Symmetry

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310 KiB

Open AccessArticle

Centrally Extended Conformal Galilei Algebras and Invariant Nonlinear PDEs

by Naruhiko Aizawa and Tadanori Kato

Symmetry 2015, 7(4), 1989-2008; https://doi.org/10.3390/sym7041989 - 3 Nov 2015

Cited by 1 | Viewed by 4962

Abstract

We construct, for any given \( \ell = \frac{1}{2} + {\mathbb N}_0, \) second-order \textit{nonlinear} partial differential equations (PDEs) which are invariant under the transformations generated by the centrally extended conformal Galilei algebras. This is done for a particular realization of the algebras [...] Read more.

We construct, for any given \( \ell = \frac{1}{2} + {\mathbb N}_0, \) second-order \textit{nonlinear} partial differential equations (PDEs) which are invariant under the transformations generated by the centrally extended conformal Galilei algebras. This is done for a particular realization of the algebras obtained by coset construction and we employ the standard Lie point symmetry technique for the construction of PDEs. It is observed that the invariant PDEs have significant difference for \( \ell > \frac{1}{3}. \) Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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204 KiB

Open AccessArticle

An Application of Equivalence Transformations to Reaction Diffusion Equations

by Mariano Torrisi and Rita Tracinà

Symmetry 2015, 7(4), 1929-1944; https://doi.org/10.3390/sym7041929 - 23 Oct 2015

Cited by 17 | Viewed by 4126

Abstract

In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension [...] Read more.

In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

2151 KiB

Open AccessArticle

Q-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems

by Oleksii Pliukhin

Symmetry 2015, 7(4), 1841-1855; https://doi.org/10.3390/sym7041841 - 16 Oct 2015

Cited by 9 | Viewed by 4143

Abstract

A wide range of reaction–diffusion systems with constant diffusivities that are invariant under Q-conditional operators is found. Using the symmetries obtained, the reductions of the corresponding systems to the systems of ODEs are conducted in order to find exact solutions. In particular, [...] Read more.

A wide range of reaction–diffusion systems with constant diffusivities that are invariant under Q-conditional operators is found. Using the symmetries obtained, the reductions of the corresponding systems to the systems of ODEs are conducted in order to find exact solutions. In particular, the solutions of some reaction–diffusion systems of the Lotka–Volterra type in an explicit form and satisfying Dirichlet boundary conditions are obtained. An biological interpretation is presented in order to show that two different types of interaction between biological species can be described. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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857 KiB

Open AccessArticle

Lie Group Method for Solving the Generalized Burgers’, Burgers’–KdV and KdV Equations with Time-Dependent Variable Coefficients

by Mina B. Abd-el-Malek and Amr M. Amin

Symmetry 2015, 7(4), 1816-1830; https://doi.org/10.3390/sym7041816 - 13 Oct 2015

Cited by 8 | Viewed by 4971

Abstract

In this study, the Lie group method for constructing exact and numerical solutions of the generalized time-dependent variable coefficients Burgers’, Burgers’–KdV, and KdV equations with initial and boundary conditions is presented. Lie group theory is applied to determine symmetry reductions which reduce the [...] Read more.

In this study, the Lie group method for constructing exact and numerical solutions of the generalized time-dependent variable coefficients Burgers’, Burgers’–KdV, and KdV equations with initial and boundary conditions is presented. Lie group theory is applied to determine symmetry reductions which reduce the nonlinear partial differential equations to ordinary differential equations. The obtained ordinary differential equations were solved analytically and the solutions are obtained in closed form for some specific choices of parameters, while others are solved numerically. In the obtained results we studied effects of both the time t and the index of nonlinearity on the behavior of the velocity, and the solutions are graphically presented. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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217 KiB

Open AccessArticle

Classical and Quantum Burgers Fluids: A Challenge for Group Analysis

by Philip Broadbridge

Symmetry 2015, 7(4), 1803-1815; https://doi.org/10.3390/sym7041803 - 9 Oct 2015

Cited by 6 | Viewed by 4634

Abstract

The most general second order irrotational vector field evolution equation is constructed, that can be transformed to a single equation for the Cole–Hopf potential. The exact solution to the radial Burgers equation, with constant mass influx through a spherical supply surface, is constructed. [...] Read more.

The most general second order irrotational vector field evolution equation is constructed, that can be transformed to a single equation for the Cole–Hopf potential. The exact solution to the radial Burgers equation, with constant mass influx through a spherical supply surface, is constructed. The complex linear Schrödinger equation is equivalent to an integrable system of two coupled real vector equations of Burgers type. The first velocity field is the particle current divided by particle probability density. The second vector field gives a complex valued correction to the velocity that results in the correct quantum mechanical correction to the kinetic energy density of the Madelung fluid. It is proposed how to use symmetry analysis to systematically search for other constrained potential systems that generate a closed system of vector component evolution equations with constraints other than irrotationality. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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304 KiB

Open AccessArticle

Symbolic and Iterative Computation of Quasi-Filiform Nilpotent Lie Algebras of Dimension Nine

by Mercedes Pérez, Francisco Pérez and Emilio Jiménez

Symmetry 2015, 7(4), 1788-1802; https://doi.org/10.3390/sym7041788 - 1 Oct 2015

Viewed by 4463

Abstract

This paper addresses the problem of computing the family of two-filiform Lie algebra laws of dimension nine using three Lie algebra properties converted into matrix form properties: Jacobi identity, nilpotence and quasi-filiform property. The interest in this family is broad, both within the [...] Read more.

This paper addresses the problem of computing the family of two-filiform Lie algebra laws of dimension nine using three Lie algebra properties converted into matrix form properties: Jacobi identity, nilpotence and quasi-filiform property. The interest in this family is broad, both within the academic community and the industrial engineering community, since nilpotent Lie algebras are applied in traditional mechanical dynamic problems and current scientific disciplines. The conditions of being quasi-filiform and nilpotent are applied carefully and in several stages, and appropriate changes of the basis are achieved in an iterative and interactive process of simplification. This has been implemented by means of the development of more than thirty Maple modules. The process has led from the first family formulation, with 64 parameters and 215 constraints, to a family of 16 parameters and 17 constraints. This structure theorem permits the exhaustive classification of the quasi-filiform nilpotent Lie algebras of dimension nine with current computational methodologies. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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228 KiB

Open AccessArticle

New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation

by Valentyn Tychynin

Symmetry 2015, 7(4), 1751-1767; https://doi.org/10.3390/sym7041751 - 29 Sep 2015

Cited by 5 | Viewed by 3809

Abstract

Additional nonlocal symmetries of diffusion-convection equations and the Burgers equation are obtained. It is shown that these equations are connected via a generalized hodograph transformation and appropriate nonlocal symmetries arise from additional Lie symmetries of intermediate equations. Two entirely different techniques are used [...] Read more.

Additional nonlocal symmetries of diffusion-convection equations and the Burgers equation are obtained. It is shown that these equations are connected via a generalized hodograph transformation and appropriate nonlocal symmetries arise from additional Lie symmetries of intermediate equations. Two entirely different techniques are used to search nonlocal symmetry of a given equation: the first is based on usage of the characteristic equations generated by additional operators, another technique assumes the reconstruction of a parametrical Lie group transformation from such operator. Some of them are based on the nonlocal transformations that contain new independent variable determined by an auxiliary differential equation and allow the interpretation as a nonlocal transformation with additional variables. The formulae derived for construction of exact solutions are used. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

255 KiB

Open AccessArticle

An Elementary Derivation of the Matrix Elements of Real Irreducible Representations of so(3)

by Rutwig Campoamor-Stursberg

Symmetry 2015, 7(3), 1655-1669; https://doi.org/10.3390/sym7031655 - 14 Sep 2015

Cited by 6 | Viewed by 4013

Abstract

Using elementary techniques, an algorithmic procedure to construct skew-symmetric matrices realizing the real irreducible representations of so(3) is developed. We further give a simple criterion that enables one to deduce the decomposition of an arbitrary real representation R of so(3) into real irreducible [...] Read more.

Using elementary techniques, an algorithmic procedure to construct skew-symmetric matrices realizing the real irreducible representations of so(3) is developed. We further give a simple criterion that enables one to deduce the decomposition of an arbitrary real representation R of so(3) into real irreducible components from the characteristic polynomial of an arbitrary representation matrix. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

297 KiB

Open AccessArticle

Symmetries, Lagrangians and Conservation Laws of an Easter Island Population Model

by M.C. Nucci and G. Sanchini

Symmetry 2015, 7(3), 1613-1632; https://doi.org/10.3390/sym7031613 - 8 Sep 2015

Cited by 22 | Viewed by 5267

Abstract

Basener and Ross (2005) proposed a mathematical model that describes the dynamics of growth and sudden decrease in the population of Easter Island. We have applied Lie group analysis to this system and found that it can be integrated by quadrature if the [...] Read more.

Basener and Ross (2005) proposed a mathematical model that describes the dynamics of growth and sudden decrease in the population of Easter Island. We have applied Lie group analysis to this system and found that it can be integrated by quadrature if the involved parameters satisfy certain relationships. We have also discerned hidden linearity. Moreover, we have determined a Jacobi last multiplier and, consequently, a Lagrangian for the general system and have found other cases independently and dependently on symmetry considerations in order to construct a corresponding variational problem, thus enabling us to find conservation laws by means of Noether’s theorem. A comparison with the qualitative analysis given by Basener and Ross is provided. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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256 KiB

Open AccessArticle

From Conformal Invariance towards Dynamical Symmetries of the Collisionless Boltzmann Equation

by Stoimen Stoimenov and Malte Henkel

Symmetry 2015, 7(3), 1595-1612; https://doi.org/10.3390/sym7031595 - 7 Sep 2015

Cited by 13 | Viewed by 4070

Abstract

Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the 2D conformal algebra. In the case without external forces, the symmetry of the conformally-invariant transport equation [...] Read more.

Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the 2D conformal algebra. In the case without external forces, the symmetry of the conformally-invariant transport equation is first generalized by considering the particle momentum as an independent variable. This new conformal representation can be further extended to include an external force. The construction and possible physical applications are outlined. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

298 KiB

Open AccessArticle

A (1+2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions

by Maksym Didovych

Symmetry 2015, 7(3), 1463-1474; https://doi.org/10.3390/sym7031463 - 24 Aug 2015

Cited by 6 | Viewed by 4213

Abstract

This research is a natural continuation of the recent paper “Exact solutions of the simplified Keller–Segel model” (Commun Nonlinear Sci Numer Simulat 2013, 18, 2960–2971). It is shown that a (1+2)-dimensional Keller–Segel type system is invariant with respect infinite-dimensional Lie algebra. All possible [...] Read more.

This research is a natural continuation of the recent paper “Exact solutions of the simplified Keller–Segel model” (Commun Nonlinear Sci Numer Simulat 2013, 18, 2960–2971). It is shown that a (1+2)-dimensional Keller–Segel type system is invariant with respect infinite-dimensional Lie algebra. All possible maximal algebras of invariance of the Neumann boundary value problems based on the Keller–Segel system in question were found. Lie symmetry operators are used for constructing exact solutions of some boundary value problems. Moreover, it is proved that the boundary value problem for the (1+1)-dimensional Keller–Segel system with specific boundary conditions can be linearized and solved in an explicit form. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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588 KiB

Open AccessArticle

Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems

by Roman Cherniha and John R. King

Symmetry 2015, 7(3), 1410-1435; https://doi.org/10.3390/sym7031410 - 17 Aug 2015

Cited by 16 | Viewed by 5037

Abstract

A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case) is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary [...] Read more.

A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case) is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary value problems with standard boundary conditions, followed as particular cases from our definition. Simple examples of direct applicability to the nonlinear problems arising in applications are demonstrated. Moreover, the successful application of the definition for the Lie and conditional symmetry classification of a class of (1 + 2)-dimensional nonlinear boundary value problems governed by the nonlinear diffusion equation in a semi-infinite domain is realised. In particular, it is proven that there is a special exponent, k ≠ -2, for the power diffusivity u^kwhen the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case k ≠ -2. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity uk and a constant non-zero flux on the boundary (such problems are common in applications and describing a wide range of phenomena) to (1 + 1)-dimensional problems. The structure and properties of the problems obtained are briefly analysed. Finally, some results demonstrating how Lie invariance of the boundary value problem in question depends on the geometry of the domain are presented. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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252 KiB

Open AccessArticle

Bäcklund Transformations for Integrable Geometric Curve Flows

by Changzheng Qu, Jingwei Han and Jing Kang

Symmetry 2015, 7(3), 1376-1394; https://doi.org/10.3390/sym7031376 - 3 Aug 2015

Cited by 7 | Viewed by 4815

Abstract

We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym [...] Read more.

We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym flows in the three-dimensional Euclidean geometry and the Sawada-Kotera flow in the affine geometry, etc. Using the fact that two different curves in a given geometry are governed by the same integrable equation, we obtain Bäcklund transformations relating to these two integrable geometric flows. Some special solutions of the integrable systems are used to obtain the explicit Bäcklund transformations. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

216 KiB

Open AccessArticle

Conservation Laws of Discrete Evolution Equations by Symmetries and Adjoint Symmetries

by Wen-Xiu Ma

Symmetry 2015, 7(2), 714-725; https://doi.org/10.3390/sym7020714 - 22 May 2015

Cited by 60 | Viewed by 5158

Abstract

A direct approach is proposed for constructing conservation laws of discrete evolution equations, regardless of the existence of a Lagrangian. The approach utilizes pairs of symmetries and adjoint symmetries, in which adjoint symmetries make up for the disadvantage of non-Lagrangian structures in presenting [...] Read more.

A direct approach is proposed for constructing conservation laws of discrete evolution equations, regardless of the existence of a Lagrangian. The approach utilizes pairs of symmetries and adjoint symmetries, in which adjoint symmetries make up for the disadvantage of non-Lagrangian structures in presenting a correspondence between symmetries and conservation laws. Applications are made for the construction of conservation laws of the Volterra lattice equation. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

Review

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365 KiB

Open AccessReview

Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions

by Malte Henkel

Symmetry 2015, 7(4), 2108-2133; https://doi.org/10.3390/sym7042108 - 13 Nov 2015

Cited by 13 | Viewed by 4797

Abstract

Dynamical symmetries are of considerable importance in elucidating the complex behaviour of strongly interacting systems with many degrees of freedom. Paradigmatic examples are cooperative phenomena as they arise in phase transitions, where conformal invariance has led to enormous progress in equilibrium phase transitions, [...] Read more.

Dynamical symmetries are of considerable importance in elucidating the complex behaviour of strongly interacting systems with many degrees of freedom. Paradigmatic examples are cooperative phenomena as they arise in phase transitions, where conformal invariance has led to enormous progress in equilibrium phase transitions, especially in two dimensions. Non-equilibrium phase transitions can arise in much larger portions of the parameter space than equilibrium phase transitions. The state of the art of recent attempts to generalise conformal invariance to a new generic symmetry, taking into account the different scaling behaviour of space and time, will be reviewed. Particular attention will be given to the causality properties as they follow for co-variant n-point functions. These are important for the physical identification of n-point functions as responses or correlators. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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