Non-local symmetry, interaction solutions and conservation lawsof the $$(1+1)$$-dimensional Wu–Zhang equation

Pramana ◽  
2021 ◽  
Vol 95 (3) ◽  
Author(s):  
Yanxia Wang ◽  
Ben Gao
Keyword(s):  
Local Symmetry ◽  
Interaction Solutions ◽  
Non Local

Nonlocal Symmetries, Conservation Laws and Interaction Solutions of the Generalised Dispersive Modified Benjamin–Bona–Mahony Equation

2018 ◽  
Vol 73 (5) ◽  
pp. 399-405 ◽  
Author(s):  
Xue-Wei Yan ◽  
Shou-Fu Tian ◽  
Min-Jie Dong ◽  
Xiu-Bin Wang ◽  
Tian-Tian Zhang
Keyword(s):  
Direct Method ◽  
Symmetry Transformation ◽  
Expansion Method ◽  
Local Symmetry ◽  
Dispersive Media ◽  
Jacobi Elliptic Function ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Non Local ◽  
Truncated Painlevé Expansion

AbstractWe consider the generalised dispersive modified Benjamin–Bona–Mahony equation, which describes an approximation status for long surface wave existed in the non-linear dispersive media. By employing the truncated Painlevé expansion method, we derive its non-local symmetry and Bäcklund transformation. The non-local symmetry is localised by a new variable, which provides the corresponding non-local symmetry group and similarity reductions. Moreover, a direct method can be provided to construct a kind of finite symmetry transformation via the classic Lie point symmetry of the normal prolonged system. Finally, we find that the equation is a consistent Riccati expansion solvable system. With the help of the Jacobi elliptic function, we get its interaction solutions between solitary waves and cnoidal periodic waves.

Symmetry reduction and new interaction solutions for the negative-order potential KdV equation

2020 ◽  
pp. 2150108
Author(s):  
Hengchun Hu ◽  
Zhenya Zhang
Keyword(s):  
Kdv Equation ◽  
Symmetry Transformation ◽  
Expansion Method ◽  
Local Symmetry ◽  
Similarity Solutions ◽  
Seed Solution ◽  
Interaction Solutions ◽  
New Exact Solutions ◽  
Negative Order ◽  
Non Local

New soliton–cnoidal interaction solutions for the negative-order potential KdV equation are studied with the help of the consistent tanh expansion method. The non-local symmetry for the negative-order potential KdV equation is derived from the truncated Painlevé expansion method. The non-local symmetry is transformed into the standard Lie point symmetry by introducing new dependent variables and the finite symmetry transformation is also presented to construct new exact solutions with non-zero seed solution. The similarity solutions of the enlarged negative-order potential KdV system are obtained through different constant selections.

scholarly journals Fuzzy Differential Subordinations Based Upon the Mittag-Leffler Type Borel Distribution

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1023
Author(s):  
Hari Mohan Srivastava ◽  
Sheza M. El-Deeb
Keyword(s):  
Fuzzy Systems ◽  
Special Functions ◽  
Univalent Functions ◽  
Local Symmetry ◽  
Open Unit ◽  
Fuzzy Membership Functions ◽  
Non Local ◽  
Differential Subordinations ◽  
Two Parameter ◽  
Borel Type

In this paper, we investigate several fuzzy differential subordinations that are connected with the Borel distribution series Bλ,α,β(z) of the Mittag-Leffler type, which involves the two-parameter Mittag-Leffler function Eα,β(z). Using the above-mentioned operator Bλ,α,β, we also introduce and study a class Mλ,α,βFη of holomorphic and univalent functions in the open unit disk Δ. The Mittag-Leffler-type functions, which we have used in the present investigation, belong to the significantly wider family of the Fox-Wright function pΨq(z), whose p numerator parameters and q denominator parameters possess a kind of symmetry behavior in the sense that it remains invariant (or unchanged) when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. Here, in this article, we have used such special functions in our study of a general Borel-type probability distribution, which may be symmetric or asymmetric. As symmetry is generally present in most works involving fuzzy sets and fuzzy systems, our usages here of fuzzy subordinations and fuzzy membership functions potentially possess local or non-local symmetry features.

Residual Symmetries and Interaction Solutions for the Classical Korteweg-de Vries Equation

2017 ◽  
Vol 72 (3) ◽  
pp. 217-222 ◽  
Author(s):  
Jin-Xi Fei ◽  
Wei-Ping Cao ◽  
Zheng-Yi Ma
Keyword(s):  
Vries Equation ◽  
Expansion Method ◽  
Point Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Dependent Variables ◽  
Non Linear ◽  
De Vries ◽  
Finite Transformation ◽  
Non Local

AbstractThe non-local residual symmetry for the classical Korteweg-de Vries equation is derived by the truncated Painlevé analysis. This symmetry is first localised to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we then obtain the finite transformation for the localised residual symmetry. Based on the consistent tanh expansion method, some exact interaction solutions among different non-linear excitations are explicitly presented finally. Some special interaction solutions are investigated both in analytical and graphical ways at the same time.

New Exact Solutions of the CDGSK Equation Related to a Non-local Symmetry

1994 ◽  
Vol 11 (10) ◽  
pp. 593-596 ◽  
Author(s):  
Senyue Lou ◽  
Hangyu Ruan ◽  
Weizhong Chen ◽  
Zhenli Wang ◽  
Lili Chen
Keyword(s):  
Exact Solutions ◽  
Local Symmetry ◽  
New Exact Solutions ◽  
Non Local

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

ELNES of Iron Compounds

1990 ◽  
Vol 48 (2) ◽  
pp. 28-29
Author(s):  
Hiroki Kurata ◽  
Kazuhiro Nagai ◽  
Seiji Isoda ◽  
Takashi Kobayashi
Keyword(s):  
Energy Loss ◽  
Metal Ions ◽  
Energy Resolution ◽  
Transition Metal Oxides ◽  
Local Symmetry ◽  
Iron Compounds ◽  
Fine Structures ◽  
Electron Energy Loss ◽  
Fe Ions ◽  
Electron Energy Loss Spectra

Electron energy loss spectra of transition metal oxides, which show various fine structures in inner shell edges, have been extensively studied. These structures and their positions are related to the oxidation state of metal ions. In this sence an influence of anions coordinated with the metal ions is very interesting. In the present work, we have investigated the energy loss near-edge structures (ELNES) of some iron compounds, i.e. oxides, chlorides, fluorides and potassium cyanides. In these compounds, Fe ions (Fe2+ or Fe3+) are octahedrally surrounded by six ligand anions and this means that the local symmetry around each iron is almost isotropic.EELS spectra were obtained using a JEM-2000FX with a Gatan Model-666 PEELS. The energy resolution was about leV which was mainly due to the energy spread of LaB6 -filament. The threshole energies of each edges were measured using a voltage scan module which was calibrated by setting the Ni L3 peak in NiO to an energy value of 853 eV.

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