rogue waves     

Nonlinear tunneling of bright and dark rogue waves in combined nonlinear Schrödinger and Maxwell-Bloch systems

We derive the analytical rogue wave solutions for the generalized inhomogeneous nonlinear Schrödinger–Maxwell–Bloch (GINLS-MB) equation describing the pulse propagation in erbium-doped fibre system. Then by suitably choosing the inhomogeneous parameters, we delineate the tunneling properties of rogue waves through dispersion and nonlinearity barriers or wells. Finally, we demonstrate the propagating characteristics of optical solitons by considering their tunneling through periodic barriers by the proper choice of external potential.     

Weak or strong nonlinearity: the vital issue

Marine hydrodynamics is characterised by both weak nonlinearities, as seen for example in drift forces, and strong nonlinearities, as seen for example in wave breaking. In many cases their relative importance is still a controversial matter. The phenomenon of particle escape, seen in linear theory, appears to offer a guide to when strongly nonlinear effects will start to become important, and what will happen when they do. In the case of the “ringing” of vertical cylinders in steep waves, particle escape is shown to correspond approximately to local wave breaking, which leads to the cavitation responsible for “ringing”. Another example is rogue waves, where recent results from weakly nonlinear theory are disappointing, and also fail to explain the rogue waves seen in relatively shallow water, as in the data from the Draupner and Gorm platforms. Recent laboratory experiments, too, show wave crests continuing to grow in height after all frequency components have come into phase, which is inconsistent with weakly nonlinear theory. Particle escape, which is more frequent in shallow water, offers a simple alternative explanation for these observations, as well as for the violent motion at the wave crests, which often confuses rogue-wave data. Extreme wave crests have long been known to be strongly nonlinear, so it appears possible that rogue waves are primarily a strongly nonlinear phenomenon. Fully nonlinear computations of two interacting regular waves are presented, to explore further the connection between particle escape and wave breaking. They are combined with Monte-Carlo simulations of particle escape in hurricane conditions, and the very few measurements of large breaking waves during hurricanes. It is concluded that large breaking waves will have occurred about once per hour, and once per 100 h, respectively, in the recent hurricanes LILI and IVAN. These findings call into question the use of non-breaking wave models in the design codes for fixed steel offshore structures.     

Solitary wave propagation in tunable liquid crystal-filled dual-core photonic crystal fibers

In this paper, we introduce the mathematical model and the necessary constraint conditions of solitary wave propagation through dual core photonic crystal fibres (PCF) filled with liquid crystal. Using the Ansatz method, we derived analytically the necessary formulas to support soliton wave propagation in the new proposed type of coupled bi-refringent fibres. We evaluated dispersion and coupling coefficients by solving wave equations using the vector finite element method. These values can be tuned either by applying external perturbations such as heat or electric field or even altering the geometry of the PCF. Based on the constraint conditions derived from analytic formulas and numerical simulation, we determined the wavelength range for which the soliton waves will be supported without any distortion. The wavelength range can be adjusted by tuning the PCF.     

Electromagnetic waves generated by opposite acoustic waves

An acoustic wave of a combined frequency (formed upon the superposition of the opposite acoustic waves of close frequencies) from a moving source generates electromagnetic waves of the same frequency with the amplitude increasing in the longitudinal direction. The problem is solved for the first time, assuming the absence of electric charges and neglecting the frequency dispersion. It is shown that the running acoustic wave is accompanied by weak electromagnetic waves. This effect may find new applications, in particular, in the space energetic.     

Analysis of Modulational Instability in Monomode Optical Fibres with High-order Nonlinearity

Abstract

The modulational instability (MI) in monomode optical fibres with fifth-order nonlinearity, fibre loss, higher-order dispersion, and the temporal variation of third-order nonlinearity is studied theoretically. The conditions for the existence of the MI and the maximal modulational growth are given and discussed in detail. The results obtained show that the key factor dominating the producing condition of the MI is the power P of the continuous wave initially launched into the optical fibres. If P falls into 3/10<P/P ₀ <1/2 where P ₀ is defined as characteristic power, the MI can be produced in the range of not only anomalous group velocity dispersion but also the normal in which the final evolution state of the modulated wave is dark soliton.     

Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion

Modulational instability (MI) in negative refractive metamaterials with saturable nonlinearity, fourth-order dispersion (FOD), and second-order nonlinear dispersion (SOND) is investigated by using standard linear stability analysis and the Drude electromagnetic model. The expression for the MI gain spectrum is obtained, which clearly reveals the influence of the saturation of the nonlinearity, FOD, and SOND parameters on the temporal MI. The evolution of the MI in negative refractive metamaterials is numerically investigated. Special attention is paid to study the effects of the higher-order dispersion terms on the formation and evolution of the solitons induced by MI. It is shown that as the third-order dispersion term increases, the solitons travel toward the right. Moreover, the magnitude of the FOD term influences considerably the number of wave trains induced by MI.     

Second harmonic generation of shear waves in crystals

Nonlinear self-interaction of shear waves in electro-elastic crystals is investigated based on the rotationally invariant state function. Theoretical analyses are conducted for cubic, hexagonal, and trigonal crystals. The calculations show that nonlinear self-interaction of shear waves has some characteristics distinctly different from that of longitudinal waves. First, the process of self-interaction to generate its own second harmonic wave is permitted only in some special wave propagation directions for a shear wave. Second, the geometrical nonlinearity originated from finite strain does not contribute to the second harmonic generation (SHG) of shear waves. Therefore, unlike the case of longitudinal wave, the second-order elastic constants do not involve in the nonlinear parameter of the second harmonic generation of shear waves. Third, unlike the nonlinearity parameter of the longitudinal waves, the nonlinear parameter of the shear wave exhibits strong anisotropy, which is directly related to the symmetry of the crystal. In the calculations, the electromechanical coupling nonlinearity is considered for the 6 mm and 3 m symmetry crystals. Complement to the SHG of longitudinal waves already in use, the SHG of shear waves provides more measurements for the determination of third-order elastic constants of solids. The method is applied to a Z-cut lithium niobate (LiNbO/sub 3/) crystal, and its third-order elastic constant c/sub 444/ is determined.     

Classification of dispersion equations for homogeneous, dielectric-magnetic, uniaxial materials

The geometric representation at a fixed frequency of the wave vector (or dispersion) surface omega(k) for lossless, homogeneous, dielectric-magnetic uniaxial materials is explored for the case when the elements of the relative permittivity and permeability tensors of the material can have any sign. Electromagnetic plane waves propagating inside the material can exhibit dispersion surfaces in the form of ellipsoids of revolution, hyperboloids of one sheet, or hyperboloids of two sheets. Furthermore, depending on the relative orientation of the optic axis, the intersections of these surfaces with fixed planes of propagation can be circles, ellipses, hyperbolas, or straight lines. The understanding obtained is used to study the reflection and refraction of electromagnetic plane waves due to a planar interface with an isotropic medium.     

Nonlinear oscillations in a thin ring — II. Four-wave resonant interactions. Self-modulation

Summary The first part of the work has concluded that bending waves in a ring, being the lowfrequency modes of resonant triplets, are stable against small perturbations. Consequently, on the one hand, a bending wave should behave as a linear quasi-harmonic wavetrain. The first-order approximation analysis predicts that the triple-mode interactions cannot play a primary role in the evolution of bending waves. On the other hand, one may anticipate that the intense bending wavetrain can be subject to the self-modulation during the long-time evolution. It means that these cannot be stable for a long time. This paper confirms such expectations, when exiting the framework of the first-order nonlinear analysis. To describe the nonlinear dynamics of the ring in detail, one should allow for higher-order approximation effects in a model. Such effects are associated with the concurrence between the diffusion of wave packets, because of different group velocities, and the amplitude-dependent dispersion of phase velocities, caused by the nonlinearity. Within the second-order approximation analysis, the amplitude modulation is experienced for intense bending waves. As a result, the soliton-like envelopes can be formed from unstable bending wavetrains.     

Resonance radiation of electromagnetic waves by a diffraction grating filled with metamaterial

We consider the diffraction of a plane inhomogeneous H-polarized wave, generated by a modulated sheet electron beam, on a periodic reflective grating with rectangular grooves filled by a metamaterial with the effective permittivity characterized by a frequency dispersion. The problem is solved using the method of reexpansion. The phenomenon of resonance radiation of electromagnetic waves by the grating is discovered and the conditions for its manifestation are established.     

Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod

Summary In this paper, we study nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. The aim is to derive simplified model equations in the far field which include both nonlinearity and dispersion. We consider disturbances in an initially pre-stressed rod. For long finite-amplitude waves, the Korteweg-de Vries (KdV) equation arises as the model equation. However, in a critical case, the coefficient of the dispersive term in the KdV equation vanishes. As a result, the dispersion cannot balance the nonlinearity. On the other hand, the latter has the tendency to make the wave profile steeper and steeper. The attention is then focused on finite-length and finite-amplitude waves. A new nonlinear dispersive equation which includes extra nonlinear terms involving second-order and third-order derivatives is derived as the model equation. In the case that the rod is composed of a compressible neo-Hookean material, that equation is further reduced to the Benjamin-Bona-Mahony (BBM) equation, which is known as an alternative to the KdV equation for modelling long finite-amplitude waves. To the author's knowledge, it is the first time that the BBM equation is found to arise as a model equation for finite-length and finite-amplitude waves.     

Nonlinear surface waves at ferrite-metamaterial waveguide structure

A new ferrite slab made of a metamaterial (MTM), surrounded by a nonlinear cover cladding and a ferrite substrate, was shown to support unusual types of electromagnetic surface waves. We impose the boundary conditions to derive the dispersion relation and others necessary to formulate the proposed structure. We analyse the dispersion properties of the nonlinear surface waves and we calculate the associated propagation index and the film–cover interface nonlinearity. In the calculation, several sets of the permeability of the MTM are considered. Results show that the waves behaviour depends on the values of the permeability of the MTM, the thickness of the waveguide and the film–cover interface nonlinearity. It is also shown that the use of the singular solutions to the electric field equation allows to identify several new properties of surface waves which do not exist in conventional waveguide.     

Wave speeds, shear bands and the second-order work for incrementally nonlinear constitutive models

The paper discusses the consequences of the incremental nonlinearity of a constitutive model of a solid for the analyses of characteristic wave speeds, acceleration waves, the second-order work criterion and shear band formation. Incremental nonlinearity may entail qualitative changes in the results as compared to incrementally linear models. Certain well-known correlations cannot be established if the constitutive equation is assumed to be incrementally nonlinear. In particular, the spectra of the characteristic wave speeds and the acceleration wave speeds become continuous and are described by different equations. The second-order work criterion as a sufficient condition of uniqueness of the incremental boundary value problem loses its applicability in bifurcation analyses, unless the applicability can be proved for a particular type of nonlinearity. The singularity of the acoustic tensor in the general nonlinear case correlates neither with the vanishing of the second-order work nor with the shear band formation.     

Periodic and solitary wave solutions for ultrashort pulses in negative-index materials

Abstract

We present a detailed analysis for the existence of dark and bright solitary waves as also fractional-transform solutions in a nonlinear Schrödinger equation model for competing cubic–quintic and higher-order nonlinearities with dispersive permittivity and permeability. Parameter domains are delineated in which these ultrashort optical pulses exist in negative-index materials (NIMs). For example, dark solitons exist for the case of normal second-order dispersion, anomalous third-order dispersion, self-focusing Kerr nonlinearity, and non-Kerr nonlinearities, while the bright solitons exist for the case of anomalous second-order dispersion, normal third-order dispersion, self-focusing Kerr nonlinearity, and non-Kerr nonlinearities. This is contrary to the situation in ordinary materials.     

Analytic study on soliton-effect pulse compression in dispersion-shifted fibers with symbolic computation

The soliton-effect pulse compression of ultrashort solitons in a dispersion-shifted fiber (DSF) is investigated based on solving the higher-order nonlinear Schrödinger equation with the effects of third-order dispersion (TOD), self-steepening (SS) and stimulated Raman scattering (SRS). By using Hirota's bilinear method with a set of parametric conditions, the analytic one-, two- and three-soliton solutions of this model are obtained. According to those solutions, the higher-order soliton is shown to be compressed in the DSF for the pulse with width in the range of a few picoseconds or less. An appealing feature of the soliton-effect pulse compression is that, in contrast to the second-order soliton compression due to the combined effects of negative TOD and SRS, the third-order soliton can significantly enhance the soliton compression in the DSF with small values of the group-velocity dispersion (GVD) at the operating wavelength.     

Rogue waves for a coupled nonlinear Schrödinger system in a multi-mode fibre

In this paper, we investigate the rogue waves for an integrable coupled nonlinear Schrödinger (CNLS) system with the self-phase modulation, cross-phase modulation and four-wave mixing term, which can describe the propagation of optical waves in a multi-mode fibre. We construct a generalized Darboux transformation (GDT) for the CNLS system and find a gauge transformation which converts the Lax pair into the constant-coefficient differential equations. Solving those equations, we can obtain the vector solutions of the Lax pair. Using the GDT, we derive an iterative formula for the nth-order rogue-wave solutions for the CNLS system. We derive the first- and second-order rogue-wave solutions for the CNLS system and analyse the profiles for the rogue waves with respect to the self-phase modulation term a, cross-phase modulation term c and four-wave mixing term b, respectively. The rogue waves become thinner with the increase in the value for the real part of b and that the effect of a or c on the rogue waves is the same as the one of the real part of b.