Ample soliton waves for the crystal lattice formation of the conformable time-fractional (N + 1) Sinh-Gordon equation by the modified Khater method and the Painlevé property

2020 ◽  
Vol 38 (3) ◽  
pp. 2745-2752 ◽  
Author(s):  
Haiyong Qin ◽  
Raghda A.M. Attia ◽  
Mostafa M.A. Khater ◽  
Dianchen Lu
Keyword(s):  
Crystal Lattice ◽  
Gordon Equation ◽  
Painlevé Property ◽  
Lattice Formation

Integrable and solvable systems, and relations among them

1985 ◽  
Vol 315 (1533) ◽  
pp. 451-457 ◽  
Keyword(s):  
Gordon Equation ◽  
Soliton Solutions ◽  
The Self ◽  
Sine Gordon Equation ◽  
Field Equations ◽  
Four Dimensions ◽  
Painlevé Property ◽  
All Solutions ◽  
Special Cases ◽  
Integrable Dynamical Systems

There are several different classes of differential equations that may be described as ‘integrable’ or ‘solvable’. For example, there are completely integrable dynamical systems; equations such as the sine—Gordon equation, which admit soliton solutions; and the self-dual gauge-field equations in four dimensions (with generalizations in arbitrarily large dimension). This lecture discusses two ideas that link all of these together: one is the Painlevé property, which says (roughly speaking) that all solutions to the equations are meromorphic; the other is that many of the equations are special cases (i.e. reductions) of others.

Insight into the crystal lattice formation of brookite in aqueous ammonia media: the electrolyte effect

CrystEngComm ◽  
2011 ◽  
Vol 13 (12) ◽  
pp. 4167 ◽  
Author(s):  
Yanchao Jiao ◽  
Bin Zhao ◽  
Feng Chen ◽  
Jinlong Zhang
Keyword(s):  
Crystal Lattice ◽  
Aqueous Ammonia ◽  
Lattice Formation ◽  
Insight Into

Void-Lattice Formation in Natural Fluorite by Electron Irradiation

1976 ◽  
Vol 34 ◽  
pp. 614-615 ◽  
Author(s):  
L.E. Murr
Keyword(s):  
Electron Microscope ◽  
Electron Irradiation ◽  
Heavy Ion ◽  
High Energy ◽  
Stoichiometric Ratio ◽  
Direct Observations ◽  
Transmission Electron ◽  
Anion Vacancies ◽  
Cation And Anion Vacancies ◽  
Lattice Formation

The production of void lattices in metals as a result of displacement damage associated with high energy and heavy ion bombardment is now well documented. More recently, Murr has shown that a void lattice can be developed in natural (colored) fluorites observed in the transmission electron microscope. These were the first observations of a void lattice in an irradiated nonmetal, and the first, direct observations of color-center aggregates. Clinard, et al. have also recently observed a void lattice (described as a high density of aligned "pores") in neutron irradiated Al2O3 and Y2O3. In this latter work, itwas pointed out that in order that a cavity be formed,a near-stoichiometric ratio of cation and anion vacancies must aggregate. It was reasoned that two other alternatives to explain the pores were cation metal colloids and highpressure anion gas bubbles.Evans has proposed that void lattices result from the presence of a pre-existing impurity lattice, and predicted that the formation of a void lattice should restrict swelling in irradiated materials because it represents a state of saturation.

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