scholarly journals CELLULAR AUTOMATON SUPERCOLLIDERS

2011 ◽  
Vol 22 (04) ◽  
pp. 419-439 ◽  
Author(s):  
GENARO J. MARTÍNEZ ◽  
ANDREW ADAMATZKY ◽  
CHRISTOPHER R. STEPHENS ◽  
ALEJANDRO F. HOEFLICH
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Elastic Collision ◽  
Collective Excitations ◽  
Compact Groups ◽  
Dimensional Torus ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Spatially Extended ◽  
Binary Strings

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular automaton analogous of localizations or quasi-local collective excitations traveling in a spatially extended nonlinear medium. They can be considered as binary strings or symbols traveling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyze what types of interaction occur between gliders traveling on a cellular automaton "cyclotron" and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in nonlinear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analyzed via implementation of cyclic tag systems.

Optical Pulse Deformation in Second Order Nonlinear Media

2003 ◽  
Vol 12 (02) ◽  
pp. 221-234 ◽  
Author(s):  
E. van Groesen
Keyword(s):  
Material Properties ◽  
Optical Pulse ◽  
Order Approximation ◽  
Second Order ◽  
Optical Pulses ◽  
Third Order ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Initial Signal ◽  
Third Order Approximation

The deformation of optical pulses in one dimensional lossless second order nonlinear media is considered. Using a KdV-type of equation, with dispersion determined by the material properties, the deformation of a bichromatic initial signal is studied. An explicit expression for a third order approximation is used and the maximal temporal amplitude MTA is investigated. This MTA is obtained by looking at the maximum over time of the amplitude at each position. It is shown that modulations of the carrier waves and of the envelopes of the bound and free third order terms determine respectively the oscillations and the recurrence of the MTA curve. We will illustrate the explicit formula with numerical displays for the characteristic cases.

scholarly journals On optimal system, exact solutions and conservation laws of the modified equal-width equation

2018 ◽  
Vol 3 (2) ◽  
pp. 409-418 ◽  
Author(s):  
Chaudry Masood Khalique ◽  
Oke Davies Adeyemo ◽  
Innocent Simbanefayi
Keyword(s):  
Wave Propagation ◽  
Conservation Laws ◽  
Optimal System ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Dimensional Wave

AbstractIn this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

Characteristic Forms of Differential Equations for Wave Propagation in Nonlinear Media

1981 ◽  
Vol 48 (4) ◽  
pp. 743-748 ◽  
Author(s):  
T. C. T. Ting
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Constitutive Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Compressible Fluids ◽  
Elastic Solids ◽  
Nonlinear Media ◽  
Nonlinear Materials ◽  
One Dimensional

Characteristic forms of differential equations for wave propagation in nonlinear media are derived directly from equations of motion and equations which combine the constitutive equations and the equations of continuity. Both Lagrangian coordinates and Eulerian coordinates are considered. The constitutive equations considered here apply to a large class of nonlinear materials. The characteristic forms derived here clearly show which components of the stress and velocity are involved in the differentiation along the bicharacteristics. Moreover, the reduction to one-dimensional cases from three-dimensional problems is obvious for the characteristic forms obtained here. Examples are given and compared with the known solution in the literature for wave propagation in linear isotropic elastic solids and isentropic compressible fluids.

Localized states in symmetric three-layered structure consisting of linear layer between focusing media separated by interfaces with nonlinear response

2019 ◽  
Vol 33 (11) ◽  
pp. 1850127
Author(s):  
S. E. Savotchenko
Keyword(s):  
Nonlinear Response ◽  
Mathematical Formulation ◽  
Localized States ◽  
Stationary States ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Symmetric Structure ◽  
Linear Layer ◽  
Dimensional Boundary ◽  
Interface Parameters

We analyze the localization in three-layered symmetric structure consisting of linear layer between focusing nonlinear media separated by nonlinear interfaces. The mathematical formulation of the model is a one-dimensional boundary value problem for the nonlinear Schrödinger equation. We find nonlinear localized states of two types of symmetry. We derive the energies of obtained stationary states in explicit form. We obtain the localization energies as exact solutions of dispersion equations choosing the amplitude of the interface oscillations as a free parameter. We analyze the conditions of their existence depending on the combination of signs of interface parameters.

GAP SOLITONS AND SLOW LIGHT

2002 ◽  
Vol 11 (03) ◽  
pp. 239-259 ◽  
Author(s):  
CLAUDIO CONTI ◽  
GAETANO ASSANTO ◽  
STEFANO TRILLO
Keyword(s):  
Slow Light ◽  
Frequency Doubling ◽  
Optical Nonlinearity ◽  
Band Gaps ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Light Localization ◽  
Gap Solitons ◽  
Basic Concepts ◽  
Model Equations

Optical nonlinearity and feedback through Bragg periodicity are the basic ingredients for light localization into gap solitons. We review the basic concepts and model equations for gap solitons in Kerr and quadratic nonlinear media encompassing a one-dimensional Bragg resonance. With specific regard to frequency doubling media, we generalize the concept of a photonic crystal to band-gaps of a nonlinear origin, and finally address the slow character of quadratic gap-solitons with reference to their excitation.

scholarly journals Stable one-dimensional periodic waves in Kerr-type saturable and quadratic nonlinear media

2004 ◽  
Vol 6 (5) ◽  
pp. S279-S287 ◽  
Author(s):  
Yaroslav V Kartashov ◽  
Alexey A Egorov ◽  
Victor A Vysloukh ◽  
Lluis Torner
Keyword(s):  
Periodic Waves ◽  
Nonlinear Media ◽  
One Dimensional
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