Localized waves in nonlinear oscillator chains

This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.     

Finite-Wavelength Stability¶of Capillary-Gravity Solitary Waves

We consider the Euler equations describing nonlinear waves on the free surface of a two-dimensional inviscid, irrotational fluid layer of finite depth. For large surface tension, Bond number larger than 1/3, and Froude number close to 1, the system possesses a one-parameter family of small-amplitude, traveling solitary wave solutions. We show that these solitary waves are spectrally stable with respect to perturbations of finite wave-number. In particular, we exclude possible unstable eigenvalues of the linearization at the soliton in the long-wavelength regime, corresponding to small frequency, and unstable eigenvalues with finite but bounded frequency, arising from non-adiabatic interaction of the infinite-wavelength soliton with finite-wavelength perturbations. Received: 7 February 2001 / Accepted: 6 October 2001     

Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators

We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schrödinger equation (M. Remoissenet, Phys. Rev. B 33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlinearity 17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchgässner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss, K. Kirchgässner, Commun. Math. Phys. 211, 439–464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchgässner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schrödinger approximation.     

Conjugate Gradient method for finding fundamental solitary waves

The Conjugate Gradient method (CGM) is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices. In this paper, we modify this method so that it can find fundamental solitary waves of nonlinear Hamiltonian equations. The main obstacle that such a modified CGM overcomes is that the operator of the equation linearized about a solitary wave is not sign definite. Instead, it has a finite number of eigenvalues on the opposite side of zero than the rest of its spectrum. We present versions of the modified CGM that can find solitary waves with prescribed values of either the propagation constant or power. We also extend these methods to handle multi-component nonlinear wave equations. Convergence conditions of the proposed methods are given, and their practical implications are discussed. We demonstrate that our modified CGMs converge much faster than, say, Petviashvili’s or similar methods, especially when the latter converge slowly.     

孤子内波导致海洋环境噪声垂直指向性异常分析

依据近场波数积分、远场耦合简正波相结合的二维噪声场模型,侧重理论研究孤子内波所在扇区,环境噪声垂直阵响应的变化,分析了某些孤子内波情形下垂直阵环境噪声水平凹槽变深这一异常现象的原因:孤子内波离垂直阵较近时,远离内波的海面噪声源多,其激发的简正波能量由低号耦合到高号,在垂直阵处高号简正波能量对环境噪声场贡献增大,导致环境噪声水平凹槽加深;对于大尺度、多波包孤子内波,其范围相对较大,内波所在区的局部简正波本征值和本征函数产生的变化影响显著,使低号简正波衰减变快,而高号衰减慢,导致接收阵处高号简正波能量增加,低号简正波变弱,这样,无论孤子内波群靠近或离接收阵远,都将使垂直阵环境噪声水平凹槽加深。     

On the acoustic modes in a duct containing a parabolic shear flow

The exact solution of the acoustic wave equation in an unidirectional shear flow with a parabolic velocity profile is obtained, representing sound propagation in a plane, parallel walled duct, with two boundary layers over rigid or impedance walls. It is shown that there are four cases, depending on the critical level(s) where the Doppler shifted frequency vanishes: (i) for propagation upstream the critical levels are outside the duct (case II); (ii) for propagation downstream there may be two (case IV), one (case I) or no (case III) critical level inside the duct. The acoustic wave equation is transformed in each of the four cases to particular forms of the extended hypergeometric equation, which has power series solutions, some involving logarithmic singularities. In the cases where critical levels occur, at real or ‘imaginary’ distance, matching of two or three pairs of solutions, valid over regions each overlapping the next, is needed. The particular case of the parabolic velocity profile is used to address general properties of sound in unidirectional shear flows. For example, it is shown that for ducted shear flows, there exist a pair of even and odd eigenfunctions, in the absence of critical levels. It is also proved, in more than one instance, that there is no single set of eigenvalues and eigenfunctions valid across one or two shear layers. This leads to the general conjecture, considering the acoustics of shear flows in ducts, that critical levels separate regions with distinct sets of eigenvalues and eigenfunctions.     

Interaction of Three Electromagnetic Fields with a Resonance Medium in EPR. The Case of a Small Local Field in Comparison with the Saturating One

A system of Bloch equations modified with allowance for the presence of a dipole–dipole reservoir for the case where the local magnetic field is small in comparison with the saturating one is suggested. The system is used for solving the problem of interaction of three electromagnetic fields: a saturating field, a probe one, and the third - a combination field resulting from the interaction of the first two in a resonance medium. The imaginary and real parts of the system susceptibility at the probe-field frequency have been investigated in detail at both different frequencies of interacting waves and coinciding ones (degenerate case). For the degenerate case, the dependence of the coefficient of the parametric connection of waves on the frequency is considered. The results of the present work are compared with those obtained by us earlier for the case where the local magnetic field is much in excess of the saturating one (Provotorov's case). It is shown that in the problem considered the amplification of weak waves when they pass through the layer of an absorbing resonance medium is inaccessible.     

A new bifurcation in non-smooth continuous system inducing semi-limit cycle

In this paper, we presented a study on a non-smooth continuous system with emphasis on a special bifurcation. As the parameter varies, a series of concentric closed orbits appear near the equilibrium point. Moreover, the outermost closed orbit attracts all the trajectories outside. It is called as a semi-limit cycle as the trajectories at only one side of this orbit are attracted. By using the theory of generalized Jacobian matrix, it is revealed that this bifurcation can be featured by a pair of complex conjugate eigenvalues reaching exactly but not crossing the imaginary axis. The bifurcation can somewhat be considered to be a degenerate case of the Hopf bifurcation, in which the eigenvalues cross the imaginary axis totally. This study enriches the knowledge of bifurcation analysis for non-smooth dynamical systems.     

Solitary wave propagation in quantum electron-positron plasmas

Existence of large amplitude stationary solitary wave structures in an unmagnetized electron-positron (e-p) plasma is studied using a quantum hydrodynamic (QHD) model that includes the quantum force (tunnelling) associated with the Bohm potential and the Fermi-dirac pressure law. It is found that in a quasi-neutral pair (e-p) plasma, where the dispersion is only due to the the quantum tunnelling effects, the large amplitude stationary solitary structure exists only when the normalized Mach speed,M <√2. Such solitary structures do not exist in absence of the Bohm potential term in an unmagnetized quasineutral pair (e-p) plasma. The system is shown to support only rarefactive stationary solitary waves. For such waves the amplitude, being independent of the quantum parameter H (the ratio of the electron plasmon to electron Fermi energy), decreases with the Mach number M, whereas the width increases with both M and H. The present theory is applicable to analyze the formation of localized coherent solitary structures at quantum scales in dense astrophysical objects as well as in intense laser fields.     

Nonlinear second harmonic generation by light wave-plasma interaction in oscillating magnetic field

The nonlinear generation of second harmonic electromagnetic waves in a thin inhomogeneous (dense and rarefied) plasma layer (of lengthd) by obliquely and normally incident light waves is analyzed. We consider the effect of an external time-dependent magnetic field on the generation and amplification of waves. Two cases are considered, when the magnetic field oscillates at a frequency (i) equal to and (ii) double that of the incident wave. For normal incidence, waves are not radiated in case (i), while in case (ii) the second harmonics are radiated equally from the plasma boundaries atx=0 andx=d. For a rarefied plasma, the second harmonics are radiated with equal amplitudes in both cases.     

On the generation of solitons and breathers in the modified Korteweg-de Vries equation

We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg-de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem. (c) 2000 American Institute of Physics.     

Effect of interaction of electromagnetic fields with a resonant medium in epr when the saturating field is small compared with the local field

We propose a system of Bloch equations, modified to take into account the presence of a dipole-dipole reservoir (DDR), for the case when the saturating magnetic field is small compared with the local field. We take into account the transverse and longitudinal magnetizations in the equation for the DDR, in contrast to our previous papers in which we took into account only the longitudinal magnetization. Using the system obtained, we solve the problem of the interaction of three fields, where one is the saturating field, the second is the probe field, and the third is a combination field that is the result of the interaction of the first two fields in a resonant medium. We have studied the imaginary and real parts of the susceptibility of the system at the probe field frequency, both when the interacting waves have different frequencies and when they have matching frequencies (the degenerate case). We have compared the results with those we obtained previously. For the degenerate case, we consider the frequency dependence of the parametric coupling coefficient of the waves. We show that weak waves can be enhanced as they pass through a layer of a resonant absorbing medium. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 73, No. 4, pp. 462–466, July–August, 2006.     

A perturbation method for computing the simplest normal forms of dynamical systems

A previously developed perturbation method is generalized for computing the simplest normal form (at each level of computation, the minimum number of terms are retained) of general n-dimensional differential equations. This “direct” approach, combining the normal form theory with center manifold theory in one unified procedure, can be used to systematically compute the simplest (or unique) normal form. Two particular singularities of the Jacobian of the system are considered in this paper: the first one is associated with one pair of purely imaginary eigenvalues (Hopf-type singularity), and the other corresponds to a simple zero and a pair of purely imaginary eigenvalues (Hopf-zero-type singularity). The approach can be easily formulated and implemented using a computer algebra system. Maple programs have been developed in this paper which can be “automatically” executed by a user without the knowledge of computer algebra. A physical oscillator model is studied in detail to show the computational efficiency of the “direct” method, and the advantage of using the simplest normal form, which greatly simplifies the analysis on dynamical systems, in particular, for bifurcations and stability.     

Backward three-wave optical solitons

Backward symbiotic solitary waves in quadratic media with absorption losses are generated through the nonlinear non-degenerate three-wave interaction. We study these solitary waves in the particular case of a doubly backward quasi-phase matching configuration. The same mechanism responsible for nanosecond solitary wave morphogenesis in the c.w. pumped Brillouin-fiber-ring laser may act for picosecond pulse generation in a quadratic c.w. pumped optical parametric oscillator (OPO). The resonant condition is automatically satisfied in stimulated Brillouin backscattering when the fiber-ring laser contains a large number of longitudinal modes beneath the gain curve. However, in order to achieve quasi-phase matching between the three optical waves in the Χ⁽²⁾ medium, a nonlinear susceptibility inversion grating of sub-μm period is required. Such a quadratic medium supports solitary waves that result from energy exchanges between dispersionless waves of different velocities. We show, by a stability analysis of the non-degenerate backward OPO in the QPM decay interaction between a c.w. pump and backward signal and idler waves that the inhomogeneous stationary solution exhibits a Hopf bifurcation with a single control parameter. Above OPO threshold, the nonlinear dynamics yields self-structuration of a backward symbiotic solitary wave, which is stable for a finite temporal walk-off (i.e. different group velocities) between signal and idler waves. We also study the dynamics of singly backward mirrorless OPO’s (BMOPO’s) pumped by an incoherent field, in line with the recent experimental demonstration of this OPO configuration. We show that this system is characterized, as a general rule, by the generation of a highly coherent backward field, despite the high degree of incoherence of the pump field. This remarkable property finds its origin in two distinct phase-locking mechanisms that originate respectively in the convection and the dispersion properties of the fields. In both cases we show that the incoherence of the pump is transferred to the co-moving field, which thus allows the backward field to evolve towards a highly coherent state. We propose realistic experimental conditions that may be implemented with currently available technology and in which backward coherent wave generation from incoherent excitation may be observed and studied.     

Two cold atoms in a time‐dependent harmonic trap in one dimension

We analyze the dynamics of two atoms with a short‐ranged pair interaction in a one‐dimensional harmonic trap with time‐dependent frequency. Our analysis is focused on two representative cases: (i) a sudden change of the trapping frequency from one value to another, and (ii) a periodic trapping frequency. In case (i), the dynamics of the interacting and the corresponding non‐interacting systems turn out to be similar. In the second case, however, the interacting system can behave quite differently, especially close to parametric resonance. For instance, in the regions where such resonance occurs we find that the interaction can significantly reduce the rate of energy increase. The implications for applications of our findings to cool or heat the system are also discussed.     

Dust‐ion‐acoustic solitary waves in a magnetized dusty pair‐ion plasma with Cairns‐Gurevich electrons and opposite polarity dust particles

The nonlinear dust‐ion‐acoustic (DIA) solitary structures have been studied in a dusty plasma, including the Cairns‐Gurevich distribution for electrons, both negative and positive ions, and immobile opposite polarity dust grains. The external magnetic field directed along the z‐axis is considered. By using the standard reductive perturbation technique and the hydrodynamics model for the ion fluid, the modified Zakharov–Kuznetsov equation was derived for small but finite amplitude waves and was provided the solitary wave solution for the parameters relevant. Using the appropriate independent variable, we could find the modified Korteweg–de Vries equation. By plotting some figures, we have discussed and emphasized how the different plasma values, such as the trapping parameter, the positive (or negative) dust number density, the non‐thermal electron parameter, and the ion cyclotron frequency, can influence the solitary wave structures. In addition, using the bifurcation theory of planar dynamical systems, we have extracted the centre and saddle points and illustrated the phase portrait of such a system for some particular plasma parameters. Finally, we have graphically investigated the behaviour of the solitary energy wave by changing the plasma values as well as by calculating the instability criterion; we have also discussed the growth rate of the solitary waves. The results could be useful for studying the physical mechanism of nonlinear propagation of DIA solitary waves in laboratory and space plasmas where non‐thermal electrons, pair‐ions, and dust particles can exist.     

Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation

We present an explicit analytical form of first and second order rogue waves for distributive nonlinear Schrödinger equation (NLSE) by mapping it to standard NLSE through similarity transformation. Upon obtaining the rogue wave solutions, we study the propagation of rogue waves through a periodically distributed system for the two cases when Wronskian of dispersion and nonlinearity is (i) zero, (ii) not equal to zero. For the former case, we discuss a mechanism to control their propagation and for the latter case we depict the interesting features of rogue waves as they propagate through dispersion increasing and decreasing fiber.     

Quantum Jarzynski equality with multiple measurement and feedback for isolated system

In this paper, we derive the Jarzynski equality (JE) for an isolated quantum system in three different cases: (i) the full evolution is unitary with no intermediate measurements, (ii) with intermediate measurements of arbitrary observables being performed, and (iii) with intermediate measurements whose outcomes are used to modify the external protocol (feedback). We assume that the measurements will involve errors that are purely classical in nature. Our treatment is based on path probability in state space for each realization. This is in contrast with the formal approach based on projection operator and density matrices. We find that the JE remains unaffected in the second case, but gets modified in the third case where the mutual information between the measured values with the actual eigenvalues must be incorporated into the relation.     

Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator

We show analytically the existence of nondegenerate symbiotic solitary waves in quadratic media with absorption losses. We study these new solitary waves in the particular case of a backward quasi-phase-matching configuration. Our numerical simulations reveal that, when it is used inside a singly resonant optical parametric oscillator, this configuration leads to the spontaneous formation of new solitary waves.