Waves In Nonlinear Lattices Research Articles

New impressive representations for the soliton behaviors arising from the (2+1)-Boussinesq equation

In the present work, we will detect unpredicted conducts “which weren't realized before” to the soliton solutions of the (2+1)-Boussinesq equation by using three distinct algorithms. The proposed equation is the famous nonlinear one which distinguishes the waves of coastal and ocean engineering that involve the nonlinearity and dispersion terms. Moreover, it is a developed form of the standard Boussinesq equation which describes the solution interaction mechanism of shallow-water waves that involve many waves and shallow water influences refraction, diffraction, shoaling and weak nonlinearity properties arising in fluid dynamics. It also plays principal role in many physics branches, such as propagation of long waves in shallow water, vibrations in a nonlinear string and ion sound waves in plasma and one-dimensional nonlinear lattice waves. The three algorithms that will select for this purpose are the (G′G) -expansion method, the extended direct algebraic method (EDAM) and the extended simple equation method (ESEM). These three distinct manners are applied in the same time and parallel. We will show comparison between our new soliton solution behaviours with that constructed before.

Supersonic voidions in 2D Morse lattice

Abstract Dynamics of M × 1-crowdion clusters ( M = 1 , 2 or 3) and vacancy clusters is analyzed in 2D triangular lattice with Morse interactions. Topological defects are created in the lattice by kicking three neighboring atoms in the three neighboring close-packed atomic rows along the rows. Initial velocity of the atoms in the inner row, vinn, can be positive or negative, while the velocities of both atoms in the outer rows, vout, are equal and positive. The case of v inn = v out was analysed earlier and it was found that if the initial velocity is sufficiently large then a 3 × 1-crowdion cluster and a three-vacancy are formed. Two major effects are reported in the case of vinn ≠ vout analyzed here. (i) Due to the difference in the initial velocities, topological defects can be produced in the lattice with smaller initial energy than in the case of v inn = v out . (ii) In certain range of the initial velocities, a vacancy can appear at a distance of dozens interatomic distances from the initiation point, moving along the close-packed row with supersonic speed. Such moving vacancy is called voidion and mechanism of its formation and motion is revealed. Our results uncover new mechanisms of mass transfer by solitary waves in nonlinear lattices.

Exact travelling wave solutions of a variety of Boussinesq-like equations

In this paper, we obtain exact traveling wave solutions of a variety of Boussinesq-like equations by using two distinct methods with symbolic computation. The Boussinesq equations play an important role in physical applications, such as in nonlinear lattice waves, acoustic waves, iron sound waves in a plasma, and vibrations in a nonlinear string. More precisely, the modified tanh-coth method is employed to obtain single soliton solutions, and the extended Jacobi elliptic function method is applied to derive doubly periodic wave solutions. Further, it is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions. The employed approaches are quite efficient for the determination of the solutions, and are practically well suited for solving nonlinear evolution equations arising in physics.

Triggering waves in nonlinear lattices: Quest for anharmonic phonons and corresponding mean-free paths

Guided by a stylized experiment we develop a self-consistent anharmonic phonon concept for nonlinear lattices which allows for explicit ``visualization.'' The idea uses a small external driving force which excites the front particles in a nonlinear lattice slab and subsequently one monitors the excited wave evolution using molecular dynamics simulations. This allows for a simultaneous, direct determination of the existence of the phonon mean-free path with its corresponding anharmonic phonon wave number as a function of temperature. The concept for the mean-free path is very distinct from known prior approaches: the latter evaluate the mean-free path only indirectly, via using both a scale for for the phonon relaxation time and yet another one for the phonon velocity. Notably, the concept here is neither limited to small lattice nonlinearities nor to small frequencies. The scheme is tested for three strongly nonlinear lattices of timely current interest which either exhibit normal or anomalous heat transport.

Solitons and Heat Transfer in Nonlinear Lattices with Cubic On-Site and Quartic Interaction Potentials

This paper deals with the transfer of soliton-like heat waves in nonlinear lattices with cubic on-site potential and quartic interparticle interaction potential. A model Hamiltonian was proposed using the second quantized operators and the same was averaged using a suitable wavefunction. The equations were derived numerically in the discrete form for the field amplitude. Moreover the resulting equations were analyzed analytically using the continuous approximation technique and the properties of heat transfer were examined theoretically.

Solitary heat waves in nonlinear lattices with squared on-site potential

A model Hamiltonian is proposed for heat conduction in a nonlinear lattice with squared on-site potential using the second quantized operators and averaging the same using a suitable wave function, equations are derived in discrete form for the field amplitude and the properties of heat transfer are examined theoretically. Numerical analysis shows that the propagation of heat is in the form of solitons. Furthermore, a systemized version of tanh method is carried out to extract solutions for the resulting nonlinear equations in the continuum case and the effect of inhomogeneity is studied for different temperatures.

Special Section on Energy Localization and Waves in Nonlinear Lattices — Memorial Issue Dedicated to Professor Shozo Takeno —

Special Section on Energy Localization and Waves in Nonlinear Lattices — Memorial Issue Dedicated to Professor Shozo Takeno —

Resonant emission of solitons from impurity-induced localized waves in nonlinear lattices

We propose a mechanism for soliton creation from resonantly excited localized waves via supratransmission in band gaps of nonlinear lattices. A nonlinear localized wave, which is formed by and vibrates around an impurity with an intrinsic frequency, is found to undergo a local resonance when subject to an external forcing. Under the resonance, an instability develops that leads to the efficient emission of solitons at a much lower rate than that in uniform lattices with no impurity.

Exact solutions for periodic and solitary matter waves in nonlinear lattices

We produce three vast classes of exact periodic and solitonic solutions to the one-dimensional Gross-Pitaevskii equation (GPE) with the pseudopotential in the form of a nonlinear lattice (NL), induced by a spatially periodic modulation of the local nonlinearity. It is well known that NLs in Bose-Einstein condensates (BECs) may be created by means of the Feshbach-resonance technique. The model may also include linear potentials with the same periodicity. The NL modulation function, the linear potential (if any), and the corresponding exact solutions are expressed in terms of the Jacobi's elliptic functions of three types, cn, dn, and sn, which give rise to the three different classes of the solutions. The potentials and associated solutions are parameterized by two free constants and an additional sign parameter in the absence of the linear potential. In the presence of the latter, the solution families feature two additional free parameters. The families include both sign-constant and sign-changing NLs. Density maxima of the solutions may coincide with either minima or maxima of the periodic pseudopotential. The solutions reduce to solitons in the limit of the infinite period. The stability of the solutions is tested via systematic direct simulations of the GPE. As a result, stability regions are identified for the periodic solutions and solitons. The periodic patterns of cn type, and the respective limit-form solutions in the form of bright solitons, may be stable both in the absence and presence of the linear potential. On the contrary, the stability of the two other solution classes, of the dn and sn types, is only possible with the linear potential.

THE PAINLEVÉ INTEGRABILITY AND N-SOLITONIC SOLUTION IN TERMS OF THE WRONSKIAN DETERMINANT FOR A VARIABLE-COEFFICIENT VARIANT BOUSSINESQ MODEL OF NONLINEAR WAVES

A variable-coefficient variant Boussinesq (VCVB) model describes the propagation of long waves in shallow water, the nonlinear lattice waves, the ion sound waves in plasmas, and the vibrations in a nonlinear string. With the help of symbolic computation, a VCVB model is investigated for its integrability through the Painlevé analysis. Then, by truncating the Painlevé expansion at the constant level term with two singular manifolds, the dependent variable transformations are obtained through which the VCVB model is bilinearized. Furthermore, the corresponding N-solitonic solutions with graphic analysis are given by the Hirota method and Wronskian technique. Additionally, a bilinear Bäcklund transformation is constructed for the VCVB model, by which a sample one-solitonic solution is presented.

Waves in Nonlinear Lattices: Ultrashort Optical Pulses and Bose-Einstein Condensates

The nonlinear Schrödinger equation i (partial differential)(z)A(z,x,t)+(inverted Delta)(2)(x,t)A+[1+m(kappax)]|A|2A=0 models the propagation of ultrashort laser pulses in a planar waveguide for which the Kerr nonlinearity varies along the transverse coordinate x, and also the evolution of 2D Bose-Einstein condensates in which the scattering length varies in one dimension. Stability of bound states depends on the value of kappa=beamwidth/lattice period. Wide (kappa>>1) and kappa=O(1) bound states centered at a maximum of m(x) are unstable, as they violate the slope condition. Bound states centered at a minimum of m(x) violate the spectral condition, resulting in a drift instability. Thus, a nonlinear lattice can only stabilize narrow bound states centered at a maximum of m(x). Even in that case, the stability region is so small that these bound states are "mathematically stable" but "physically unstable."

Heating and phase transitions of dust-plasma crystals in a flowing plasma

A three dimensional particle simulation is used to study hexagonal dust crystals in flowing plasmas. The flowing velocity ${\mathrm{v}}_{0}$ is mesothermal (${\mathrm{v}}_{\mathrm{Ti}}$${\mathrm{v}}_{0}$${\mathrm{v}}_{\mathrm{Te}}$), which is a typical situation for dust particles confined in industrial plasmas and plasma experiments. Unlike for the Yukawa system (Debye-H\"uckel shielded dust), we find parameter regimes where a hexagonal dust crystal is stable. This is due to the wakefield generated around the individual particles which causes attractive interparticle forces along the axis of the plasma flow. These attractive forces cause a rather strong binding of the crystal since the simulation shows that solid-fluid transitions occur at a much lower \ensuremath{\Gamma} value (higher dust temperature) than for the Yukawa system. Stability of a hexagonal dust crystal is found to depend strongly upon the gas neutral pressure since stable crystal structure is obtained for dust-neutral collision frequencies ${\ensuremath{\gamma}}_{\mathrm{d}}$ above some threshold value ${\ensuremath{\gamma}}_{\mathrm{d}0}$. For ${\ensuremath{\gamma}}_{\mathrm{d}}$${\ensuremath{\gamma}}_{\mathrm{d}0}$ the ion flow will excite crystal waves (phonons) where the amplitude of these waves grows as a function of time. This instability will either saturate due to nonlinear lattice waves or continue until the dust crystal structure eventually melts.

Scattering of Envelope Solitons by a Mass Impurity in Nonlinear Lattices

Scatterings of nonlinear lattice waves by a mass impurity are studied. The waves are assumed to be nonlinear modulations of the monochromatic waves. Due to the impurity there appear the incident, reflected and transmitted waves. We show that the three waves are described by independent Nonlinear Schrödinger (NLS) equations respectively. Using the continuity conditions of the waves at the impurity site, we analytically construct the transmitted and reflected waves from the incident wave. As an application, scattering of an incident NLS envelope soliton is investigated. We find that at most one soliton is generated both in the reflected wave and in the transmitted wave.

Waves in Nonlinear Lattice

In this article waves in nonlinear lattice or in nonlinear medium are studied. One of the aims is to seek for the point of view to deal with the great majority of phenomena related to nonlinear waves in general. For one dimensional nonlinear lattice analytic and computer-experimental treatments have been developed. It has been found that a certain kind of pulse-like waves (solitons) is the fundamental motion in nonlinear lattice vibration. If two or more solitons collide, they interact nonlinearly, pass through one another and, when they separate, return to their original forms. Thus solitons are conserved and behave like particles.