Conservation laws of evolution systems

Zharinov, V. V.

doi:10.1007/bf01035536

Cited by 35 publications

(33 citation statements)

References 7 publications

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“…Although in many simple cases conservation laws can be investigated in the above empiric framework, for deeper analysis one often needs to consider more rigorous definitions, that can be found, e.g., in [5][6][7].…”

Section: Basic Notions On Conservation Laws and Potential Symmetrymentioning

confidence: 99%

Potential systems for PDEs having several conservation laws

Иванова

2009

J Eng Math

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A generalization of the usual procedure for constructing potential systems for systems of partial differential equations with multidimensional spaces of conservation laws is considered. More precisely, for the construction of potential systems with a multi-dimensional space of local conservation laws, instead of using only basis conservation laws, their arbitrary linear combinations are used that are inequivalent with respect to the equivalence group of the class of systems or symmetry group of the fixed system. It appears that the basis conservation laws can be equivalent with respect to groups of symmetry or equivalence transformations, or vice versa; in this sense the number of independent linear combinations of conservation laws can be grater than the dimension of the space of conservation laws. The first possibility leads to an unnecessary, often cumbersome, investigation of equivalent systems, the second one makes possible that a great number of inequivalent potential systems are missed. Examples of all these possibilities are given.

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Section: Basic Notions On Conservation Laws and Potential Symmetrymentioning

confidence: 99%

Potential systems for PDEs having several conservation laws

Иванова

2009

J Eng Math

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“…According to the general principles of the algebraic geometric approach (group analysis) [1], [3], [5], [6], (infinitesimal) symmetries of a given evolution system ∂ t u = f are evolution derivations commuting with the full evolution operator Ev f . Specifically, we say that a function g ∈ F defines a symmetry of the evolution system ∂ t u = f if the commutator [Ev f , ev g ] = 0.…”

Section: Symmetriesmentioning

confidence: 99%

Evolution systems on a lattice

Zharinov

2008

Theor Math Phys

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In the framework of the algebraic geometric approach to differential-difference equations, we study symmetries and conservation laws of evolutionary systems on multidimensional lattices. We describe conservation laws in terms of their characteristics.We here extend results previously obtained in [1] for evolution systems in the Euclidean space to evolution systems on multidimensional integer lattices. For this, we adapt the basic principles of the algebraic geometric (or group) analysis of partial differential equations (see, e.g., [2]-[6]) to the discrete case. We introduce the concept of the difference jet, we study functions on the space of difference jets and operations defined on these functions, and we introduce some elements of the calculus of variations on a lattice. We use the developed formalism to describe conservation laws of evolution systems on multidimensional lattices in terms of their characteristics, which are defined in the same way as for evolution systems in the Euclidean space.In what follows, we assume that all linear operations are over the field R and that all infinite-dimensional vector spaces carry the natural locally convex topology. All linear maps that we introduce are continuous, but we do not prove this explicitly. We use the following standard notation: N = {1, 2, . . . }, Z + = {0, 1, 2, . . . }, and Z = {0, ±1, ±2, . . . }. We let Map(X; Y ) denote the set of all maps from the set X to the set Y . 1. The jet bundle over a lattice 1.1. The lattice. Let D ∈ N. A lattice (more precisely, a D-dimensional lattice) is the set L = Z D . There are two algebraic structures defined on a lattice L: the structure of an Abelian group with respect to addition and the structure of a Z-module. As a Z-module, L has a basis {e 1 , . . . , e D } ⊂ L, where e 1 = (1, 0, . . . , 0), . . . , e D = (0, . . . , 0, 1), and hence L m = (µ 1 , . . . , µ D ) = α µ α e α , µ α ∈ Z, 1 ≤ α ≤ D.For m = (µ 1 , . . . , µ D ) ∈ L, we set |m| = α |µ α | ∈ Z + . For brevity, we write {|m| ≤ p} instead of {m ∈ L; |m| ≤ p}, where p ∈ Z + .The action of the group L is defined on the lattice L, L n → T n ∈ Map(L; L), m → T n m = m + n.The maps T n are called translations.

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“…Although the above definitions are suitable for the first intuitive illustration of notion of conservation laws, to obtain complete and correct understanding we should use a more rigorous definition of conservation laws (see, e.g., [32,40]) namely, for any system L of differential equations the set CV(L) of its conserved vectors is a linear space and the subset CV 0 (L) of trivial conserved vectors is a linear subspace That is why we assume the description of the set of conservation laws as finding CL(L) that is equivalent to construction of either a basis if dim CL(L) < ∞ or a system of generatrices in the infinitedimensional case. The elements of CV(L) that belong to the same equivalence class giving a conservation law F are all considered as conserved vectors of this conservation law and we additionally identify elements from CL(L) with their representatives in CV(L).…”

Section: Conservation Lawsmentioning

confidence: 99%

Lie group analysis of two-dimensional variable-coefficient Burgers equation

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Sophocleous

Tracinà

2010

Z. Angew. Math. Phys.

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The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.

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Conservation laws of evolution systems

Cited by 35 publications

References 7 publications

Potential systems for PDEs having several conservation laws

Potential systems for PDEs having several conservation laws

Evolution systems on a lattice

Lie group analysis of two-dimensional variable-coefficient Burgers equation

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