Splitting PDE parameterized symmetries

and Parameter-continuous symmetry transformations

The determination of symmetries for partial differential equation systems (PDE) is relevant in several contexts, the most obvious of which is of course the determination of the PDE solutions. For instance, generally speaking, the knowledge of a N-dimensional Lie symmetry group can be used to reduce the number of independent variables of PDE by N. So if PDE depends only on N independent variables, that amounts to completely solving it. If only N-1 symmetries are known or can be successfully used then PDE becomes and ODE; etc., all advantageous situations. In Maple, a complete set of symmetry commands, to perform each step of the symmetry approach or several of them in one go, is part of the PDEtools  package.

Besides the dependent and independent variables, PDE frequently depends on some constant parameters, and besides the PDE symmetries for arbitrary values of those parameters, for some particular values of them, PDE transforms into a completely different problem, admitting different symmetries. The question then is: how can you determine those particular values of the parameters and the corresponding different symmetries? That was the underlying subject of a recent question in Mapleprimes. The answer to those questions is relatively simple and yet not entirely obvious for most of us, motivating this post, organized briefly around one example.

To reproduce the input/output below you need Maple 2019 and to have installed the Physics Updates v.449 or higher.

Consider the family of Korteweg-de Vries equation for involving three constant parameters . For convenience (simpler input and more readable output) use the diff_table  and declare  commands

This pde admits a 4-dimensional symmetry group, whose infinitesimals - for arbitrary values of the parameters - are given by

Looking at pde (1) as a nonlinear problem in u, a, b and q, it splits into four cases for some particular values of the parameter:

The legend above indicates the pivots and the tree of cases, depending on whether each pivot is equal or different from 0. At the end there is the algebraic sequence of cases. The first case is the general case, for which the symmetry infinitesimals were computed as  above, but clearly the other three cases admit more general symmetries. Consider for instance the second case, pass the  to ignore the parameterizing equation , and you get

These infinitesimals are indeed much more general than , in fact so general that (5) is almost unreadable ... Specialize the three arbitrary functions into something "easy" just to be able follow - e.g. take _F1 to be just the + operator, _F2 the * operator and

This symmetry is of course completely different than computed for the general case.

The symmetry (7) can be verified against  or directly against pde after substituting .

Summarizing: "to split PDE symmetries into cases according to the values of the PDE parameters, split the PDE into cases with respect to these parameters (command PDEtools:-casesplit ) then compute the symmetries for each case"

Parameter continuous symmetry transformations

A different, however closely related question, is whether pde admits "symmetries with respect to the parameters a, b and q", so whether exists continuous transformations of the parameters a, b and q that leave pde invariant in form.

Beforehand, note that since the parameters are constants with regards to the dependent and independent variables (here ), such continuous symmetry transformations cannot be used directly to compute a solution for pde. They can, however, be used to reduce the number of parameters. And in some contexts, that is exactly what we need, for example to entirely remove the splitting into cases due to their presence, or to proceed applying a solving method that is valid only when there are no parameters (frequently the case when computing exact solutions to "PDE & Boundary Conditions").

To compute such "continuous symmetry transformations of the parameters" that leave pde invariant one can always think of these parameters as "additional independent variables of pde". In terms of formulation, that amounts to replacing the dependency in the dependent variable, i.e. replace  by

Compute now the infinitesimals: note there are now three additional ones, related to continuous transformations of and  - for readability, avoid displaying the redundant functionality  on the left-hand sides of these infinitesimals

This result is more general than what is convenient for algebraic manipulations, so specialize the seven arbitrary functions of  and keep only the first symmetry that result from this specialization: that suffices to illustrate the removal of any of the three parameters a, b, or q

To remove the parameters, as it is standard in the symmetry approach, compute a transformation to canonical coordinates, with respect to the parameter a. That means a transformation that changes the list of infinitesimals, or likewise its infinitesimal generator representation,

into  or its equivalent generator representation  

That same transformation, when applied to , entirely removes the parameter a.

The transformation is computed using CanonicalCoordinates and the last argument indicates the "independent variable" (in our case a parameter) that the transformation should remove. We choose to remove a

Invert this transformation in order to apply it

The next step is not necessary, but just to understand how all this works, verify its action over the infinitesimal generator

Now that we see the transformation (17) is the one we want, just use it to change variables in

As expected, this result depends only on two parameters, , and , and the one equivalent to a (that is , see the transformation used (17)), is not present anymore.

To remove b or q we use the same steps, (15), (17) and (19), just changing the parameter to be removed, indicated as the last argument  in the call to CanonicalCoordinates . For example, to eliminate b (represented in the new variables by ), input

and as expected this result does not contain To remove a second parameter, the whole cycle is repeated starting with computing infinitesimals, for instance for (22). Finally, the case of function parameters is treated analogously, by considering the function parameters as additional dependent variables instead of independent ones.