Exterior Differential Systems with Symmetry

Anderson, Ian; Fels, Mark E.

doi:10.1007/s10440-005-1136-y

Cited by 35 publications

(213 citation statements)

References 12 publications

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“…In a different direction, the formulation of group foliations of nonlinear PDEs by using exterior differential systems has been studied in Ref. [19,20].…”

Section: Methods Of Group Foliationmentioning

confidence: 99%

Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables

Anco

Feng

Wolf

2015

Journal of Mathematical Analysis and Applications

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Abstract. Explicit solutions are obtained for a class of semilinear radial Schrödinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new solutions that are not invariant under any symmetries of this class of equations. Many of the solutions have interesting analytical behavior connected with blow-up and dispersion. Several interesting nonlinearity powers arise in these solutions, including the case of the critical (pseudo-conformal) power. In contrast, standard symmetry reduction methods lead to nonlinear ordinary differential equations for which few if any explicit solutions can be derived by standard integration methods.

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“…In a different direction, the formulation of group foliations of nonlinear PDEs by using exterior differential systems has been studied in Ref. [19,20].…”

Section: Methods Of Group Foliationmentioning

confidence: 99%

Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables

Anco

Feng

Wolf

2015

Journal of Mathematical Analysis and Applications

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“…In view of this Lemma, in fact, one readily gets the following Proposition 3 Let E be defined by (1), Ω be the volume form (9) and ω i 's defined by (7). The vector fields {Y 1 , ..., Y k } on E determine a solvable structure for D =< D x > iff the following two conditions are satisfied:…”

Section: New Applications Of Solvable Structures To the Integration Omentioning

confidence: 99%

“…In this case, one cannot completely integrate D, but just reduce its codimension by one. This partial reduction is a particular case of that described in the paper [1] and in terms of the forms ω i 's can be described as follows. Since Y 1 is a symmetry of D, then it is also a symmetry of the Pfaffian system I generated by ω 1 , ..., ω n−1 .…”

Section: Solvable Structuresmentioning

confidence: 99%

Local and nonlocal solvable structures in the reduction of ODEs

Ferraioli¹,

Morando²

2008

J. Phys. A: Math. Theor.

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Abstract.Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure.

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“…Later work of Johnson, Ovsiannikov, and others [9,19,47] showed renewed interest. More recently, group foliation has been used to study equations of mathematical physics [30,34], and reformulated using the language of exterior differential systems [3], demonstrating potential for further development and application.…”

Section: Introductionmentioning

confidence: 99%

“…Central to this adaptation is the introduction of the pseudogroup jet differential expressions which, after pull-back by a moving frame, generalize Cartan's structure equations of a moving frame for Lie group actions [14], and play the role of Mansfield's 'curvature matrix' equation in the reconstruction process. The reconstruction step is also related to the reconstruction procedure appearing in symmetry reduction of exterior differential systems [2,3,48].…”

Section: Introductionmentioning

confidence: 99%

Group Foliation of Differential Equations Using Moving Frames

Thompson

Valiquette

2015

Forum math. Sigma

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We incorporate the new theory of equivariant moving frames for Lie pseudogroups into Vessiot's method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudogroup jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and noninvariant solutions of differential equations admitting a symmetry group.2010 Mathematics Subject Classification: 35B06, 53A55 (primary); 35C05, 58H05 (secondary)

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Exterior Differential Systems with Symmetry

Abstract: We use the theory of reduction of exterior differential systems with symmetry to study the problem of using a symmetry group of a differential equation to find noninvariant solutions. (2000): 58A15, 34A26. Mathematics Subject Classifications

Cited by 35 publications

References 12 publications

Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables

Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables

Local and nonlocal solvable structures in the reduction of ODEs

Group Foliation of Differential Equations Using Moving Frames

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