scholarly journals Symmetry-Preserving Difference Models of Some High-Order Nonlinear Integrable Equations

2021 ◽  
Vol 28 (4) ◽  
pp. 452-465
Author(s):  
Zhonglong Zhao
Keyword(s):  
High Order ◽  
Difference Model ◽  
Linear Combinations ◽  
Integrable Equations

AbstractIn this paper, a procedure for constructing the symmetry-preserving difference models by means of the potential systems is employed to investigate some kinds of integrable equations. The invariant difference models for the Benjamin–Ono equation and the nonlinear dispersive $$K\left( {m,n} \right)$$ K m , n equation are investigated. Four cases of $$K\left( {m,n} \right)$$ K m , n equations which yield compactons are studied. The invariant difference models preserving all the symmetries are obtained. Furthermore, some linear combinations of the symmetries are used to construct the invariant difference models. The invariant difference model of the Hunter–Saxton equation is constructed. The idea of this paper can be further extended to discrete some other high-order nonlinear integrable equations.

scholarly journals ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

2012 ◽  
Vol 19 (04) ◽  
pp. 1250028
Author(s):  
TING SU ◽  
HUIHUI DAI ◽  
XIAN GUO GENG
Keyword(s):  
Optical Fibers ◽  
Variable Coefficient ◽  
High Order ◽  
Explicit Solutions ◽  
Compatibility Conditions ◽  
Dressing Method ◽  
Integrable Equations ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Hirota Equations

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.

An efficient routine for computing symmetric real spherical harmonics for high orders of expansion

2005 ◽  
Vol 38 (3) ◽  
pp. 501-504 ◽  
Author(s):  
Andrzej Kudlicki ◽  
Małgorzata Rowicka ◽  
Mirosław Gilski ◽  
Zbyszek Otwinowski
Keyword(s):  
Electron Microscopy ◽  
Spherical Harmonics ◽  
Efficient Method ◽  
Protein Crystallography ◽  
High Order ◽  
Linear Combinations ◽  
Crystallographic Symmetry ◽  
Stable Computation ◽  
Very High

A numerically efficient method of constructing symmetric real spherical harmonics is presented. Symmetric spherical harmonics are real spherical harmonics with built-in invariance with respect to rotations or inversions. Such symmetry-invariant spherical harmonics are linear combinations of non-symmetric ones. They are obtained as eigenvectors of an appropriate operator, depending on symmetry. This approach allows for fast and stable computation up to very high order symmetric harmonic bases, which can be used in e.g. averaging of non-crystallographic symmetry in protein crystallography or refinement of large viruses in electron microscopy.

Determination of the Burgers vector of a dislocation in a Fe-Mn alloy by weak-beam image in HVEM

1978 ◽  
Vol 36 (1) ◽  
pp. 330-331
Author(s):  
Y. Ishida ◽  
H. Ishida ◽  
K. Kohra ◽  
H. Ichinose
Keyword(s):  
High Order ◽  
Simulation Technique ◽  
The Other ◽  
Image Simulation ◽  
Diffraction Vector ◽  
Burgers Vector ◽  
Weak Beam ◽  
Beam Image ◽  
Edge Component

IntroductionA simple and accurate technique to determine the Burgers vector of a dislocation has become feasible with the advent of HVEM. The conventional image vanishing technique(1) using Bragg conditions with the diffraction vector perpendicular to the Burgers vector suffers from various drawbacks; The dislocation image appears even when the g.b = 0 criterion is satisfied, if the edge component of the dislocation is large. On the other hand, the image disappears for certain high order diffractions even when g.b ≠ 0. Furthermore, the determination of the magnitude of the Burgers vector is not easy with the criterion. Recent image simulation technique is free from the ambiguities but require too many parameters for the computation. The weak-beam “fringe counting” technique investigated in the present study is immune from the problems. Even the magnitude of the Burgers vector is determined from the number of the terminating thickness fringes at the exit of the dislocation in wedge shaped foil surfaces.

The usefulness of high index low symmetry zone axes for holz line analysis

1987 ◽  
Vol 45 ◽  
pp. 384-385
Author(s):  
C. M. Sung ◽  
D. B. Williams
Keyword(s):  
Lattice Parameter ◽  
High Order ◽  
Zone Axis ◽  
High Index ◽  
High Symmetry ◽  
Second Phases ◽  
Line Patterns ◽  
Low Symmetry ◽  
Line Analysis

Researchers have tended to use high symmetry zone axes (e.g. <111> <114>) for High Order Laue Zone (HOLZ) line analysis since Jones et al reported the origin of HOLZ lines and described some of their applications. But it is not always easy to find HOLZ lines from a specific high symmetry zone axis during microscope operation, especially from second phases on a scale of tens of nanometers. Therefore it would be very convenient if we can use HOLZ lines from low symmetry zone axes and simulate these patterns in order to measure lattice parameter changes through HOLZ line shifts. HOLZ patterns of high index low symmetry zone axes are shown in Fig. 1, which were obtained from pure Al at -186°C using a double tilt cooling holder. Their corresponding simulated HOLZ line patterns are shown along with ten other low symmetry orientations in Fig. 2. The simulations were based upon kinematical diffraction conditions.

Are HOLZ lines kinematic in off-zone-axis orientation?

1993 ◽  
Vol 51 ◽  
pp. 692-693
Author(s):  
J. M. Zuo ◽  
A. L. Weickenmeier ◽  
R. Holmestad ◽  
J. C. H. Spence
Keyword(s):  
Structure Determination ◽  
Standard Procedure ◽  
High Order ◽  
Zone Axis ◽  
Great Promise ◽  
Line Position ◽  
Measured Intensity ◽  
Constant Intensity ◽  
Axis Orientation ◽  
Diffraction Condition

The application of high order reflections in a weak diffraction condition off the zone axis center, including those in high order laue zones (HOLZ), holds great promise for structure determination using convergent beam electron diffraction (CBED). It is believed that in this case the intensities of high order reflections are kinematic or two-beam like. Hence, the measured intensity can be related to the structure factor amplitude. Then the standard procedure of structure determination in crystallography may be used for solving unknown structures. The dynamic effect on HOLZ line position and intensity in a strongly diffracting zone axis is well known. In a weak diffraction condition, the HOLZ line position may be approximated by the kinematic position, however, it is not clear whether this is also true for HOLZ intensities. The HOLZ lines, as they appear in CBED patterns, do show strong intensity variations along the line especially near the crossing of two lines, rather than constant intensity along the Bragg condition as predicted by kinematic or two beam theory.

Relativistic high-order harmonic generation

2003 ◽  
Vol 50 (3-4) ◽  
pp. 375-386
Author(s):  
D. B. MilosÕeviĆ ◽  
W. Becker
Keyword(s):  
Harmonic Generation ◽  
High Order ◽  
High Order Harmonic Generation ◽  
High Order Harmonic
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