Dispersion-induced dynamical transition in parametric solitary waves

We investigate the influence of dispersion on parametric solitary waves. We show that dispersion is responsible for a transition towards a new type of dynamical solitary wave characterized by the presence of traveling phase defect arrays within their envelopes. The transition is described analytically through an original extension of the Kolmogorov-Petrovskii-Piskunov approach to front propagation into unstable states.     

Dispersion management for solitons in a Korteweg-de Vries system

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.     

Rarefactive ion-acoustic electrostatic solitary structures in nonthermal plasmas

An investigation has been made of ion-acoustic solitary waves in an unmagnetized nonthermal plasma whose constituents are an inertial ion fluid and nonthermally distributed electrons. The properties of stationary solitary structures are briefly studied by the pseudo-potential approach, which is valid for arbitrary amplitude waves, and by the reductive perturbation method which is valid for small but finite amplitude limit. The time evolution of both compressive and rarefactive solitary waves, which are found to coexist in this nonthermal plasma model, is also examined by solving numerically the full set of fluid equations. The temporal behaviour of positive (compressive) solitary waves is found to be typical, i.e., the positive initial disturbance breaks up into a series of solitary waves with the largest in front. However, the behaviour of negative (rarefactive) solitary waves is quite different. These waves appear to be unstable and produce positive solitary waves at a later time. The relevancy of this investigation to observations in the magnetosphere of density depressions is briefly pointed out. Received 12 October 1999     

负折射介质中孤波间相互作用

以描述负折射介质中超短脉冲传输的归一化非线性薛定谔方程为模型,采用对称分步傅里叶算法研究了负折射介质中亮、暗孤波间的相互作用.数值模拟发现:当孤波的初始频移为零时,亮孤波间的相互作用与常规介质中类似;当孤波的初始频移不为零时,其传输速度和相互作用明显受三阶色散和自陡峭效应的影响,主要表现为相互排斥.而负折射介质中暗孤波间的相互作用与常规介质中的相互作用类似,无论暗孤波是否存在初始频移,暗孤波间的相互作用在三阶色散和自陡峭的影响下都表现为相互排斥.结果表明,通过调节三阶色散和自陡峭系数可以在一定程度上抑制负折射介质中亮、暗孤波间的相互作用.该研究结果为负折射介质在未来高速通信中的应用提供了理论依据.     

负折射介质中孤波间相互作用

以描述负折射介质中超短脉冲传输的归一化非线性薛定谔方程为模型,采用对称分步傅里叶算法研究了负折射介质中亮、暗孤波间的相互作用.数值模拟发现:当孤波的初始频移为零时,亮孤波间的相互作用与常规介质中类似;当孤波的初始频移不为零时,其传输速度和相互作用明显受三阶色散和自陡峭效应的影响,主要表现为相互排斥.而负折射介质中暗孤波间的相互作用与常规介质中的相互作用类似,无论暗孤波是否存在初始频移,暗孤波间的相互作用在三阶色散和自陡峭的影响下都表现为相互排斥.结果表明,通过调节三阶色散和自陡峭系数可以在一定程度上抑制负折射介质中亮、暗孤波间的相互作用.该研究结果为负折射介质在未来高速通信中的应用提供了理论依据.     

Interfacial waves with free-surface boundary conditions: an approach via a model equation

In a two-fluid system where the lower fluid is bounded below by a rigid bottom and the upper fluid is bounded above by a free surface, two kinds of solitary waves can propagate along the interface and the free surface: classical solitary waves characterized by a solitary pulse or generalized solitary waves with nondecaying oscillations in their tails in addition to the solitary pulse. The classical solitary waves move faster than the generalized solitary waves. The origin of the nonlocal solitary waves can be understood from a physical point of view. The dispersion relation for the above system shows that short waves can propagate at the same speed as a “slow” solitary wave. The interaction between the solitary wave and the short waves creates a nonlocal solitary wave. In this paper, the interfacial-wave problem is reduced to a system of ordinary differential equations by using a classical perturbation method, which takes into consideration the possible resonance between short waves and “slow” solitary waves. In the past, classical Korteweg–de Vries type models have been derived but cannot deal with the resonance. All solutions of the new system of model equations, including classical as well as generalized solitary waves, are constructed. The domain of validity of the model is discussed as well. It is also shown that fronts connecting two conjugate states cannot occur for “fast” waves. For “slow” waves, fronts exist but they have ripples in their tails.     

Generation of Solitary Rossby Waves by Unstable Topography

The effect of topography on generation of the solitary Rossby waves is researched.Here,the topography,as a forcing for waves generation,is taken as a function of longitude variable x and time variable t,which is called unstable topography.With the help of a perturbation expansion method,a forced mKdv equation governing the evolution of amplitude of the solitary Rossby waves is derived from quasi-geostrophic vorticity equation and is solved by the pseudospectral method.Basing on the waterfall plots,the generational features of the solitary Rossby waves under the influence of unstable topography and stable topography are compared and some conclusions are obtained.     

Vortex rings and solitary waves in trapped Bose–Einstein condensates

We discuss nonlinear excitations in an atomic Bose–Einstein condensate which is trapped in a harmonic potential. We focus on axially symmetric solitary waves propagating along a cylindrical condensate. A quasi one-dimensional dark soliton is the only nonlinear mode for a condensate with weak interactions. For sufficiently strong interactions of experimental interest solitary waves are hybrids of one-dimensional dark solitons and three-dimensional vortex rings. The energy-momentum dispersion of these solitary waves exhibits characteristics similar to a mode proposed sometime ago by Lieb in a strictly 1D model, as well as some rotonlike features. We subsequently discuss interactions between solitary waves. Head-on collisions between dark solitons are elastic. Slow vortex rings collide elastically but faster ones form intermediate structures during collisions before they lose energy to the background fluid. Solitary waves and their interactions have been observed in experiments. However, some of their intriguing features still remain to be experimentally identified.     

On estimates for short wave stability and long wave instability in three-layer Hele-Shaw flows

We consider the linear stability of three-layer Hele-Shaw flows with each layer having constant viscosity and viscosity increasing in the direction of a basic uniform flow. While the upper bound results on the growth rate of long waves are well known from our earlier works, lower bound results on the growth rate of short stable waves are not known to date. In this paper, we obtain such a lower bound. In particular, we show the following results: (i) the lower bound for stable short waves is also a lower bound for all stable waves, and the exact dispersion curve for the most stable eigenvalue intersects the dispersion curve based on the lower bound at a wavenumber where the most stable eigenvalue is zero; (ii) the upper bound for unstable long waves is also an upper bound for all unstable waves, and the exact dispersion curve for the most unstable eigenvalue intersects the dispersion curve based on the upper bound at a wavenumber where the most unstable eigenvalue is zero. Numerical results are provided which support these findings.     

Exact solutions and localized excitations of a （3＋ 1）-dimensional Gross-Pitaevskii system

Periodic wave solutions and solitary wave solutions to a generalized (3+1)-dimensional Gross-Pitaevskii equation with time-modulated dispersion, nonlinearity, and potential are derived in terms of an improved homogeneous balance principle and a mapping approach. These exact solutions exist under certain conditions via imposing suitable constraints on the functions describing dispersion, nonlinearity, and potential. The dynamics of the derived solutions can be manipulated by prescribing specific time-modulated dispersions, nonlinearities, and potentials. The results show that the periodic waves and solitary waves with novel behaviors are similar to the similaritons reported in other nonlinear systems.     

Exact solutions and localized excitations of (3+1)-dimensional Gross—Pitaevskii system

Periodic wave solutions and solitary wave solutions to a generalized (3+1)-dimensional Gross--Pitaevskii equation with time-modulated dispersion, nonlinearity, and potential are derived in terms of an improved homogeneous balance principle and a mapping approach. These exact solutions exist under certain conditions via imposing suitable constraints on the functions describing dispersion, nonlinearity, and potential. The dynamics of the derived solutions can be manipulated by prescribing specific time-modulated dispersions, nonlinearities, and potentials. The results show that the periodic waves and solitary waves with novel behaviors are similar to similaritons reported in other nonlinear systems.     

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.     

Modulational instability of relativistic ion-acoustic waves in aplasma with trapped electrons

The association between the modified Korteweg-de Vries solitary wave and the modulationally unstable envelope solitary wave in a weakly relativistic unmagnetized plasma with trapped electrons is discussed. The effect of trapped electrons modifies the nonlinearity of the nonlinear Schrodinger equation and gives rise to the propagation of the modulationally unstable ion-acoustic solitary wave. The amplitude of the envelope solitary wave increases while the number of trapped electrons decreases. The velocity of the solitary wave decreases with increasing ionic temperature and increasing particle velocities. The ion oscillation mode, which satisfies the nonlinear dispersion relation, is also derived. The theory is applied to explain space observations of the solar energetic flows in interplanetary space and of the energetic particle events in the Earth's magnetosphere     

强耦合尘埃等离子体中正电扰动下的尘埃声激波

研究了强耦合尘埃等离子体的尘埃声波的线性色散关系和尘埃声孤波的非线性传播。考虑一个包含电子、离子、正电扰动尘埃颗粒的完全电离的三成分模型等离子体。假定其电子、离子数密度服从玻尔兹曼分布,而大质量的尘埃成分用一组经典流体方程描述,对系统方程进行线性化,得到了尘埃声波的线性色散关系,发现离子的集中参数对色散关系的影响很大。用约化摄动法对系统方程进行展开,得到了描述小振幅孤波的伯格斯方程。基于伯格斯方程研究了尘埃声孤波的基本特性,发现尘埃颗粒的强耦合效应对尘埃声孤波有很大的修正作用。该研究结果有助于理解尘埃空间等离子体中局域波的一些特性。     

Electrodynamics and dispersion properties of a magnetoplasma containing elongated and rotating dust grains

The electrodynamics and dispersion properties of a magnetized dusty plasma containing elongated and rotating charged dust grains are examined. Starting from an appropriate Lagrangian for dust grains, a kinetic equation for the dust grain and the corresponding equations of motion are derived. Expressions for the dust charge and dust current densities are obtained with the finite size (the dipole moment) of elongated and rotating dust grains taken into account. These charge and current densities are combined with the Maxwell-Vlasov system of equations to derive dispersion relations for the electromagnetic and electrostatic waves in a dusty magnetoplasma. The dispersion relations are analyzed to demonstrate that the dust grain rotation introduces new classes of instabilities involving various low-frequency waves in a dusty magnetoplasma. Examples of various unstable low-frequency waves include the electron whistler, the dust whistler, dust cyclotron waves, AlfvÉn waves, electromagnetic ion-cyclotron waves, as well as lower-hybrid, electrostatic ion cyclotron, modified dust ion-acoustic waves, etc. Also found is a new type of unstable waves whose frequency is close to the dust grain rotation frequency. The present results should be useful in understanding the properties of low-frequency waves in cosmic and laboratory plasmas that are embedded in an external magnetic field and contain elongated and rotating charged dust grains.     

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研究了强耦合尘埃等离子体的尘埃声波的线性色散关系和尘埃声孤波的非线性传播。考虑一个包含电子、离子、正电扰动尘埃颗粒的完全电离的三成分模型等离子体。假定其电子、离子数密度服从玻尔兹曼分布,而大质量的尘埃成分用一组经典流体方程描述,对系统方程进行线性化,得到了尘埃声波的线性色散关系,发现离子的集中参数对色散关系的影响很大。用约化摄动法对系统方程进行展开,得到了描述小振幅孤波的伯格斯方程。基于伯格斯方程研究了尘埃声孤波的基本特性,发现尘埃颗粒的强耦合效应对尘埃声孤波有很大的修正作用。该研究结果有助于理解尘埃空间等离子体中局域波的一些特性。     

Anomalous dispersion in the Belousov-Zhabotinsky reaction: Experiments and modeling

We report results on dispersion relations and instabilities of traveling waves in excitable systems. Experiments employ solutions of the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction confined to thin capillary tubes which create a pseudo-one-dimensional system. Theoretical analyses focus on a three-variable reaction-diffusion model that is known to reproduce qualitatively many of the experimentally observed dynamics. Using continuation methods, we show that the transition from normal, monotonic to anomalous, single-overshoot dispersion curves is due to an orbit flip bifurcation of the solitary pulse homoclinics. In the case of “wave stacking”, this anomaly induces attractive pulse interaction, slow solitary pulses, and faster wave trains. For “wave merging”, wave trains break up in the wake of the slow solitary pulse due to an instability of wave trains at small wavelength. A third case, “wave tracking” is characterized by the non-existence of solitary waves but existence of periodic wave trains. The corresponding dispersion curve is a closed curve covering a finite band of wavelengths.     

Dark-in-the-Bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms

We present new type of Dark-in-the-Bright solution also called dipole soliton for the higher order nonlinear Schrödinger (HNLS) equation with non-Kerr nonlinearity under some parametric conditions and subject to constraint relation among the parameters in optical context. This equation could be a model equation of pulse propagation beyond ultrashort range in optical communication systems. The solitary wave solution is composed of the product of bright and dark solitary waves. This type of pulse shape to be formed both the group velocity dispersion and third-order dispersion must be compensated. We also investigated the stability of the solitary wave solution under some initial perturbation on the parametric conditions. We have shown that the shape of pulse remains unchanged up to 20 normalized lengths even under some very small violation in parametric conditions.     

On the stability of nonlinear waves in integrable models

A new method of stability investigation is presented for solutions of nonlinear equations integrable with the help of the inverse scattering transform (IST). The stability problem for periodic nonlinear waves in weakly dispersive media is solved with respect to transverse perturbations. It is shown that for positive dispersion media one-dimensional waves are unstable, and for negative dispersion such waves are stable.