scholarly journals Two-dimensional solitons in saturable media with a quasi-one-dimensional lattice potential

2006 ◽  
Vol 73 (3) ◽  
Author(s):  
Thawatchai Mayteevarunyoo ◽  
Boris A. Malomed
Keyword(s):  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Lattice Potential

scholarly journals DIFFERENTIAL EQUATION APPROACH FOR ONE- AND TWO-DIMENSIONAL LATTICE GREEN'S FUNCTION

2007 ◽  
Vol 21 (02n03) ◽  
pp. 139-154 ◽  
Author(s):  
J. H. ASAD
Keyword(s):  
Differential Equation ◽  
Green’S Function ◽  
Recurrence Relation ◽  
Green's Function ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
The One ◽  
Lattice Green's Function ◽  
Lattice Green’S Function

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

Stress Driven Patterning of Epitaxial Films

1993 ◽  
Vol 318 ◽  
Author(s):  
Michael A. Grinfeld
Keyword(s):  
Surface Energy ◽  
Surface Morphology ◽  
Film Thickness ◽  
Critical Thickness ◽  
Epitaxial Films ◽  
Layer By Layer ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Periodic Spacing

ABSTRACTWe study possible morphologies of epitaxial films atop attractive substrates appearing as a result of competition of misfit stresses, van der Waals forces and surface energy. Corresponding formula for the critical thickness of the dislocation-free Stranski-Krastanov pattern is established for the isotropic deformable films and substrates. If the film thickness exceeds the critical magnitude the layer-by-layer pattern switches to islanding. At the first stage the islands have a shape of striae (i.e. long parallel trenches with periodic spacing). We discuss also i)the circumstances in which surface morphology of the film corresponds to a two-dimensional superlattice of islands rather than a one dimensional lattice of striae and ii)the influence of a buffer inter-layer.

scholarly journals New types of one-dimensional discrete breathers in a two-dimensional lattice

2020 ◽  
Vol 10 (2) ◽  
pp. 185-188
Author(s):  
Alexander Semenov ◽  
Ramil Murzaev ◽  
Yuri Bebikhov ◽  
Aleksey Kudreyko ◽  
Sergey Dmitriev
Keyword(s):  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Discrete Breathers

A modified two-dimensional triangular lattice model under honk environment

2020 ◽  
Vol 31 (06) ◽  
pp. 2050089
Author(s):  
Cong Zhai ◽  
Weitiao Wu
Keyword(s):  
Stability Analysis ◽  
Hydrodynamic Model ◽  
High Dimensional ◽  
Two Dimensional ◽  
Lattice Hydrodynamic Model ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
The Stability ◽  
Honk Effect ◽  
Real Traffic

The honk effect is not uncommon in the real traffic and may exert great influence on the stability of traffic flow. As opposed to the linear description of the traditional one-dimensional lattice hydrodynamic model, the high-dimensional lattice hydrodynamic model is a gridded analysis of the real traffic environment, which is a generalized form of the one-dimensional lattice model. Meanwhile, the high-dimensional traffic flow exposed to the open-ended environment is more likely to be affected by the honk effect. In this paper, we propose an extension of two-dimensional triangular lattice hydrodynamic model under honk environment. The stability condition is obtained via the linear stability analysis, which shows that the stability region in the phase diagram can be effectively enlarged under the honk effect. Modified Korteweg–de Vries equations are derived through the nonlinear stability analysis method. The kink–antikink solitary wave solution is obtained by solving the equation, which can be used to describe the propagation characteristics of density waves near the critical point. Finally, the simulation example verifies the correctness of the above theoretical analysis.

The one-dimensional minesweeper game: What are your chances of winning?

2016 ◽  
Vol 27 (11) ◽  
pp. 1650127 ◽  
Author(s):  
M. Rodríguez-Achach ◽  
H. F. Coronel-Brizio ◽  
A. R. Hernández-Montoya ◽  
R. Huerta-Quintanilla ◽  
E. Canto-Lugo
Keyword(s):  
Numerical Simulations ◽  
Critical Density ◽  
Computer Game ◽  
Simple System ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Np Completeness ◽  
The One ◽  
Good Agreement

Minesweeper is a famous computer game consisting usually in a two-dimensional lattice, where cells can be empty or mined and gamers are required to locate the mines without dying. Even if minesweeper seems to be a very simple system, it has some complex and interesting properties as NP-completeness. In this paper and for the one-dimensional case, given a lattice of n cells and m mines, we calculate the winning probability. By numerical simulations this probability is also estimated. We also find out by mean of these simulations that there exists a critical density of mines that minimize the probability of winning the game. Analytical results and simulations are compared showing a very good agreement.

On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums

2008 ◽  
Vol 464 (2100) ◽  
pp. 3327-3352 ◽  
Author(s):  
Ross C McPhedran ◽  
I.J Zucker ◽  
Lindsay C Botten ◽  
Nicolae-Alexandru P Nicorovici
Keyword(s):  
Critical Line ◽  
Quadratic Function ◽  
Quadratic Functions ◽  
Square Lattice ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Lattice Sums ◽  
The One ◽  
Complex Powers

We consider a general class of two-dimensional lattice sums consisting of complex powers s of inverse quadratic functions. We consider two cases, one where the quadratic function is negative definite and another more restricted case where it is positive definite. In the former, we use a representation due to H. Kober, and consider the limit u →∞, where the lattice becomes ever more elongated along one period direction (the one-dimensional limit). In the latter, we use an explicit evaluation of the sum due to Zucker and Robertson. In either case, we show that the one-dimensional limit of the sum is given in terms of ζ (2 s ) if Re( s )>1/2 and either ζ (2 s −1) or ζ (2−2 s ) if Re( s )<1/2. In either case, this leads to a Riemann property of these sums in the one-dimensional limit: their zeros must lie on the critical line Re( s )=1/2. We also comment on a class of sums that involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. We show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it; others are identically zero on it, while still others have no zeros on it.

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