scholarly journals GROUP FOLIATION OF DIFFERENTIAL EQUATIONS USING MOVING FRAMES

2015 ◽  
Vol 3 ◽  
Author(s):  
ROBERT THOMPSON ◽  
FRANCIS VALIQUETTE
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Moving Frames ◽  
Equivariant Moving Frames ◽  
Group Foliation

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.

A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

1993 ◽  
Vol 48 (4) ◽  
pp. 535-550 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Classification Problem ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Point Symmetry Group ◽  
Relativistic Case ◽  
Lie Point Symmetry

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

Approximately preserving symmetries in the numerical integration of ordinary differential equations

1999 ◽  
Vol 10 (5) ◽  
pp. 419-445 ◽  
Author(s):  
ARIEH ISERLES ◽  
ROBERT McLACHLAN ◽  
ANTONELLA ZANNA
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Symmetry Group ◽  
Numerical Integration ◽  
Adjoint Method ◽  
General Procedure ◽  
Original Method ◽  
Time Step ◽  
Numerical Integrator ◽  
Solution Sequence

We present a general procedure for recursively improving the invariance of a numerical integrator under a symmetry group. If h is a symmetry, we construct the adjoint method h−1h. In each time step we apply either the original method or the adjoint method, according to a prescription based on the Thue–Morse sequence. The outcome is a solution sequence which displays progressively smaller symmetry errors, to any desired order in the time-step. The method can also be used to force the solution to stay close to a desired submanifold of phase space, while retaining structural properties of the original method.

Symmetry Reductions and Group-Invariant Radial Solutions to the n-Dimensional Wave Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Wei Feng ◽  
Songlin Zhao
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Symmetry Group ◽  
Group Method ◽  
Radial Solutions ◽  
Radial Wave ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Wave ◽  
Radial Wave Equation

AbstractIn this paper, we derive explicit group-invariant radial solutions to a class of wave equation via symmetry group method. The optimal systems of one-dimensional subalgebras for the corresponding radial wave equation are presented in terms of the known point symmetries. The reductions of the radial wave equation into second-order ordinary differential equations (ODEs) with respect to each symmetry in the optimal systems are shown. Then we solve the corresponding reduced ODEs explicitly in order to write out the group-invariant radial solutions for the wave equation. Finally, several analytical behaviours and smoothness of the resulting solutions are discussed.

Differential Invariants of the (2+1)-Dimensional Breaking Soliton Equation

2016 ◽  
Vol 71 (9) ◽  
pp. 855-862
Author(s):  
Zhong Han ◽  
Yong Chen
Keyword(s):  
Recurrence Relations ◽  
Differential Invariant ◽  
Lie Symmetry ◽  
Differential Invariants ◽  
Moving Frame ◽  
Moving Frames ◽  
Soliton Equation ◽  
Invariant Algebra ◽  
Equivariant Moving Frames

AbstractWe construct the differential invariants of Lie symmetry pseudogroups of the (2+1)-dimensional breaking soliton equation and analyze the structure of the induced differential invariant algebra. Their syzygies and recurrence relations are classified. In addition, a moving frame and the invariantization of the breaking soliton equation are also presented. The algorithms are based on the method of equivariant moving frames.

Diverging type-D metrics

1977 ◽  
Vol 355 (1680) ◽  
pp. 31-52 ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Symmetry Group ◽  
Two Dimensional ◽  
Field Equations ◽  
Killing Vectors ◽  
Type D

This paper contains an investigation of algebraically special spaces with two commuting Killing vectors. It is shown that the field equations for these spaces can be reduced to two ordinary differential equations, one of which is quasi-linear in one of the variables. The metric is type D iff it possesses a two dimensional, abelian, orthogonally transitive symmetry group. Finally, the type D metrics of Kinnersley are expressed in various coordinates, including those of Plebanski and Demianski.

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