Another possible model equation for long waves in nonlinear dispersive systems

In the same spirit in which Benjamin, Bona, and Mahoney modified the Korteweg-de Vries equation (U_x+U_t+UU_x+U_xxx=0) to obtain the so-called BBM equation, U_x+U_t+UU_x?U_xxt=0, we propose a different modification: U_x+U_t+UU_x+U_xtt=0. The advantages in this equation are 1) the system is conservative since it can be derived from the Lagrangian density $\text{L=}\text{1}\text{2}\text{θ}{}_{\mathrm{x}}\text{θ}{}_{\mathrm{t}}\text{+}\text{1}\text{2}\text{θ}{}^2{}_{\mathrm{x}}\text{+}\text{1}\text{6}\text{θ}{}^3{}_{\mathrm{x}}\text{?}\text{1}\text{2}\text{θ}{}^2{}_{\mathrm{xt}}\text{,}\text{where}\text{θ}{}_{\mathrm{x}}\text{≡ U;2)}$ for large wave-numbers |k|, the infinitesimal-wave phase speed falls off like $\text{1}\text{|k|}$, in accord with physical intuition; 3) since the equation is of second order in t, both U and U_t can be independently specified for t = 0. Several conservation laws satisfied by solutions to this equation are given.     

Envelope-soliton propagation for three interacting coherent excitations in a dispersive medium

An initial value problem is set up to describe propagation of a low-frequency wave-field interacting with two almost transparent wave-fields in a dispersive medium. With no linear loss, perfect phase-matching, and equal group velocities for the two high-frequency wave-packets, it is shown how the solution of the above problem can evolve to well-known soliton solutions of the sine-Gordon equation. Other attempts for solving the more general problem in which all the group velocities are different are also discussed.     

Secular-free solution up to third order of a model nonlinear equation for dispersive waves

The perturbation method of Lindstedt is applied to study the non linear effect of a nonlinear equation $$\nabla ^2 {\rm E} - \frac{1}{{c^2 }}\frac{{\partial ^2 {\rm E}}}{{\partial t^2 }} - \frac{{\omega _0^2 }}{{c^2 }}{\rm E} + \frac{{2v}}{{c^2 }}\frac{{\partial {\rm E}}}{{\partial t}} + E^2 \left[ {\frac{{\partial {\rm E}}}{{\partial t}} \times A} \right] = 0,$$ where (A. E)=0 andA,c, ω ₀ andν are constants in space and time. Amplitude dependent frequency shifts and the solution up to third order are derived.     

Nonlocal models for nonlinear, dispersive waves

Considered herein are model equations for the unidirectional propagation of small-amplitude, nonlinear, dispersive, long waves such as those governed by the classical Korteweg-de Vries equation. Of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial, so that the operator modelling dispersive effect is nonlocal. Particular cases in view here are the Benjamin-Ono equation and the intermediate long-wave equation which arise in internal-wave theory, and Smith's equation which governs certain types of continental-shelf waves.

The initial-value problem for these equations is shown to be globally well posed in the classical sense, including continuous dependence upon the initial data and, in certain cases upon the modelling of nonlinear and dispersive effects. Whilst the results are stated for the specific equations listed above, the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear, dispersive evolution equations. Particularly worthy of note is our theorem implying that solutions of the intermediate long-wave equation converge strongly to solutions of the Korteweg-de Vries equation, or to solutions of the Benjamin-Ono equation, in appropriate asymptotic limits.     

New equation for the description of inelastic interaction of nonlinear localized waves in dispersive media

A wave equation for the simulation of nonlinear plane solitary perturbations of the free surface of a shallow fluid has been derived. In contrast to the modified Boussinesq equation, the new one correctly describes the interaction of counter-propagating small-amplitude waves. It has been shown analytically that collisions of solitons are inelastic even in the first-order perturbation theory and the nonlinear dynamics of such collisions is qualitatively different from that described by the modified Boussinesq equation.     

A model for the propagation of nonlinear dispersive strain waves in a solid with defect clusters

A model for the propagation of nonlinear dispersive one-dimensional longitudinal strain waves in an isotropic solid with quadratic nonlinearity of elastic continuum is developed with taking into account the interaction with atomic defect clusters. The governing nonlinear dispersive-dissipative equation describing the evolution of longitudinal strain waves is derived. An approximate solution of the model equation was derived which describes asymmetrical distortion of geometry of the solitary strain wave due to the interaction between the strain field and the field of clusters. The contributions of the finiteness of the relaxation times of cluster-induced atomic defects to the linear elastic modulus and the lattice dissipation and dispersion parameters are determined. The amplitudes and width of the nonlinear waves depend on the elastic constants and on the properties of the defect subsystem (atomic defects, clusters) in the medium. The explicit expression is received for the sound velocity dependence upon the fractional cluster volume, which is in good agreement with experiment. The critical value of cluster volume fraction for the influence on the strain wave propagation is determined.     

Breaking of stationary waves in nonlinear dispersive media

A nontrivial behaviour of a nonlinear wave under influence of small disturbing factors like dissipation, smooth inhomogeneity of medium parameters, etc. is considered by the example of sine-Gordon equation. The stage of slow “adiabatic” variation of the parameters of quasi-stationary wave is shown to change at some finite distance due to strong instability. The wave form becomes essentially nonstationary (breaking of stationary wave structure). The breaking condition is defined by the extremum of the wave adiabatic invariant. The behaviour of a wave at the nonadiabatic stage is described using a Galerkin procedure.     

Coupled nonlinear waves in a composite magnetic-semiconducting medium

In this Letter we investigate the propagation of coupled nonlinear Alfven-spin and helicon-spin waves in a composite medium having the properties of both magnetic and semiconducting materials. We arrive at the nonlinear evolution equations for these coupled waves using the reductive perturbation method for which we obtain soliton solutions. We also discuss the limiting cases for the two materials separately.     

Effect of gravitational radiation upon electromagnetic waves in a dispersive medium

We show that linearized gravitational radiation produces fluctuations in intensity and position of a distant source if the ray travels in a dispersive medium. The effect, however, depends upon the nongeodesic character of the ray and does not occur in an electrostatic plasma. When the index of refraction n is greater than unity a Cerenkov type resonance produces scintillation proportional to D _O ^3/2 (D _O being the distance of the source) and a dancing proportional to D _O ^1/2 if, instead, n<1 the scintillation behaves like D _O and the dancing does not diverge as D _O . The calculation is performed in detail for a random and isotropic spectrum of gravitational waves W(). This effect allows one to set an upper limit to W() at the frequency at which the fluctuations are observed, but for the rarified interstellar and intergalactic plasmas these limits are not very interesting.     

A new model equation for nonlinear Rossby waves and some of its solutions

In this paper, we investigate the $(2+1)$ dimensional nonlinear Rossby waves under non-traditional approximation. Using the asymptotic methods of multiple scales and weak nonlinear perturbation expansions, we derive a new modified Zakharov–Kuznetsov equation from the barotropic potential vorticity equation with the complete Coriolis parameter, the topography and the dissipation. Based on the new auxiliary equation method, new exact solutions of the new mZK equation are obtained when the dissipation is absent. However, the new auxiliary equation method fails to solve the new mZK equation with the dissipative term. Therefore, the weak nonlinear method and the homotopy perturbation method are developed to solve the obtained new mZK equation. Through numerical simulations, the results show the effects of different parameters on Rossby waves.     

通过被动介质耦合的两螺旋波的同步

采用Br模型研究了通过被动介质耦合的两二维可激发系统中螺旋波的同步,被动介质由可激发元素组成,这些元素之间不存在耦合.数值模拟结果表明,被动介质对螺旋波的同步有很大影响,当两系统中的初态螺旋波相同时,被动介质可导致稳定螺旋波发生漫游,螺旋波转变为螺旋波对或反靶波;当两系统中的初态螺旋波不同步时,在适当的参数下,两螺旋波可以实现同步、相同步,此外还观察到两螺旋波波头相互排斥、多螺旋波共存、同步的时空周期斑图、系统演化到静息态等现象.在被动介质中,一般可观察到波斑图,但是在某些情况下,被动介质会出现同步振荡现象.这些结果有助于人们理解心脏系统中出现的时空斑图.     

Traveling waves in a nonlinear medium with cubic nonlinearity

Unlike linear nondispersive media, which allow propagation of wave packets of arbitrary forms, nonlinear media admit only certain profiles of traveling waves. Here we examine media with Duffing oscillators, i.e., with bound electrons for which an equilibrium disturbance causes forces proportional to the first and third powers of deviation. We show that the linearly polarized traveling plane waves such media can transmit have profiles modulated as Jacobi elliptic functions. When discussing propagation across an interface between different media, only incidence from the side of the linear medium is considered. Even in this case, to launch a traveling wave in the nonlinear medium, a severe restriction must be imposed on the incident wave’s amplitude.     

Finite-amplitude standing acoustic waves in a cubically nonlinear medium

The acoustic field in a resonator filled with a cubically nonlinear medium is investigated. The field is represented as a linear superposition of two strongly distorted counterpropagating waves. Unlike the case of a quadratically nonlinear medium, the counterpropagating waves in a cubically nonlinear medium are coupled through their mean (over a period) intensities. Free and forced standing waves are considered. Profiles of discontinuous oscillations containing compression and expansion shock fronts are constructed. Resonance curves, which represent the dependences of the mean field intensity on the difference between the boundary oscillation frequency and the frequency of one of the resonator modes, are calculated. The structure of the profiles of strongly distorted “forced” waves is analyzed. It is shown that discontinuities are formed only when the difference between the mean intensity and the detuning takes certain negative values. The discontinuities correspond to the jumps between different solutions to a nonlinear integro-differential equation, which, in the case of small dissipation, degenerates into a third-degree algebraic equation with an undetermined coefficient. The dependence of the intensity of discontinuous standing waves on the frequency of oscillations of the resonator boundary is determined. A nonlinear saturation is revealed: at a very large amplitude of the resonator wall oscillations, the field intensity in the resonator ceases depending on the amplitude and cannot exceed a certain limiting value, which is determined by the nonlinear attenuation at the shock fronts. This intensity maximum is reached when the frequency smoothly increases above the linear resonance. A hysteresis arises, and a bistability takes place, as in the case of a concentrated system at a nonlinear resonance.     

Transmission of waves by a nonlinear random medium

We study analytically and numerically the effect of nonlinearity on transmission of waves through a random medium. We introduce and analyze quantities associated with the scattering problem that clarify the lack of uniqueness due to the nonlinearity as well as the localization of waves due to the random inhomogeneities. We show that nonlinearity tends to delocalize the waves and that for very large scattering regions the average transmitted energy is small.     

Spatiotemporal instability in dispersive nonlinear Kerr medium with a finite response time

We investigated the spatiotemporal modulation instability (MI) in a medium with a non-instantaneous Kerr response in which the nonlinear contribution to the refractive index is governed by a relaxing Debye type equation. The expression for the MI gain spectrum in non-instantaneous Kerr medium is obtained, which clearly reveals the influence of the finite time of the nonlinear response (relaxation effect) on the spatiotemporal MI. It is shown that, due to the relaxation effect, the spatiotemporal MI can appear for the case of anomalous dispersion and defocusing nonlinearity, while it cannot appear in instantaneous Kerr medium for the same case, and a new MI gain spectrum appears, adding to the conventional MI gain spectrum similar to that in an instantaneous Ker medium. In addition, the bandwidth of MI gain spectrum is extended and the maximum MI gain is reduced with increasing of the relaxation time. Interestingly, spatiotemporal MI can appear for any spatial frequencies in non-instantaneous Kerr medium for any combination of dispersion and nonlinearity. We have performed an experiment of MI in carbon disulfide (CS₂), a typical non-instantaneous Kerr medium, which shows quantitative agreement with the theoretical analyses.     

Statistical properties of a wave with random frequency modulation in a dispersive medium

The probability distributions of frequency, intensity, and other parameters of a wave with random frequency modulation propagating in a dispersive medium are determined and studied. In particular, the probability distributions for photon density are found for the limiting cases of linear and quadratic medium dispersion characteristics. It is demonstrated that in the latter case the photon density probability distribution has a singularity produced by the absence of time focusing of the wave in the vicinity of dispersion curve inflection points. The statistics of wave frequency in a multifrequency regime are considered in detail.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 21, No. 12, pp. 1797–1802, December, 1978.The authors thank A. N. Malakhov for his interest in the study.