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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. [ABSTRACT FROM AUTHOR]

It is shown that a simple modification of the standard Lagrangian underlying the dynamics of Newtonian lattices enables one to infer the hidden Lagrangian structure of certain classes of first order in time evolution equations which lack the conventional Lagrangian structure. Implication to other setups is outlined and exemplified. [ABSTRACT FROM AUTHOR]

We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka–Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract Exploiting the finite span of compactons we use the K (m , n) Equation to develop an approximate description of early and late stages of their interaction. [ABSTRACT FROM AUTHOR]

Using a Lotka–Volterra type system on a hexagonal lattice we derive and study a novel, strongly nonlinear dispersive equation u t = ∂ x ( u + Δ u ) n , n > 1 , the n -Cubic equation, which supports the formation and propagation of planar compactons endowed with extended regularity at their perimeter. Compactons may be uni-modal or, if n is odd, multi-modal as well. Both evolution and interaction of compactons are presented and discussed. [ABSTRACT FROM AUTHOR]

In this letter we re-address a class of genuinely nonlinear third order dispersive equations; C 1 ( m , a , b ) : u t + ( u m ) x + 1 b [ u a ( u b ) x x ] x = 0 , which among other solitary structures admit compactons, and demonstrate that certain subclasses of these equations may be cast into Hamiltonian and Lagrangian formulations resulting in new conservation laws, some of which are nonlocal . In particular, the new nonlocal conservation law of the K ( n , n ) equations enables us to prove that the response to a certain class of excitations cannot contain only compactons. [ABSTRACT FROM AUTHOR]

We study formation of patterns in reaction processes with a logarithmic-diffusion: u t = ( ln ⁡ u ) x x + R ( u ) . For the generic R = u ( 1 − u ) case the problem of travelling waves, TW, is mapped into a linear one with the propagation speed λ selected by a boundary condition, b.c. at the far away upstream. Dirichlet b.c. relaxes the process into a steady state , whereas convective b.c. u x + h u = 0 , leads the system into a heating (cooling) TW for h < 1 ( 1 < h ) or, if h = 1 , into an equilibrium. We derive explicit solutions of symmetrically expanding waves and of formations which collapse in a finite time. Both are shown to be attractors of classes of initial excitations. For a bi-stable reaction R = − u ( α − u ) ( 1 − u ) we show that for α < 1 / 3 the system may evolve into a TW, an equilibrium, an expanding formation or to collapse. The 1 / 3 < α regime admits either a cooling TW or a collapse. Few other transport processes are outlined in the appendix. [ABSTRACT FROM AUTHOR]

A third order dispersive equation u t + ( u m ) x + 1 b [ u a ∇ 2 u b ] x = 0 is used to explore two very different classes of compact patterns. In the first, the prevailing singularity at the edge induces traveling compactons, solitary waves with a compact support. In the second, the singularity induced at the perimeter of the initial excitation, entraps the dynamics within the domain's interior (nonetheless, certain very singular excitations may escape it). Here, overlapping compactons undergo interaction which may result in an interchange of their positions, or form other structures, all confined within their initial support. We conjecture , and affirm it empirically, that whenever the system admits more than one type of compactons, only the least singular compactons may be evolutionary. The entrapment due to singularities is also unfolded and confirmed numerically in a class of diffusive equations u t = u k ∇ 2 u n with k > 1 and n > 0 with excitations entrapped within their initial support observed to converge toward a space–time separable structure. A similar effect is also found in a class of nonlinear Klein–Gordon Equations. [ABSTRACT FROM AUTHOR]

We study a quintic dispersive equation u t = [ a u 2 + b ( u u x x + β u x 2 ) + c ( u u 4 x + 2 q 1 u x u 3 x + q 2 u x x 2 ) ] x and show that if β = q 1 = − q 2 , it may be cast into v t = [ v L ω u ] x , where v = u ω , ω = 2 β + 1 and L ω is a fourth order linear operator. This enables to construct traveling patterns via superposition of solutions. A plethora of bell-shaped, multi-humped and asymmetric compacton, is found. Their interaction ranges from being almost elastic to a noisy one, including fusion of bell-shaped compactons and anti-compactons into robust asymmetric structures. A stationary, zero-mass, doublet-like compacton is found to be an attractor of topologically similar, zero-mass, excitations. [ABSTRACT FROM AUTHOR]

A Klein–Gordon model on a chain or a rectangular lattice endowed with cubic inter-site and on-site forces is presented. It supports a plethora of strictly compact stable discrete breathers which are pinned at all times to their initial support. In the continuum limit radially symmetric uni-nodal and multi-nodal compact breathers emerge. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: The standard derivation of Schrödinger''s equation from a Lorentz-invariant Feynman path integral consists in taking first the limit of infinite speed of light and then the limit of short time slice. In this order of limits the light cone of the path integral disappears, giving rise to an instantaneous spread of the wave function to the entire space. We ascribe the failure of the propagation in time according to Schrödinger''s equation to retain the light cone of the path integral to the very nature of the limiting process: it is a regular expansion of a singular approximation problem, because the time-dependent boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit of the time-sliced relativistic path integral, which produces Schrödinger''s equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. This produces an intermediate model between non-relativistic and relativistic mechanics of a single particle quantum particle. These boundary conditions relieve the solutions of Schrödinger''s equation in the entire space of several annoying, seemingly unrelated unphysical artifacts, including non-analytic wave functions, spontaneous appearance of discontinuities, non-existence of moments when the initial wave function has a jump discontinuity (e.g., a collapsed wave function after a measurement), and so on. The practical implications of the present formulation are yet to be seen. [Copyright &y& Elsevier]

Abstract: We study the quasi-linear complex Klein–Gordon equation and present two classes of strictly localized compact stationary breathers. In the first class breathers vibrate at an anharmonic rate but the site potential has to be quartic. In the second class a more general, Q-ball type, site potentials are admitted but vibrations are harmonic. Notably, unlike the Q-balls supporting models, if the chosen potential has a top then multi-nodal modes cannot accumulate there: only a finite number of multi-nodal modes is possible, each constrained by its own spectrum of harmonic vibrations. [Copyright &y& Elsevier]

Abstract: We study the formation and interaction of discretons, solitary waves with an almost compact support (tails decaying at a super-exponential rate), on a hexagonal lattice and its spatial extension. Discretons are shown to be robust and their interaction though not entirely, is quite clean. [Copyright &y& Elsevier]

Abstract: We introduce and study a new class of nonlinear dispersive equations: , where is the dispersive flux with typical being monomials in u and (which amalgamates all KdV type equations with a monomial nonlinear dispersion) and show that it admits either traveling or stationary compactons. In the second case initial datum given on a compact support evolves into a sequence of stationary compactons, with the spatio-temporal evolution being confined to the initial support. We also discuss an N-dimensional extension which induces N-dimensional compactons convected in x-direction. Two families of explicit solutions of N-dimensional compactons are also presented. [Copyright &y& Elsevier]

Abstract: We study the phase dynamics of a chain of autonomous, self-sustained, dispersively coupled oscillators. 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–compact formations of glued together kink–antikink pairs that propagate with a unique speed, but may assume an arbitrary width. We demonstrate that lattice solitary waves, though not exactly compact, have tails which decay at a superexponential rate. They are robust and collide nearly elastically and together with wave sources are the building blocks of the dynamics that emerges from typical initial conditions. In finite lattices, after a long time, the dynamics becomes chaotic. Numerical studies of the complex Ginzburg–Landau lattice show that the non-dispersive coupling causes a damping and deceleration, or growth and acceleration, of compactons. A simple perturbation method is applied to study these effects. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

We discuss the problems involved in deriving correct quasi-continuum approximations of Hamiltonians describing the motion of dense lattices directly from the underlying discrete Hamiltonian. This assures that the resulting equations of motion have a Hamiltonian structure and are free from ultra-violet instabilities. [Copyright &y& Elsevier]

We introduce gradient-tempered Ginzburg-Landau, GL , free energy functional J = ∫ (G (u x) − W (u)) d x , where W (u) = λ 2 (2 u 2 − u 4) is the bulk energy endowed with a stiffness coefficient a 2 (u) which vanishes both at the phases and on the interface: a 2 (u) = | u | 2 γ | 1 − u 2 | 2 β , 1 2 < β , γ ≤ 1. It induces compact drops and interphase transitions within a finite domain. The assumed interface energy G = 1 + a 2 u x 2 − 1 , tempers the divergence of gradients in the ultra-violet limit. Consequently, when λ 2 , the bulk energy strength parameter crosses a critical value, depending on their amplitude, the forming drops may develop at their edges sharp jumps and turn into fingers across which the energy remains finite. In the Tempered Allen-Cahn, u t = − δ δ x J equation, the resulting flux-saturating diffusion delays the resolution of initial discontinuities and if 1 ≤ γ , it blocks excitation's spread beyond its initial span. A number of explicit spherical solutions is also presented. [ABSTRACT FROM AUTHOR]

A computer-aided study of Z-pinch plasma transport reveals a high level of structured behavior. The equations asymptote into time and space separable forms, with the profiles of all the variables determined solely by the time dependence of the plasma current. Plasma profiles (normalized with respect to their values on the axis and to the plasma radius) are almost independent of the level of Bremsstrahlung radiation and thus can be determined using the self-similar pattern of a nonradiating plasma. Radiation causes the plasma column to collapse within a finite time and is accompanied by a rapid growth in the density and pressure. The temporal growth of the temperature initially follows that of the current, but if the current crosses the Pease limit, the temperature also rapidly grows. [ABSTRACT FROM AUTHOR]

The equations governing the diffusion and transport of a fully collisional plasma across a strong magnetic field in a bounded domain are analyzed. Following a relatively short relaxation time, the diffusion exhibits universal properties independent of the choice of initial data. Mathematically this appears as a time asymptotic solution which is space-time separable. The temporal decay rate is a nonlinear eigenvalue which is found via the solution of a related eigenvalue problem. This determines the spatial distribution of both the particle density and the pressure. Some of the transport effects caused by Bremsstrahlung radiation, particle, and heat injection are considered, and the conditions under which the system evolves into an equilibrium are examined. [ABSTRACT FROM AUTHOR]

An amplitude equation is derived, which describes the evolution of a disturbed film interface H(&tao;Z,Y) flowing down an infinite vertical cylindrical column. Using a new approach, which accounts for fast spatial changes, the nonlinear evolution of the interface is shown to be governed by H[SUB&tao;] + βHH[SUBz]+αH[SUBzz]+γ▿[SUP2]{N[(1/ω[SUP2]) H + ▿[SUP2]H]}=0, where ω is the normalized cylinder radius and α, β, and γ are constants, ▿--(δ[SUBz], δ[SUBY]), and -N = [1+ε[SUP4](VH)[SUP2]][SUP-3/2]. It is shown numerically that for some linearly unstable equilibria the evolving waves break in a finite time. [ABSTRACT FROM AUTHOR]

The nonlinear evolution of interfacial waves separating liquids of different viscosity and density (Rayleigh-Taylor instability) in a 2-D channel is studied. Using a new approach, which accounts for large gradients, the nonlinear evolution of the interface, y = ∊A (τ,&xgr;),∊,&Le; 1, is shown to be governed by the regularized Kuramoto-Sivashinsky equation A[SUBτ] +βAA[SUB&xgr;] + {αA+ γA[SUB&xgr;&xgr;]/(1 + ∊[SUP4]A[SUP2,SUB&xgr;])[SUP&frac32;] = O, where the constants α,β, and γ are determined at equilibrium, &xgr; is the slow coordinate along the channel, &xg; = ∊(x - C[SUB0]t), and τ = ∊[SUP2]t. It is shown numerically that for ∊[SUP2]&ges;0. 1β linearly unstable waves (while always of finite amplitude) are propelled by convection toward breaking in a finite time. [ABSTRACT FROM AUTHOR]

A new amplitude equation governing the velocity evolution of ion-acoustic waves in a collisionless plasma is derived: Vtt+(V2)xt=[(1-V2)Vx] x+βVxxtt. From this equation the existence of a critical amplitude threshold is immediately deduced, which leads to a double layer and above which solitary waves cannot exist. Above this threshold shock waves form and density undergoes explosive growth. Multidimensional waves and waves due to the magnetic drift are also considered. [ABSTRACT FROM AUTHOR]

Corrigendum to “On quintic equations with a linear window” [Phys. Lett. A 380 (2016) 135–141]. [ABSTRACT FROM AUTHOR]

Presents a study which considered flexible helical fiber with an arbitrary stress-strain relation. How the solution describes a new class of spatial motion of an elastic string; Methodology; Conclusions.

The capillary instability of a thin liquid layer on a cylinder is studied. Using an integral approach and lubrication approximation, transport equations governing the spatiotemporal evolution of a film thickness and the temperature along the film are obtained. Evolution of the system under both isothermal and nonisothermal conditions is studied numerically. It is shown that nonlinear interaction of the linearly unstable modes begets an additional mode with a wavelength equal to that of the fastest growing wave. This, in turn, causes the formation of satellite drops along with the main ones. Application of these results in a possible continuous technology of high-temperature superconductor wire fabrication is discussed. [ABSTRACT FROM AUTHOR]