Reduction of differential equations and conditional symmetry

Zhdanov, Renat; Tsifra, I. M.

doi:10.1007/bf02384233

Cited by 8 publications

(8 citation statements)

References 6 publications

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“…Based on this definition is our proof of equivalence of the above two approaches to reduction of PDEs provided some reasonable restrictions are met (see, also [30]). The present paper is a natural continuation of our earlier papers [20,31], where some ideas presented below were indicated. We present these ideas in a rigorous mathematical form which, as we believe, should give new insights into the theory of conditional symmetries of PDEs.…”

Section: Introductionmentioning

confidence: 85%

A Precise Definition of Reduction of Partial Differential Equations

Zhdanov

Tsyfra

Popovych

1999

Journal of Mathematical Analysis and Applications

113

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We give a comprehensive analysis of interrelations between the basic concepts of Ž . the modern theory of symmetry classical and non-classical reductions of partial differential equations. Using the introduced definition of reduction of differential Ž . equations we establish equivalence of the non-classical conditional symmetry and Ž . direct Ansatz approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1 q 3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalizations of the well-known symmetry reductions of the nonlinear wave equations.

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Section: Introductionmentioning

confidence: 85%

A Precise Definition of Reduction of Partial Differential Equations

Zhdanov

Tsyfra

Popovych

1999

Journal of Mathematical Analysis and Applications

113

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“…Let us now come back to the original coordinates x, u: we will see that the set of conditions (30) is the result of the following procedure.…”

Section: Symmetric and Invariant Equationsmentioning

confidence: 99%

On the Notion of Conditional Symmetry of Differential Equations

Cicogna

Laino²

2006

Rev. Math. Phys.

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Symmetry properties of PDE's are considered within a systematic and unifying scheme: particular attention is devoted to the notion of conditional symmetry, leading to the distinction and a precise characterization of the notions of "true" and "weak" conditional symmetry. Their relationship with exact and partial symmetries is also discussed. An extensive use of "symmetry-adapted" variables is made; several clarifying examples, including the case of Boussinesq equation, are also provided.

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“…If this is the case, a conditional symmetry allows the reduction of the initial equation into a reduced form, and in this way one can obtain other invariant solutions: see [27,28,29,30,31,32]. See also [35,36,37,38,39,40,41,42,43] for careful discussions about various related problems and reduction procedures.…”

Section: "Weak" Symmetriesmentioning

confidence: 99%

“…Solutions of this form are interesting due to the presence of terms depending on x − T (t) describing generalized wave propagation. It is known that there are some delicate points related to the definition of conditional symmetries: see [35,36,37,38,39,40,41,42]; this is actually related to the introduction of a subtler classification of the notion of conditional symmetry [44,45]. We do not deal here with this problem, and we prefer to consider a particularly interesting example of partial symmetry for our system (8).…”

Section: "Weak" Symmetriesmentioning

confidence: 99%

Applications of Symmetry Methods to the Theory of Plasma Physics

Cicogna¹

2006

SIGMA

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Abstract. The theory of plasma physics offers a number of nontrivial examples of partial differential equations, which can be successfully treated with symmetry methods. We propose three different examples which may illustrate the reciprocal advantage of this "interaction" between plasma physics and symmetry techniques. The examples include, in particular, the complete symmetry analysis of system of two PDE's, with the determination of some conditional and partial symmetries, the construction of group-invariant solutions, and the symmetry classification of a nonlinear PDE.

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Reduction of differential equations and conditional symmetry

Cited by 8 publications

References 6 publications

A Precise Definition of Reduction of Partial Differential Equations

A Precise Definition of Reduction of Partial Differential Equations

On the Notion of Conditional Symmetry of Differential Equations

Applications of Symmetry Methods to the Theory of Plasma Physics

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