Modulational instability of a strip beam in a bulk type I quadratic medium

The evolution of a strip (one-dimensional) fundamental beam with propagation distance owing to spatial modulational instabilities was analyzed in a quadratic medium near type I phase matching. We obtained the gain coefficient for the modulational instability and showed that the wave evolves into a clean periodic sequence of solitary waves and does not reproduce the incident beam.     

等离子体中非线性朗谬尔波的哈密顿描述

研究了在双离子（H^ ,O^ )成份等离子体中的非线性朗谬尔波的特性,从流体方程出发,考虑低频离子运动的完全非线性和双极势的色散,得到了描述高频电场缓变振幅与低频势扰动的耦合方程组。利用哈密顿方法,在小振幅情况下,对方程组解耦合,利用Sagdeev势方法,对孤立波的性状进行了讨论,结果表明,双离子成份等离子体中双极势的孤立子的幅度相对电子,离子等离子体的双极势孤立子的幅度要大,而相应的高频电场强度的幅度相对要小,相应的孤立子的速度传播区也较小。     

Dissipation of Nonlinear Wave in Inhomogeneous Dust Plasma Crystal

A Korteweg-de Vires-type (KdV-type) equation and a modifiedNonlinear Schrödinger equation (NLSE) for the dust lattice wave(DLW) are derived in a weakly inhomogeneous dust plasma crystal. Itseems that the amplitude and the velocity of the dust lattice solitary waves decay exponentially with increasing time in a dust lattice. The modulational instability of this dust lattice envelope waves is investigated as well. It is found that the waves are modulational stable under certain conditions. On the other hand, the waves are modulational unstable if the conditions are not satisfied.     

Modulation of dust acoustic waves with non-adiabatic dust charge fluctuations

The modulational instability of dust acoustic waves in a dusty plasma with non-adiabatic dust charge fluctuation is studied. Using the perturbation method, a modified nonlinear Schrödinger equation containing a damping term that comes from the effect of dust charge variation is derived. It is found that the modulational instability of the wave packet and the propagation characters of the envelope solitary waves are modified significantly by the non-adiabatic dust charge fluctuation.     

Discrete nonlinear Schrödinger equation with arbitrary nonlocality: Linear stability analysis and localized solution

The modulational instability of a plane wave for a discrete nonlinear Schrödinger equation with arbitrary nonlocality is analyzed. This model describes light propagation in a thin film planar waveguide arrays of nematic liquid crystals subjected to a periodic transverse modulation by a low frequency electric field. It is shown that nonlocality can both suppress and promote the growth rate and bandwidth of instability, depending on the type of a response function of a discrete medium. A solitary wave (breather-like) solution is built by the variational approximation and its stability is demonstrated.     

The modulation instability revisited

The modulational instability (or “Benjamin-Feir instability”) has been a fundamental principle of nonlinear wave propagation in systems without dissipation ever since it was discovered in the 1960s. It is often identified as a mechanism by which energy spreads from one dominant Fourier mode to neighboring modes. In recent work, we have explored how damping affects this instability, both mathematically and experimentally. Mathematically, the modulational instability changes fundamentally in the presence of damping: for waves of small or moderate amplitude, damping (of the right kind) stabilizes the instability. Experimentally, we observe wavetrains of small or moderate amplitude that are stable within the lengths of our wavetanks, and we find that the damped theory predicts the evolution of these wavetrains much more accurately than earlier theories. For waves of larger amplitude, neither the standard (undamped) theory nor the damped theory is accurate, because frequency downshifting affects the evolution in ways that are still poorly understood.     

Varieties of solitons and solitary waves

A summary is presented of the principal types of completely integrable partial differential equations having soliton solutions. Each type is derived from an appropriate physical model of an electromagnetic wave problem, with the intention to show how known mathematical results apply to a coherent class of physical problems in electromagnetic waves. The non-linear Schrödinger (NS) equation appears when the induced non-linear dielectric polarization is expanded in a series of powers of the electric field, only the linear and third-order polarizations are retained, and the temporal spectrum of the wave is a narrow band far removed from any resonance of the medium. The sine-Gordon equation appears from a similar optical model of propagation in a dielectric consisting of identical 2-level atomic systems, but resonance occurs between the carrier frequency of the wave and the transition frequency of the atoms. The Boussinesq and Korteweg– de Vries equations appear at different levels of approximation to a potential wave on a transmission line having a non-linear capacitance such that the charge stored is a non-linear function of the line potential. In all cases the evolution variable is the propagation distance; the transverse variable is time, but in the case of the NS equation it may alternatively be a spatial coordinate, giving rise to the possibility of spatial solitons as well as temporal solitons for NS-type problems. Two examples are derived of non-integrable Hamiltonian systems having spatial solitary waves, namely the second-order cascade interaction and vector spatial solitary waves of the third-order interaction, and a brief survey of the analytical solutions for the plane waves and solitary waves of these two types is presented. Finally, the addition of a second spatial dimension to the non-linear transmission line problem leads to the Kadomtsev–Petviashvili equations, and a further approximation for weakly modulated travelling waves leads to the Davey–Stewartson equations. Both of these completely integrable systems support combined spatial–temporal solitons.     

Beam breakup and modulational instability in a bulk type I quadratic medium

The effect of a small modulation superimposed on a strip (1D) solitary wave propagating in a bulk quadratic medium was investigated both analytically and numerically near Type I phase matching. General, exact results were obtained. By using first order perturbation theory, we obtained the gain coefficient for the modulational instability and the modulation cut-off frequency and we investigated their dependence on various beam and material parameters. The wave evolves into a clean periodic sequence of solitary waves and does not reproduce the incident beam. We showed that the beam breakup observed experimentally is due entirely to noise induced modulational instability.     

Nonlinear ion-acoustic structures in a nonextensive electron–positron–ion–dust plasma: Modulational instability and rogue waves

The nonlinear propagation of planar and nonplanar (cylindrical and spherical) ion-acoustic waves in an unmagnetized electron–positron–ion–dust plasma with two-electron temperature distributions is investigated in the context of the nonextensive statistics. Using the reductive perturbation method, a modified nonlinear Schrödinger equation is derived for the potential wave amplitude. The effects of plasma parameters on the modulational instability of ion-acoustic waves are discussed in detail for planar as well as for cylindrical and spherical geometries. In addition, for the planar case, we analyze how the plasma parameters influence the nonlinear structures of the first- and second-order ion-acoustic rogue waves within the modulational instability region. The present results may be helpful in providing a good fit between the theoretical analysis and real applications in future spatial observations and laboratory plasma experiments.     

Extended Schwinger-Boson theory of the two-dimensional spin-1/2 anisotropic Heisenberg antiferromagnets

The properties of the possible solitary electromagnetic waves, propagating in two-dimensional SIS Josephson junction without dissipative losses are investigated on the basis of the local theory of the junction. A classification of the waves in the junction with respect to the Swihart velocity is made. It is shown that allowed and forbidden areas for the wave numbers, wave frequency and wave amplitude exist. The cut-off frequency for the solitary waves which velocity is greater than the Swihart velocity can be smaller than the Josephson plasma frequency and moreover these waves can propagate only in a junction that is large in the direction perpendicular to the propagation direction. On the contrary the solitary waves which velocity is smaller than the Swihart velocity request junction size in the above direction to be smaller than a critical one. The investigated two-dimensional solitary waves can be connected with one or two quanta of the magnetic flux.     

Propagation and modulational instability of Rossby waves in stratified fluids

Perturbation analysis and scale expansion are used to derive the (2+1)-dimensional coupled nonlinear Schrödinger (CNLS) equations that can describe interactions of two Rossby waves propagating in stratified fluids. The (2+1)-dimensional equations can reflect and describe the wave propagation more intuitively and accurately. The properties of the two waves in the process of propagation can be analyzed by the solution obtained from the equations using the Hirota bilinear method, and the influence factors of modulational instability are analyzed. The results suggest that, when two Rossby waves with slightly different wave numbers propagate in the stratified fluids, the intensity of bright soliton decreases with the increases of dark soliton coefficients. In addition, the size of modulational instable area is related to the amplitude and wave number in y direction.     

Modulational instability in systems with integrating nonlinearity

A new type of modulational instability for coherent as well as partially coherent light in systems with integrating nonlinearity caused by an irreversible process is investigated both experimentally and theoretically. In such systems plane waves never reach the stationary limit and exhibit a nontrivial time dependence resulting in new features of the modulational instability. For example, the modulational frequency of the nonexponentially increasing perturbation with maximum gain decreases while the wave is propagating. The threshold for vanishing modulational instability due to a finite degree of spatial coherence depends only on system parameters and not on the light intensity.     

Polariton effect in nonlinear pulse propagation

The joint influence of the polariton effect and Kerr-like nonlinearity on the propagation of optical pulses is studied. The existence of different families of envelope solitary wave solutions in the vicinity of the polariton gap is shown. The properties of solutions depend strongly on the carrier wave frequency. In particular, solitary waves inside and outside the polariton gap exhibit different velocity and amplitude dependences on their duration.     

Intrinsic Localized Spin Wave Modes and Modulational Instability in a Two-Dimensional Heisenberg Ferromagnet

An analytical study on the properties of intrinsic localized modes and modulational instability in a quantum two-dimensional ferromagnet with single-ion uniaxial anisotropy is completed in the semiclassical limit. By making use of the semidiscrete multiple-scale method, we obtain a line localized solution and a radially symmetric localized solution, and analyze their existence conditions. Taking into account that the existence of bright localized solutions is closely linked to the modulational instability of plane waves, we analytically study the discrete modulational instability of plane spin waves. The result of the modulational instability analysis show that the uniaxial anisotropy plays a key role in the appearance of our intrinsic localized spin wave modes.     

Modulational Instability of Dust Ion Acoustic Waves in a Collisional Dusty Plasma

The modulational instability of dust ion acoustic waves in a dust plasma with ion-dust collision effects is studied. Using the perturbation method, a modified nonlinear Schrodinger equation contains a damping term that comes from the effect of the ion-dust collision is derived. It is found that the inclusion of the ion-dust collision would modify the modulational instability of the wave packet and could not admit any stationary envelope solitary waves.     

Modulational Instability of Dust Ion Acoustic Waves in a Collisional Dusty Plasma

The modulational instability of dust ion accoustic waves in a dust plasma with ion-dust collision effects is studied.Using the perturbation method,a modified nonlinear Schroedinger equation contains a damping term that comes from the effect of the ion-dust collision is derived.It is found that the inclusion of the ion-dust collision would modify the modulational instability of the wave packet and could not admit any stationary envelope solitary waves.     

Modulational instability of dark solitons in three wave resonant interaction

We study the propagation of velocity-locked dark triplet solitons in the three wave resonant interaction model. The modulational instability of the plane wave background where the solitons sit prevents the long range propagation. However even a small second order dispersion proves to greatly reduce, or suppress, the modulational instability gain, allowing for effective stable soliton propagation.     

无碰撞电流片低频电磁模不稳定性：MHD模型

利用含无电阻广义Ohm定律的可压缩磁流体力学(MHD)理论，研究了在具有剪切磁场的无碰撞电流片中低频电磁模不稳定性，假定等离子体压力各向同性，推导出了三维扰动传播波模的色散关系.色散关系的数值求解集中在电流片中间平面(z=0)和半厚度边缘(z=1)上，并分别考虑了二维传播和三维传播，以及不同的离子惯性长度情况.主要结果如下：1)对 于二维扰动传播(k_z=0)的波，在z=0平面上，Alfven波增长率最大，不稳定的波 频率 和波数范围也更宽.离中间平面越远，增长率越小，波数区域越小.同时，随着离子惯性长度 的增大，Alfven波不稳定性的增长率变大.2)对于三维扰动传播(k_z≠0)的波， 哨声是 不稳定的.在电流片中间平面上，哨声有明显的增长率；而在离子惯性区外边，哨声的增长 率还变大.3)在电流片中间(z=0)平面上，低频波主要是电流不稳定性激发的.在离中间 平面较远处，电流、密度和压力的梯度不稳定性变得更重要. 关键词： 无碰撞电流片 磁流体力学 色散关系 不稳定性     

Modulational instability of dust acoustic waves in a dusty plasma with nonthermal electrons and ions

Effects of nonadiabaticity of variable dust charge, dust fluid temperature, trapped electrons as well as nonisothermality of ions on the amplitude modulation of dust acoustic waves in an unmagnetized dusty plasma are investigated. A modified nonlinear Schr?dinger equation (NLSE) is obtained by the standard reductive perturbation technique and is solved numerically by the split-step Fourier method. The modulational instability and the envelope solitary wave structure are found to be modified somewhat by the effects of nonthermally distributed ions and trapped electrons.     

Note on wavefront dislocation in surface water waves

At singular points of a wave field, where the amplitude vanishes, the phase may become singular and wavefront dislocation may occur. In this Letter we investigate for wave fields in one spatial dimension the appearance of these essentially linear phenomena. We introduce the Chu-Mei quotient as it is known to appear in the ‘nonlinear dispersion relation’ for wave groups as a consequence of the nonlinear transformation of the complex amplitude to real phase-amplitude variables. We show that unboundedness of this quotient at a singular point, related to unboundedness of the local wavenumber and frequency, is a generic property and that it is necessary for the occurrence of phase singularity and wavefront dislocation, while these phenomena are generic too. We also show that the ‘soliton on finite background’, an explicit solution of the NLS equation and a model for modulational instability leading to extreme waves, possesses wavefront dislocations and unboundedness of the Chu-Mei quotient.