scholarly journals The monomial representations of the Clifford group

2012 ◽  
Vol 12 (5&6) ◽  
pp. 404-431
Author(s):  
D. Markus Appleby ◽  
Ingemar Bengtsson ◽  
Stephen Brierley ◽  
Markus Grassl ◽  
David Gross ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Heisenberg Group ◽  
Mutually Unbiased Bases ◽  
Permutation Matrices ◽  
Clifford Group ◽  
First Time ◽  
Monomial Representations

We show that the Clifford group---the normaliser of the Weyl-Heisenberg group---can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.

scholarly journals Some Methods about Finding the Exact Solutions of Nonlinear Modified BBM Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Fan Niu ◽  
Jianming Qi ◽  
Zhiyong Zhou
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
First Time ◽  
Bbm Equation

Finding exact solutions of nonlinear equations plays an important role in nonlinear science, especially in engineering and mathematical physics. In this paper, we employed the complex method to get eight exact solutions of the modified BBM equation for the first time, including two elliptic function solutions, two simply periodic solutions, and four rational function solutions. We used the exp − ϕ z -expansion methods to get fourteen forms of solutions of the modified BBM equation. We also used the sine-cosine method to obtain eight styles’ exact solutions of the modified BBM equation. Only the complex method can obtain elliptic function solutions. We believe that the complex method presented in this paper can be more effectively applied to seek solutions of other nonlinear evolution equations.

scholarly journals Conditional symmetries and exact solutions of a nonlinear three-component reaction-diffusion model

2020 ◽  
pp. 1-21
Author(s):  
R. M. CHERNIHA ◽  
V. V. DAVYDOVYCH
Keyword(s):  
Asymptotic Behaviour ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Hunter Gatherers ◽  
Classical Symmetry ◽  
Classical Symmetries ◽  
Reaction Diffusion Model ◽  
Three Component Reaction ◽  
Biological Interpretation ◽  
First Time

Abstract Q-conditional (non-classical) symmetries of the known three-component reaction-diffusion (RD) system [K. Aoki et al. Theor. Popul. Biol. 50, 1–17 (1996)] modelling interaction between farmers and hunter-gatherers are constructed for the first time. A wide variety of Q-conditional symmetries are found, and it is shown that these symmetries are not equivalent to the Lie symmetries. Some operators of Q-conditional (non-classical) symmetry are applied for finding exact solutions of the RD system in question. Properties of the exact solutions (in particular, their asymptotic behaviour) are identified and possible biological interpretation is discussed.

EXACT SOLUTIONS OF THE EINSTEIN–MAXWELL EQUATIONS IN CHARGED PERFECT FLUID SPHERES FOR THE GENERALIZED TOLMAN–OPPENHEIMER–VOLKOFF EQUATIONS

2013 ◽  
Vol 22 (02) ◽  
pp. 1350009 ◽  
Author(s):  
LI ZOU ◽  
FANG-YU LI ◽  
HAO WEN
Keyword(s):  
Cosmological Constant ◽  
Exact Solutions ◽  
Perfect Fluid ◽  
Maxwell Equations ◽  
Constant Density ◽  
New Exact Solutions ◽  
First Time ◽  
Classical Constant ◽  
Fluid Spheres ◽  
Charged Case

Exact solutions of the Einstein–Maxwell equations for spherically symmetric charged perfect fluid have been broadly studied so far. However, the cases with a nonzero cosmological constant are seldom focused. In the present paper, the Tolman–Oppenheimer–Volkoff (TOV) equations have been generalized from the neutral case of hydrostatic equilibrium to the charged case of hydroelectrostatic equilibrium, and base on it, for the first time we find a series of new exact solutions of Einstein–Maxwell's equations with a nonzero cosmological constant for static charged perfect fluid spheres. Moreover, two special TOV equations and two classical constant density interior solutions are also given.

scholarly journals New Exact Solutions for High Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yongan Xie ◽  
Shengqiang Tang
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Bifurcation Theory ◽  
Solitary Wave Solutions ◽  
Nonlinear Schrödinger ◽  
Light Pulses ◽  
New Exact Solutions ◽  
Wave Solutions ◽  
First Time ◽  
Quintic Nonlinear Schrödinger Equation

We study a class of high dispersive cubic-quintic nonlinear Schrödinger equations, which describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. Applying bifurcation theory of dynamical systems and the Fan sub-equations method, more types of exact solutions, particularly solitary wave solutions, are obtained for the first time.

scholarly journals Local fractional Euler’s method for the steady heat-conduction problem

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 735-738 ◽  
Author(s):  
Feng Gao ◽  
Xiao-Jun Yang
Keyword(s):  
Heat Conduction ◽  
Exact Solutions ◽  
Numerical Solution ◽  
Heat Conduction Problem ◽  
Relaxation Equation ◽  
Euler’S Method ◽  
Steady Heat Conduction ◽  
First Time ◽  
Steady Heat

In this paper, the local fractional Euler?s method is proposed to consider the steady heat-conduction problem for the first time. The numerical solution for the local fractional heat-relaxation equation is presented. The comparison between numerical and exact solutions is discussed.

Sloshing in Regular Polygonal Basins—Frequencies, Modes, and Tileability

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Helmholtz Equation ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Waves ◽  
Small Amplitude ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Shallow Water Waves ◽  
First Time

Abstract The classical theory of small amplitude shallow water waves is applied to regular polygonal basins. The natural frequencies of the basins are related to the eigenvalues of the Helmholtz equation. Exact solutions are presented for triangular, square, and circular basins while pentagonal, hexagonal, and octagonal basins are solved, for the first time, by an efficient Ritz method. The first five eigenvalues of each basin are tabulated and the corresponding mode shapes are discussed. Tileability conditions are presented. Some modes (eigenmodes) can be tiled into larger domains.

Nonlinear dynamical wave structures to the Zoomeron equation for population models

2022 ◽  
Author(s):  
Ahmet Bekir ◽  
Emad H. M. Zahran
Keyword(s):  
Exact Solutions ◽  
Fractional Order ◽  
Population Models ◽  
Nonlinear Dynamical ◽  
Wave Solutions ◽  
Biological Population ◽  
First Time

Abstract In this paper, the nonlinear dynamical exact wave solutions to the non-fractional order and the time-fractional order of the biological population models are achevied for the first time in the framwork of the Paul-Painlevé approachmethod (PPAM). When the variables appearing in the exact solutions take specific values, the solaitry wave solutions will be easily satisfied.The realized results prove the efficiency of this technique.

scholarly journals Integral Equations of Non-Integer Orders and Discrete Maps with Memory

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1177
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Exact Solutions ◽  
Integral Equations ◽  
Fractional Integral ◽  
Periodic Sequence ◽  
Fractional Integrals ◽  
Dirac Delta ◽  
Discrete Maps ◽  
Delta Functions ◽  
First Time ◽  
With Memory

In this paper, we use integral equations of non-integer orders to derive discrete maps with memory. Note that discrete maps with memory were not previously derived from fractional integral equations of non-integer orders. Such a derivation of discrete maps with memory is proposed for the first time in this work. In this paper, we derived discrete maps with nonlocality in time and memory from exact solutions of fractional integral equations with the Riemann–Liouville and Hadamard type fractional integrals of non-integer orders and periodic sequence of kicks that are described by Dirac delta-functions. The suggested discrete maps with nonlocality in time are derived from these fractional integral equations without any approximation and can be considered as exact discrete analogs of these equations. The discrete maps with memory, which are derived from integral equations with the Hadamard type fractional integrals, do not depend on the period of kicks.

scholarly journals Exact Solutions and Conservation Laws of a Generalized (1 + 1) Dimensional System of Equations via Symbolic Computation

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2916
Author(s):  
Sivenathi Oscar Mbusi ◽  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution Equations ◽  
Theoretical Physics ◽  
Dimensional System ◽  
Ansatz Method ◽  
New Exact Solutions ◽  
Time Space ◽  
First Time

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific applications, the mentioned system naturally arises. Time, space, and scaling transformation symmetries lead to novel similarity reductions and new exact solutions. The solutions obtained include solitary waves and cnoidal and snoidal waves. The familiarity of closed-form solutions of nonlinear ordinary and partial differential equations enables numerical solvers and supports stability analysis. Although many efforts have been dedicated to solving nonlinear evolution equations, there is no unified method. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied on this underlying equation. It should also be noted that the methods applied in this paper give a unique solution set that differs from the newly reported solutions. In addition, we derive the conservation laws of the underlying system. It is also worth mentioning that this is the first time that the conservation laws for the equation under study are derived.

scholarly journals Optical solitons to a perturbed Gerdjikov-Ivanov equation using two different techniques

2021 ◽  
Vol 67 (5 Sep-Oct) ◽  
Author(s):  
Maha S.M. Shehata ◽  
Hadi Rezazadeh ◽  
Anwar J.M. Jawad ◽  
Emad H.M. Zahran ◽  
Ahmet Bekir
Keyword(s):  
Exact Solutions ◽  
Large Volume ◽  
Optical Solitons ◽  
First Time

In this article the perturbed Gerdjikov-Ivanov (GI)-equation which acts for the dynamics of propagation of solitons is employed. The balanced modified extended tanh-function and the non-balanced Riccati-Bernoulli Sub-ODE methods are used for the first time to obtain the new optical solitons of this equation. The obtained results give an accuracy interpretation of the propagation of solitons. We held a comparison between our results and those are in the previous work. The efficiency of these methods for constructing the exact solutions has been demonstrated. It is shown that these different technique's reduces the large volume of calculations.

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