Automorphisms of real Lie algebras of dimension five or less

The Lie algebra version of the Krull-Schmidt Theorem is formulated and proved. This leads to a method for constructing the automorphisms of a direct sum of Lie algebras from the automorphisms of its indecomposable components. For finite-dimensional Lie algebras, there is a well-known algorithm for finding such components, so the theorem considerably simplifies the problem of classifying the automorphism groups. We illustrate this by classifying the automorphisms of all indecomposable real Lie algebras of dimension five or less. Our results are presented very concisely, in tabular form

Moving frames: difference and differential-difference Lagrangians

This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the equivariant formulation of the conservation laws arising from Noether's Theorem. The differential-difference theory is not merely an amalgam of the differential and difference theories, but has additional features that reflect the need for the group action to preserve the prolongation structure. Projectable moving frames are introduced; these cause the invariant derivative operator to commute with shifts in the discrete variables. Examples include a Toda-type equation and a method of lines semi-discretization of the Nonlinear Schr\"odinger Equation.Comment: 28 pages, 3 figure

Characteristics of Conservation Laws for Difference Equations

Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether's Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether's Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka-Volterra equation

Transformations, symmetries and Noether theorems for differential-difference equations

The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether’s theorem. We state and prove the differential-difference version of Noether’s second theorem, together with a Noether-type theorem that spans the gap between the analogues of Noether’s two theorems. These results are applied to various equations from physics

Moving frames: difference and differential-difference Lagrangians

This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the equivariant formulation of the conservation laws arising from Noether's Theorem. The differential-difference theory is not merely an amalgam of the differential and difference theories, but has additional features that reflect the need for the group action to preserve the prolongation structure. Projectable moving frames are introduced; these cause the invariant derivative operator to commute with shifts in the discrete variables. Examples include a Toda-type equation and a method of lines semi-discretization of the Nonlinear Schroedinger Equation

A New Technique for Preserving Conservation Laws

This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the literature preserve fully discrete local conservation laws whose densities are either quadratic or a Hamiltonian. The approach generalizes to time integrators with more steps and conservation laws of other kinds; higher-dimensional PDEs can be treated by iterating the new strategy. We use the Boussinesq equation as a benchmark and introduce new families of schemes of order two and four that preserve three conservation laws. We show that the new technique is practicable for PDEs with three dependent variables, introducing as an example new families of second-order schemes for the potential Kadomtsev–Petviashvili equation

Locally conservative finite difference schemes for the Modified KdV equation

Finite diffrence schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity. In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the Modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and co-workers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature

Simple bespoke preservation of two conservation laws

Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic-numeric approach feasible. To illustrate the simplified approach, we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg-de Vries (KdV) equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared to others in the literature

Moving Frames and Noether’s Finite Difference Conservation Laws I

We consider the calculation of Euler--Lagrange systems of ordinary difference equations, including the difference Noether's Theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We introduce the difference moving frame, a natural discrete moving frame that is adapted to difference equations by prolongation conditions. For any Lagrangian that is invariant under a Lie group action on the space of dependent variables, we show that the Euler-Lagrange equations can be calculated directly in terms of the invariants of the group action. Furthermore, Noether's conservation laws can be written in terms of a difference moving frame and the invariants. We show that this form of the laws can significantly ease the problem of solving the Euler--Lagrange equations, and we also show how to use a difference frame to integrate Lie group invariant difference equations. In this Part I, we illustrate the theory by applications to Lagrangians invariant under various solvable Lie groups. The theory is also generalized to deal with variational symmetries that do not leave the Lagrangian invariant. Apart from the study of systems that are inherently discrete, one significant application is to obtain geometric (variational) integrators that have finite difference approximations of the continuous conservation laws embedded \textit{a priori}. This is achieved by taking an invariant finite difference Lagrangian in which the discrete invariants have the correct continuum limit to their smooth counterparts. We show the calculations for a discretization of the Lagrangian for Euler's elastica, and compare our discrete solution to that of its smooth continuum limit

Self-Invariant Contact Symmetries

Every smooth second-order scalar ordinary differential equation (ODE) that is solved for the highest derivative has an infinite-dimensional Lie group of contact symmetries. However, symmetries other than point symmetries are generally difficult to find and use. This paper deals with a class of one-parameter Lie groups of contact symmetries that can be found and used. These symmetry groups have a characteristic function that is invariant under the group action; for this reason, they are called ‘self-invariant.’ Once such symmetries have been found, they may be used for reduction of order; a straightforward method to accomplish this is described. For some ODEs with a oneparameter group of point symmetries, it is necessary to use self-invariant contact symmetries before the point symmetries (in order to take advantage of the solvability of the Lie algebra). The techniques presented here are suitable for use in computer algebra packages. They are also applicable to higher-order ODEs