Special Issue "Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 October 2016).

Special Issue Editor

Prof. Dr. Roman M. Cherniha
E-Mail Website
Guest Editor
Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereshchenkivs'ka Street, 01601 Kyiv, Ukraine
Interests: Non-linear PDEs: Lie and conditional symmetries, exact solutions and their properties; Application of symmetry-based methods for analytical solving nonlinear initial and boundary value problems arising in mathematical physics and mathematical biology
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Special Issue Information

Dear Colleagues,

This Special Issue is a natural continuation of the previous one, “Lie Theory and Its Applications”, which was very successful https://www.mdpi.com/journal/symmetry/special_issues/lie_theory.

Nowadays, the most powerful methods for construction of exact solutions of nonlinear partial differential equations (PDEs) are symmetry based methods. These methods originated from the Lie method, which was created by the prominent Norwegian mathematician Sophus Lie in the 19th century. The method was essentially developed using modern mathematical language in the 1960s and 1970s. Although the technique of the Lie method is well-known, the method still attracts the attention of many researchers, and new results are published on a regular basis.

However, it is well-known that the Lie method is not efficient for solving PDEs with a “poor” Lie symmetry (i.e., their maximal algebra of invariance is trivial). Thus, other symmetry-based methods (conditional symmetry, weak symmetry, generalized conditional symmetry etc.) were developed during the last few decades. The best known among them is the method of nonclassical symmetries, proposed by G. Bluman and J. Cole in 1969. Nevertheless, this approach was suggested almost 60 years ago, its successful applications for solving nonlinear equations were accomplished only in the 1990s. Moreover, one may say that progress is still modest in applications of non-Lie methods to systems of PDEs and integro-differential equations, especially those arising in real world applications. Thus, this Special Issue welcomes articles devoted to these topics. Articles and reviews devoted to the theoretical foundations of symmetry based methods and their applications for solving other nonlinear equations (especially reaction-diffusion-convection equations and higher-order PDEs) and nonlinear models (especially for biomedical applications) are also welcome.

Prof. Dr. Roman M. Cherniha
Guest Editor

Manuscript Submission Information

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Keywords

  • Lie symmetry
  • nonclassical symmetry
  • Q-conditional symmetry
  • (generalized) conditional symmetry
  • invariance algebra of nonlinear PDE
  • symmetry of (initial) boundary-value problem
  • exact solution
  • invariant solution
  • non-Lie solution

Published Papers (10 papers)

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Research

Open AccessArticle
On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries
Symmetry 2017, 9(3), 33; https://doi.org/10.3390/sym9030033 - 27 Feb 2017
Cited by 16
Abstract
A conservation law theorem stated by N. Ibragimov along with its subsequent extensions are shown to be a special case of a standard formula that uses a pair consisting of a symmetry and an adjoint-symmetry to produce a conservation law through a well-known [...] Read more.
A conservation law theorem stated by N. Ibragimov along with its subsequent extensions are shown to be a special case of a standard formula that uses a pair consisting of a symmetry and an adjoint-symmetry to produce a conservation law through a well-known Fréchet derivative identity. Furthermore, the connection of this formula (and of Ibragimov’s theorem) to the standard action of symmetries on conservation laws is explained, which accounts for a number of major drawbacks that have appeared in recent work using the formula to generate conservation laws. In particular, the formula can generate trivial conservation laws and does not always yield all non-trivial conservation laws unless the symmetry action on the set of these conservation laws is transitive. It is emphasized that all local conservation laws for any given system of differential equations can be found instead by a general method using adjoint-symmetries. This general method is a kind of adjoint version of the standard Lie method to find all local symmetries and is completely algorithmic. The relationship between this method, Noether’s theorem and the symmetry/adjoint-symmetry formula is discussed. Full article
Open AccessArticle
Symmetry Analysis and Conservation Laws of the Zoomeron Equation
Symmetry 2017, 9(2), 27; https://doi.org/10.3390/sym9020027 - 21 Feb 2017
Cited by 13
Abstract
In this work, we study the (2 + 1)-dimensional Zoomeron equation which is an extension of the famous (1 + 1)-dimensional Zoomeron equation that has been studied extensively in the literature. Using classical Lie point symmetries admitted by the equation, for the first [...] Read more.
In this work, we study the (2 + 1)-dimensional Zoomeron equation which is an extension of the famous (1 + 1)-dimensional Zoomeron equation that has been studied extensively in the literature. Using classical Lie point symmetries admitted by the equation, for the first time we develop an optimal system of one-dimensional subalgebras. Based on this optimal system, we obtain symmetry reductions and new group-invariant solutions. Again for the first time, we construct the conservation laws of the underlying equation using the multiplier method. Full article
Open AccessFeature PaperArticle
A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II
Symmetry 2017, 9(1), 13; https://doi.org/10.3390/sym9010013 - 20 Jan 2017
Cited by 3
Abstract
A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry [...] Read more.
A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1)-dimensional. Exact solutions of some (1 + 1)-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1)-dimensional simplified Keller–Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2)-dimensional Neumann problem for the simplified Keller–Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view) domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found. Full article
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Open AccessArticle
Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain
Symmetry 2017, 9(1), 2; https://doi.org/10.3390/sym9010002 - 24 Dec 2016
Cited by 5
Abstract
Diffusion-limited erosion is a distinct universality class of fluctuating interfaces. Although its dynamical exponent z = 1 , none of the known variants of conformal invariance can act as its dynamical symmetry. In d = 1 spatial dimensions, its infinite-dimensional dynamic symmetry is [...] Read more.
Diffusion-limited erosion is a distinct universality class of fluctuating interfaces. Although its dynamical exponent z = 1 , none of the known variants of conformal invariance can act as its dynamical symmetry. In d = 1 spatial dimensions, its infinite-dimensional dynamic symmetry is constructed and shown to be isomorphic to the direct sum of three loop-Virasoro algebras. The infinitesimal generators are spatially non-local and use the Riesz-Feller fractional derivative. Co-variant two-time response functions are derived and reproduce the exact solution of diffusion-limited erosion. The relationship with the terrace-step-kind model of vicinal surfaces and the integrable XXZ chain are discussed. Full article
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Open AccessArticle
Reduction Operators and Exact Solutions of Variable Coefficient Nonlinear Wave Equations with Power Nonlinearities
Symmetry 2017, 9(1), 3; https://doi.org/10.3390/sym9010003 - 22 Dec 2016
Cited by 2
Abstract
Reduction operators, i.e., the operators of nonclassical (or conditional) symmetry of a class of variable coefficient nonlinear wave equations with power nonlinearities, are investigated within the framework of a singular reduction operator. A classification of regular reduction operators is performed with respect to [...] Read more.
Reduction operators, i.e., the operators of nonclassical (or conditional) symmetry of a class of variable coefficient nonlinear wave equations with power nonlinearities, are investigated within the framework of a singular reduction operator. A classification of regular reduction operators is performed with respect to generalized extended equivalence groups. Exact solutions of some nonlinear wave models, which are invariant under certain reduction operators, are also constructed. Full article
Open AccessArticle
The Method of Linear Determining Equations to Evolution System and Application for Reaction-Diffusion System with Power Diffusivities
by Lina Ji
Symmetry 2016, 8(12), 157; https://doi.org/10.3390/sym8120157 - 20 Dec 2016
Cited by 2
Abstract
The method of linear determining equations is constructed to study conditional Lie–Bäcklund symmetry and the differential constraint of a two-component second-order evolution system, which generalize the determining equations used in the search for classical Lie symmetry. As an application of the approach, the [...] Read more.
The method of linear determining equations is constructed to study conditional Lie–Bäcklund symmetry and the differential constraint of a two-component second-order evolution system, which generalize the determining equations used in the search for classical Lie symmetry. As an application of the approach, the two-component reaction-diffusion system with power diffusivities is considered. The conditional Lie–Bäcklund symmetries and differential constraints admitted by the reaction-diffusion system are identified. Consequently, the reductions of the resulting system are established due to the compatibility of the corresponding invariant surface conditions and the original system. Full article
Open AccessArticle
Noether Symmetries Quantization and Superintegrability of Biological Models
Symmetry 2016, 8(12), 155; https://doi.org/10.3390/sym8120155 - 20 Dec 2016
Cited by 2
Abstract
It is shown that quantization and superintegrability are not concepts that are inherent to classical Physics alone. Indeed, one may quantize and also detect superintegrability of biological models by means of Noether symmetries. We exemplify the method by using a mathematical model that [...] Read more.
It is shown that quantization and superintegrability are not concepts that are inherent to classical Physics alone. Indeed, one may quantize and also detect superintegrability of biological models by means of Noether symmetries. We exemplify the method by using a mathematical model that was proposed by Basener and Ross (2005), and that describes the dynamics of growth and sudden decrease in the population of Easter Island. Full article
Open AccessArticle
Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems
Symmetry 2016, 8(12), 140; https://doi.org/10.3390/sym8120140 - 26 Nov 2016
Cited by 3
Abstract
It is generally known that classical point and potential Lie symmetries of differential equations (the latter calculated as point symmetries of an equivalent system) can be different. We question whether this is true when the symmetries are extended to nonclassical symmetries. In this [...] Read more.
It is generally known that classical point and potential Lie symmetries of differential equations (the latter calculated as point symmetries of an equivalent system) can be different. We question whether this is true when the symmetries are extended to nonclassical symmetries. In this paper, we consider two classes of nonlinear partial differential equations; the first one is a diffusion–convection equation, the second one a wave, where we will show that the majority of the nonclassical point symmetries are included in the nonclassical potential symmetries. We highlight a special case were the opposite is true. Full article
Open AccessArticle
Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations
Symmetry 2016, 8(11), 128; https://doi.org/10.3390/sym8110128 - 15 Nov 2016
Cited by 2
Abstract
In this paper, we develop the symmetry-related methods to study invariant subspaces of the two-dimensional nonlinear differential operators. The conditional Lie–Bäcklund symmetry and Lie point symmetry methods are used to construct invariant subspaces of two-dimensional differential operators. We first apply the multiple conditional [...] Read more.
In this paper, we develop the symmetry-related methods to study invariant subspaces of the two-dimensional nonlinear differential operators. The conditional Lie–Bäcklund symmetry and Lie point symmetry methods are used to construct invariant subspaces of two-dimensional differential operators. We first apply the multiple conditional Lie–Bäcklund symmetries to derive invariant subspaces of the two-dimensional operators. As an application, the invariant subspaces for a class of two-dimensional nonlinear quadratic operators are provided. Furthermore, the invariant subspace method in one-dimensional space combined with the Lie symmetry reduction method and the change of variables is used to obtain invariant subspaces of the two-dimensional nonlinear operators. Full article
Open AccessArticle
Lorentz Transformations from Intrinsic Symmetries
Symmetry 2016, 8(9), 94; https://doi.org/10.3390/sym8090094 - 09 Sep 2016
Abstract
We reveal the frame-exchange space-inversion (FESI) symmetry and the frame-exchange time-inversion (FETI) symmetry in the Lorentz transformation and propose a symmetry principle stating that the space-time transformation between two inertial frames is invariant under the FESI or the FETI transformation. In combination with [...] Read more.
We reveal the frame-exchange space-inversion (FESI) symmetry and the frame-exchange time-inversion (FETI) symmetry in the Lorentz transformation and propose a symmetry principle stating that the space-time transformation between two inertial frames is invariant under the FESI or the FETI transformation. In combination with the principle of relativity and the presumed nature of Euclidean space and time, the symmetry principle is employed to derive the proper orthochronous Lorentz transformation without assuming the constancy of the speed of light and specific mathematical requirements (such as group property) a priori. We explicitly demonstrate that the constancy of the speed of light in all inertial frames can be derived using the velocity reciprocity property, which is a deductive consequence of the space–time homogeneity and the space isotropy. The FESI or the FETI symmetry remains to be preserved in the Galilean transformation at the non-relativistic limit. Other similar symmetry operations result in either trivial transformations or improper and/or non-orthochronous Lorentz transformations, which do not form groups. Full article
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