Solitary Waves and Their Linear Stability in Nonlinear Lattices

Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schrödinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, which span a large number of lattice periods. In this limit, the allowed positions of solitary waves relative to the lattice, as well as their linear stability properties, hinge upon a certain recurrence relation which contains information beyond all orders of the usual two‐scale perturbation expansion. It follows that only two such positions are permissible, and of those two solitary waves, one is linearly stable and the other unstable. For a cosine lattice, in particular, the two possible solitary waves are centered at a maximum or minimum of the lattice, with the former being stable, and the analytical predictions for the associated linear stability eigenvalues are in excellent agreement with numerical results. Furthermore, a countable set of multi‐solitary‐wave bound states are constructed analytically. In spite of rather different physical settings, the exponential asymptotics approach followed here is strikingly similar to that taken in earlier studies of solitary wavepackets involving a periodic carrier and a slowly varying envelope, which underscores the general value of this procedure for treating multiscale solitary‐wave problems.     

Periodic Oscillations and Waves in Nonlinear Weakly Coupled Dispersive Systems

Bifurcations of periodic solutions in autonomous nonlinear systems of weakly coupled equations are studied. A comparative analysis is carried out between the mechanisms of Lyapunov–Schmidt reduction of bifurcation equations for solutions close to harmonic oscillations and cnoidal waves. Sufficient conditions for the branching of orbits of solutions are formulated in terms of the Pontryagin functional depending on perturbing terms.     

On Partially Irregular Almost Periodic Solutions of Weakly Nonlinear Ordinary Differential Systems

For weakly nonlinear almost periodic ordinary differential systems, we obtain conditions for the existence of partially irregular almost periodic solutions and propose algorithms for their construction. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1123 – 1130, August, 2005.     

Solitary Waves for Nonconvex FPU Lattices

Solitary waves in a one-dimensional chain of atoms are investigated. The potential energy is required to be monotone and grow super-quadratically. The existence of solitary waves with a prescribed asymptotic strain is shown under certain assumptions on the asymptotic strain and the wave speed. It is demonstrated that the invariance of the equations allows one to transform a system with nonconvex potential energy density to the situation under consideration.     

Standing Waves for Periodic Discrete Nonlinear SchrSdinger Equations with Asymptotically Linear Terms

In this paper we study the existence of nontrivial solutions for the periodic discrete nonlinear equation Lun ωun = fn(un), where Lun = a n+1un+1 + an1un1+ bnun is the discrete Laplacian in one spatial dimension. The given real-valued sequences an , bn are assumed to be N -periodic in n, i.e., an+N = an , b n+N = bn . The nonlinearity fn(t) is N-periodic in n and asymptotically linear at infinity. We show that, if ω is in the spectrum gap of L, there is a nontrivial solution. The proof is based on the strongly indefinite functional critical points theorem developed recently.     

一维耦合格点映射中的周期解

证明了具有最临近耦合的一维耦合格点映射关于时间为周期的非线性解的存在性。这一类系统能被看成无穷个耦合振子构成的阵列。给出了估计这类关于时间为周期的解存在的临界耦合强度的一种方法。对于一些特殊的关于时间为周期的非线性解,证明了其空间的指数衰减性。     

Envelope Solitary Waves and Periodic Waves in the AB Equations

This article deals with the envelope solitary waves and periodic waves in the AB equations that serve as model equations describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra‐short pulses in nonlinear optics. An envelope solitary wave has a width proportional to its velocity and inversely proportional to its amplitude. The velocity of the envelope solitary wave is partially dependent on its amplitude in the sense that the amplitude determines the upper or lower limit of the velocity. When two envelope solitary waves collide, they survive the collision and retain their identities except for a shift in the positions of both the envelopes and the carrier waves. The periodic wave solutions in sine wave form may be stable or unstable depending upon the wave parameters. When the sine wave is destabilized by small perturbations, its long‐time evolution shows a Fermi–Pasta–Ulam‐type oscillation.     

Transverse Orbital Stability of Periodic Traveling Waves for Nonlinear Klein–Gordon Equations

In this paper, we establish the orbital stability of a class of spatially periodic wave train solutions to multidimensional nonlinear Klein–Gordon equations with periodic potential. We show that the orbit generated by the one‐dimensional wave train is stable under the flow of the multidimensional equation under perturbations which are, on one hand, coperiodic with respect to the translation or Galilean variable of propagation, and, on the other hand, periodic (but not necessarily coperiodic) with respect to the transverse directions. That is, we show their transverse orbital stability. The class of periodic wave trains under consideration is the family of subluminal rotational waves, which are periodic in the momentum but unbounded in their position.     

Propagation of Sound Waves in Partially Saturated Soils

We investigate the propagation of sound waves by means of a newly constructed model in a sandstone filled with two immiscible fluids. The speeds and attenuations of the four emerging waves (one transversal, three longitudinal) are illustrated in dependence on frequency and initial saturation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodic Traveling Waves in Diatomic Granular Chains

We study bifurcations of periodic traveling waves in diatomic granular chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter, and the periodic waves with a wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic traveling waves of homogeneous granular chains exist.     

Loops of Energy Bands for Bloch Waves in Optical Lattices

We consider stationary Bloch waves in a Bose–Einstein condensate placed in a periodic potential for varying strengths of inter‐atomic interactions. Bifurcations of the stationary states are known to occur in this context. These bifurcations generate loops in the energy bands of the Bloch waves near the ends and the center of the Brillouin zone. Using the method of Lyapunov–Schmidt reductions, we show that these bifurcations are of the supercritical pitchfork type. We also characterize the change in stability of the stationary states across the bifurcation point. Analytical results are illustrated by numerical computations.     

Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices

We employ KAM theory to rigorously investigate quasiperiodic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices. Toward this end, we apply a coherent structure ansatz to the Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation describing the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather sets to prove that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. We show that this threshold scales with the square root of the inverse of the two-body scattering length, whereas the rotation number of tori above this threshold is proportional to the amplitude. As a consequence, one obtains the same dynamical picture for lattices of all depths, as an increase in depth essentially affects only scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.     

Periodic Plane Waves for the Oregonator

The Oregonator is a set of differential equations proposed by R. J. Field and R. M. Noyes as a model for the oscillating chemical reaction first studied by B. P. Belousov and A. M. Zhabotinskii. In this paper it is shown that the associated diffusion equations have periodic plane waves for parameter values not covered in earlier work. This amounts to studying a singularly perturbed system when nothing is known about the stability of periodic solutions for the reduced system.     

Nonlinear Waves of Vorticity

This is a review of solutions of the vorticity equation for two-dimensional flow of an inviscid incompressible fluid that represent nonlinear waves. Geophysical applications are emphasized. Some of the solutions are valid in the beta-plane of Rossby. Some are related to weakly nonlinear perturbations of basic parallel flows and axisymmetric flows, to initial-value problems of hydrodynamic instability and to variational principles of minimal enstrophy or maximal entropy. Some have been found by exploiting well-known ideas of the theory of solitons. In addition to listing known solutions and presenting a synthesis of their relationship to other fluid dynamic results, we report a few new ideas and new solutions for strongly nonlinear waves.     

Periodic Travelling Waves and Compactons in Granular Chains

We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz’s contact forces. Each bead periodically undergoes a compression phase followed by free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behavior, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wave number, we observe fast instabilities of the periodic travelling waves, leading to a disordered regime.     

Weakly Nonlinear Waves in Nonideal Fluids

We study the propagation of weakly nonlinear waves in nonideal fluids, which exhibit mixed nonlinearity. A method of multiple scales is used to obtain a transport equation from the Navier–Stokes equations, supplemented by the equation of state for a van der Waals fluid. Effects of van der Waals parameters on the wave evolution, governed by the transport equation, are investigated.     

Weakly Nonlinear Internal Waves in Shear

An evolution equation in a finite depth fluid for weakly nonlinear long internal waves is derived in a stratified and sheared medium. The equation reduces to the Korteweg-deVries equation when the depth is small compared to the wavelength, and to the Benjamin-Ono equation when the depth is large compared to the wavelength. Both the cases with and without critical levels are investigated. Numerical solutions to the evolution equation are presented to illustrate the effect of shear on the evolution of a waveform.     

Geometric Aspects of Spatially Periodic Interfacial Waves

Periodic waves at the interface between two inviscid fluids of differing densities are considered from a geometric point of view. A new Hamiltonian formulation is used in the analysis and restriction of the Hamiltonian structure to space-periodic functions leads to an O -invariant Hamiltonian system. Motivated by the simplest O -invariant Hamiltonian system, the spherical pendulum, we analyze the properties of traveling waves, standing waves, interactions between standing and traveling waves (mixed waves) and time-modulated spatially periodic waves. A singularity in the bifurcation of traveling waves leads to a nonlinear resonance and this is investigated numerically.     

Statistical Inference in Single-Index and Partially Nonlinear Models

A finite series approximation technique is introduced. We first applythis approximation technique to a semiparametric single-index model toconstruct a nonlinear least squares (LS) estimator for an unknown parameterand then discuss the confidence region for this parameter based on theasymptotic distribution of the nonlinear LS estimator. Meanwhile, acomputational algorithm and a small sample study for this nonlinear LSestimator are developed. Additionally, we apply the finite seriesapproximation technique to a partially nonlinear model and obtain some newresults.     

Damping of Periodic Waves in Physically Significant Wave Systems

Damping of periodic waves in the classically important nonlinear wave systems—nonlinear Schrödinger, Korteweg–deVries (KdV), and modified KdV—is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m=m(t) . The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m= 0 to soliton waves when m= 1 .