On a reduction and solutions of nonlinear wave equations with broken symmetry

Fushchich, W. I.; Tsifra, I. M.

doi:10.1088/0305-4470/20/2/001

Cited by 85 publications

(102 citation statements)

References 4 publications

Supporting

Mentioning

102

Contrasting

Order By: Relevance

“…Conversely, if a ∆(s, z, v (m) ) = 0 is such that a system like (30) admits the symmetry X = ∂/∂s, then condition (9) must be satisfied, and applying once again Lemma 2.1, we see that ∆ must have the form (29).…”

Section: Symmetric and Invariant Equationsmentioning

confidence: 99%

See 1 more Smart Citation

On the Notion of Conditional Symmetry of Differential Equations

Cicogna

Laino²

2006

Rev. Math. Phys.

View full text Add to dashboard Cite

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.

show abstract

Section: Symmetric and Invariant Equationsmentioning

confidence: 99%

“…If X is a weak CS of order σ, the system (39) gives rise to a system of σ reduced equations, which -if admits solutions -produces X−invariant solutions of ∆ = 0. Introducing X−adapted variables s, z, such that Xs = 1, Xz = 0, the PDE has the form (22) in the case of true CS, or (29) in the case of weak CS.…”

Section: Symmetric and Invariant Equationsmentioning

confidence: 99%

On the Notion of Conditional Symmetry of Differential Equations

Cicogna

Laino²

2006

Rev. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…The above system (8) admits also several interesting "non-exact" (here generically called "weak") symmetries; we are going to consider an example of conditional symmetry (see [27,28,29,30,31,32]), and one of partial symmetry (according to the definition given in [33], see also [34] and below). As well known, any vector field X = ξ i ∂/∂x i + ζ a ∂/∂u a is a conditional symmetry for a differential equation (or a system thereof) ∆ = 0 if the system ∆ = 0 enlarged with the invariance condition (13) admits some solution.…”

Section: "Weak" Symmetriesmentioning

confidence: 99%

“…As well known, any vector field X = ξ i ∂/∂x i + ζ a ∂/∂u a is a conditional symmetry for a differential equation (or a system thereof) ∆ = 0 if the system ∆ = 0 enlarged with the invariance condition (13) admits some solution. 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%

Applications of Symmetry Methods to the Theory of Plasma Physics

Cicogna¹

2006

SIGMA

View full text Add to dashboard Cite

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.

show abstract

“…The first paper devoted to nonclassical symmetry of partial differential equations was published by Bluman and Cole [57]. However, the real importance of these symmetries was understood much later after the explanations given in several papers [31,32,[58][59][60][61] where the method of conditional symmetries had been used in order to construct new exact solutions of a number of nonlinear partial differential equations.…”

Section: Conditional Symmetry and New Solutions Of The Yang-millsmentioning

confidence: 99%

Symmetry and Exact Solutions of the Maxwell and SU(2) Yang–Mills Equations

Zhdanov

Lahno

Modern Nonlinear Optics, Part II

View full text Add to dashboard Cite

We give the overview of solution techniques for the general conformally-invariant linear and nonlinear wave equations centered around the idea of dimensional reductions by their symmetry groups. The efficiency of these techniques is demonstrated on the examples of the SU (2) Yang-Mills and the vacuum Maxwell equations. For the Yang-Mills equations we have derived the most general form of the conformally-invariant solution and construct a number of their new analytical non-Abelian solutions in explicit form. We have completely solved the problem of symmetry reduction of the Maxwell equations by subgroups of the conformal group. This yields twelve multi-parameter families of their exact solutions, a majority of which are new and might be of considerable interest for applications.Plan.

show abstract

On a reduction and solutions of nonlinear wave equations with broken symmetry

Cited by 85 publications

References 4 publications

On the Notion of Conditional Symmetry of Differential Equations

On the Notion of Conditional Symmetry of Differential Equations

Applications of Symmetry Methods to the Theory of Plasma Physics

Symmetry and Exact Solutions of the Maxwell and SU(2) Yang–Mills Equations

Contact Info

Product

Resources

About