Solitary waves in electro-mechanical lattices.

We introduce and study both analytically and numerically a class of microelectromechanical chains aiming to turn them into transmission lines of solitons. Mathematically, their analysis reduces to the study of a spatially one-dimensional nonlinear Klein-Gordon equation with a model dependent onsite nonlinearity induced by the electrical forces. Since the basic solitons appear to be unstable for most of the force regimes, we introduce a stabilizing algorithm and demonstrate that it enables a stable and persisting propagation of solitons. Among other fascinating nonlinear formations induced by the presented models, we mention the "meson": a stable square shaped pulse with sharp fronts that expands with a sonic speed, and "flatons": flat-top solitons of arbitrary width.

Solitary phase waves in a chain of autonomous oscillators.

In the present paper, we study phase waves of self-sustained oscillators with a nearest-neighbor dispersive coupling on an infinite lattice. To analyze the underlying dynamics, we approximate the lattice with a quasi-continuum (QC). The resulting partial differential model is then further reduced to the Gardner equation, which predicts many properties of the underlying solitary structures. Using an iterative procedure on the original lattice equations, we determine the shapes of solitary waves, kinks, and the flat-like solitons that we refer to as flatons. Direct numerical experiments reveal that the interaction of solitons and flatons on the lattice is notably clean. All in all, we find that both the QC and the Gardner equation predict remarkably well the discrete patterns and their dynamics.

Breathers in strongly anharmonic lattices.

We present and study a family of finite amplitude breathers on a genuinely anharmonic Klein-Gordon lattice embedded in a nonlinear site potential. The direct numerical simulations are supported by a quasilinear Schrodinger equation (QLS) derived by averaging out the fast oscillations assuming small, albeit finite, amplitude vibrations. The genuinely anharmonic interlattice forces induce breathers which are strongly localized with tails evanescing at a doubly exponential rate and are either close to a continuum, with discrete effects being suppressed, or close to an anticontinuum state, with discrete effects being enhanced. Whereas the D-QLS breathers appear to be always stable, in general there is a stability threshold which improves with spareness of the lattice.

Quasicontinuum Fokker-Planck equation.

Building on the work [C. R. Doering, P. S. Hagan, and P. Rosenau, Phys. Rev. A 36, 985 (1987)] we present a regularized Fokker-Planck equation for discrete-state systems with more accurate short-time behavior than its standard, Kramers-Moyal counterpart. This regularization leads to a quasicontinuum Fokker-Planck equation with several key features: it preserves crucial aspects of state-space discreteness ordinarily lost in the standard Kramers-Moyal expansion; it is well posed, and it is more amenable to analytical and numerical tools currently available for continuum systems. In order to expose the basic idea underlying the regularization, it suffices for us to focus on two simple problems--the chemical reaction kinetics of a one-component system and a two-dimensional symmetric random walk on a square lattice. We then describe the path to applying this approach to more complex, discrete-state stochastic systems.

Emergence of compact structures in a Klein-Gordon model.

The Klein-Gordon model (KG) phi=P{'}(|phi|)phi/|phi| is Lorenz invariant and has a finite wave speed, yet its localized modes, whether Q balls or vortices, suffer from the same fundamental flaw as all other solitons-they extend indefinitely. Using the KG model as a case study, we demonstrate that appending the site potential, P{a}(phi|), with a subquadratic part P(|phi|)=b{2}|phi|{1+delta}+P{a}(|phi|), 0

Compactification of nonlinear patterns and waves.

We present a nonlinear mechanism(s) which may be an alternative to a missing wave speed: it induces patterns with a compact support and sharp fronts which propagate with a finite speed. Though such mechanism may emerge in a variety of physical contexts, its mathematical characterization is universal, very simple, and given via a sublinear substrate (site) force. Its utility is shown studying a Klein-Gordon -u(tt) + [phi/(u(x)]x = P'(u) equation, where phi'(sigma) = sigma + beta sigma3 and endowed with a subquadratic site potential P(u) approximately /1-u2/(alpha+1), 0 < or = alpha < 1, and the Schrödinger iZt + inverted delta2 Z = G(/Z/)Z equation in a plane with G(A) = gammaA(-delta) - sigmaA2, 0 < delta < or = 1.

Multidimensional compactons.

We study the two and three dimensional, N=2, 3, nonlinear dispersive equation CN(m,a+b): u(t)+(u(m))x + [u(a)inverted delta2ub]x=0 where the degeneration of the dispersion at the ground state induces cylindrically and spherically symmetric compactons convected in the x direction. An initial pulse of bounded extent decomposes into a sequence of robust compactons. Colliding compactons seem to emerge from the interaction intact, or almost so.

Compactification of patterns by a singular convection or stress.

A wide variety of propagating disturbances in physical systems are described by equations whose solutions lack a sharp propagating front. We demonstrate that presence of particular nonlinearities may induce such fronts. To exemplify this idea, we study both dissipative u_{t}+ partial differential_{x}f(u)=u_{xx} and dispersive u_{t}+ partial differential_{x}f(u)+u_{xxx}=0 patterns, and show that a weakly singular convection f(u)=-u;{alpha}+u;{m}, 0

Phase compactons in chains of dispersively coupled oscillators.

We study the phase dynamics of a chain of autonomous oscillators with a dispersive coupling. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons: solitary waves with a compact support and kovatons which are compact formations of glued together kink-antikink pairs that may assume an arbitrary width. These robust objects seem to collide elastically and, together with wave trains, are the building blocks of the dynamics for typical initial conditions. Numerical studies of the complex Ginzburg-Landau and Van der Pol lattices show that the presence of a nondispersive coupling does not affect kovatons, but causes a damping and deceleration or growth and acceleration of compactons.

Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit.

We demonstrate that certain strictly anharmonic one-dimensional FPU lattices with a suitable quartic site potential appended support almost-compact discrete breathers over a macroscopic localized domain that is essentially fixed independently of the sparseness of the lattice. Beyond that domain the discrete breather tails decay at a double-exponential rate in the lattice-cell index, becoming truly compact in the continuum limit. Furthermore, the discrete breather is stable for amplitudes below a sharp threshold that depends on the sparseness of the lattice. For the two-dimensional version of the problem, the continuum limit of a planar hexagonal lattice with a purely quartic interaction potential begets an isotropic multidimensional nonlinear wave equation. When a quartic site potential of the appropriate sign is appended, the continuum equation has a compactly supported radial breather solution.

Reaction and concentration dependent diffusion model.

We study the formation of patterns in the genuinely nonlinear reaction diffusion model equation u(t)+2a(u(2))(x) = (u(2))(xx)+F(x,u), where u may be viewed as an amplitude of a thermal wave in plasma or density of a biological species and F = u(1-u) or F = q(x)u(l), l = 0,2. We provide a transformation which maps the model into a purely diffusive process free of its interacting part and its intrinsic temporal and spatial scales. The well known attractors of the diffusive process enable us to completely characterize the emerging patterns which, depending on F and initialization, may be a semicompact, or a compact, traveling wave or a nontrivial equilibrium.