Determining Systems

This page is our central repository for determining systems for PDE and PDE systems.

The focus of this page is to make available several pre-computed determining systems, and some timing results for simplification of these systems through use of differential elimination.

First a few words about symmetry determining systems.

Consider a single PDE having one dependent variable phi, and n + 1 dependent variables $\bf x$ and t (here $\bf x$ = (x₁,..., x_n)). To find the determining system for the Lie symmetry generators of the PDE, we consider the expansion of a general transformation about the identity transformation, or mathematically, a transformation of the form:

($$\phi$$, t,$$\bf x$$) $$\mapsto$$ ($$\hat{\phi}$$,$$\hat{t}$$,$$\hat{\bf x}$$) = ($$\phi$$ + $$\Phi$$$$\epsilon$$ + $$\cal {O}$$($$\epsilon^{2}_{}$$), t + T$$\epsilon$$ + $$\cal {O}$$($$\epsilon^{2}_{}$$),$$\bf x$$ + $$\bf X$$$$\epsilon$$ + $$\cal {O}$$($$\epsilon^{2}_{}$$)),

where $\Phi$,T, and $\bf X$ are all unknown functions of $\phi$,t, and $\bf x$, and are called the infinitesimals.

Application of this transformation to the original PDE, enforcement of the symmetry condition (i.e. that the transformed PDE is of the same form as the original PDE), and retaining only up to the $\cal {O}$($\epsilon$) terms gives us a linear determining system of PDE for the infinitesimals.

It should be noted that although the determining system is linear in the infinitesimals, the original PDE can contain unknowns (or so-called classifying functions), which makes the overall problem of simplification of these equations a nonlinear one. In this way, determination of the form of the symmetries can proceed with an entire class of PDE.

Now to discuss a few details.

All downloadable systems are stored in tarred gzipped archives. Each archive (unless noted otherwise) contains all the systems for that form of the input PDE. The determining system is stored in the files as a Maple list of differential expressions. Once downloaded and unpacked, the files can easily be read into Maple V3-Maple 6.01 using a read command. Once read, the determining system will be in the list U, and can be further manipulated from that point. Any assumptions on expressions that do not vanish are present in the N list, and a ranking, if the default is not being used, is present in the Janet list. Either N or Janet may not be present.

It should be noted that the available systems are in completely unsimplified form. This means that there may be duplicate equations, or equations that easily reduce to zero with respect to others. The SYMMETRY package, written by Dr. Mark Hickman was used to create the defining systems from the original PDE.

The PureElim algorithm running in the DiffElim environment has some similarities to the RifSimp and Standard Form algorithms. The system is brought into simplified form with respect to a ranking of the derivatives. This allows the code to determine the leading derivative of an equation and determine the leading term of that equation. By default PureElim ranks indeterminates in the following way:

1

Rank dependent variables ahead of any unknown constants.

2

Rank by total differential degree of each derivative.

3

Rank by the number of differentiations in each independent variable in alphabetic order.

4

Rank by the dependent variable name in alphabetic order.

For a sample problem involving dependent variables Phi,X, and T, independent variables phi,x, and t, and constant c we would have

Phi_x1 > c by criteria 1,

Phi_{t, t} > Phi_x1 by criteria 2,

Phi_{t, t} > Phi_{t, x1} by criteria 3, and

Phi_{t, t} > X1_{t, t} by criteria 4.

The Janet variable alters the ranking by adding criteria before the default ranking. It is in list form, and can specify any of the above type of criteria. For a sample problem involving dependent variables Phi,X, and T, independent variables phi,x, and t, and constant c it is specified as:

Janet : = [[Phi, X, T],[phi, x, t], phi, x, t, Phi, X, T, c]. (1)

The first list element corresponds to criteria 1, saying that Phi,X, and T are preferred over other dependent variables or constants. The second list element consists of only independent variables, so it is differentiation based, and compares the number of differentiations between the indeterminates with respect to phi,x, and t total (this corresponds to the total differential degree criteria 2). The remaining list elements are easy to interpret as they are the single element instances of the two types of criteria described above.

Now that the extremely brief background is done, the systems and their timings are available through the links below.

Subsections

• The n-dimensional Laplace Equation
• The (n+1)-dimensional Harmonic Oscillator
• The (n+1)-dimensional VNLS system

If you have any questions or suggestions on the content of these pages, please contact either myself at wittkopf@cecm.sfu.ca or Greg Reid at reid@julian.uwo.ca .