PDEtools - Maple Programming Help

Home : Support : Online Help : Mathematics : Differential Equations : DEtools : Lie Symmetry Method : Commands for PDEs (and ODEs) : PDEtools/InvariantEquation

PDEtools

 InvariantEquation
 computes the differential equation invariant under a given symmetry group

 Calling Sequence InvariantEquation(S, DepVars, 'options'='value')

Parameters

 S - the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator, or a list of n of them representing an n-dimensional group of symmetries DepVars - a function or a list of them indicating the dependent variables of the problem arbitraryfunctionname - (optional) the name used to represent the arbitrary function that appears in the returned invariant equation explicit - (optional) to return the invariant equation(s) with the highest derivative(s) isolated on the left-hand-side(s) order - (optional) the order of the differential equation required. The default value is 1. jetnotation = ... - (optional) can be true (same as jetvariables), false (default), jetvariables, jetvariableswithbrackets, jetnumbers or jetODE; to respectively return or not using the different jet notations available simplifier = ... - (optional) indicates the simplifier to use instead of the default simplify/size simplifier

Description

 • Given a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator, or a list of n of these lists or operators, possibly representing an n-dimensional symmetry group, InvariantEquation returns an expression $\mathrm{\Lambda }\left({\mathrm{\phi }}_{1},{\mathrm{\phi }}_{2},\mathrm{...}\right)$, that is simultaneously invariant under all of the ${G}_{j}$ symmetry generators ($1\le j\le n$) related to the given $n$ infinitesimals. The ${\mathrm{\phi }}_{k}$ form a complete set of $m$ differential invariants, so they are functions of the independent and dependent variables of the problem (DepVars), as well as their partial derivatives up to order (if not indicated, its default value is 1), and satisfy ${G}_{j}\left({\mathrm{\phi }}_{k}\right)=0$. The equation $\mathrm{\Lambda }\left({\mathrm{\phi }}_{1},\mathrm{...},{\mathrm{\phi }}_{m}\right)=0$ is thus the most general partial differential equation simultaneously invariant under all the symmetries in S.
 • InvariantEquation works also with dynamical symmetries, that is, symmetries where the infinitesimals themselves depend on derivatives of the dependent variables of the problem up to (at most) order - 1, where order is the parameter used to indicate the differential order of the desired invariant equation.
 • Note that in the multidimensional case, depending on the form of the infinitesimals given, the problem may have no solution. Also the ${\mathrm{\phi }}_{k}$ entering $\mathrm{\Lambda }\left({\mathrm{\phi }}_{1},{\mathrm{\phi }}_{2},\mathrm{...}\right)$ are all independent of each other and their number is unique, but one can still rewrite the ${\mathrm{\phi }}_{k}$ in different ways, by combining invariants to construct equivalent but algebraically different invariants of the same order.
 • The representation $\mathrm{\Lambda }\left({\mathrm{\phi }}_{1},{\mathrm{\phi }}_{2},\mathrm{...}\right)=0$ for the invariant equation can be transformed into a system of equations by constructing the objects

${\mathrm{\phi }}_{k}=\mathrm{\Lambda }\left({\mathrm{\phi }}_{1},\mathrm{..},{\mathrm{\phi }}_{k-1},{\mathrm{\phi }}_{k+1},\mathrm{..},{\mathrm{\phi }}_{m}\right)$

 so that when the ${\mathrm{\phi }}_{k}$, isolated on the left-hand-side, is linear in the highest derivatives, of differential order equal to the value of order used by InvariantEquation, the resulting system will be an explicit PDE system linear in the highest derivatives.
 • You can request InvariantEquation to directly return a similar explicit PDE system of differential order equal to order by passing the optional argument explicit. In that case the system returned:
 – will contain as many differential equations as the number - say $N$ - of partial derivatives of order equal to order;
 – all the partial derivatives of order = order will appear on the left-hand-sides isolated;
 – the right-hand-sides will involve $N$ arbitrary functions of the form $\mathrm{_Fn}\left(\mathrm{...}\right)$ with $n=1..N$;
 – the right-hand-sides will depend on all the partial derivatives of at most one order less than order, including the unknowns themselves and the independent variables of the problem.
 • This invariant PDE system is returned as an ordered list of equations, where the ordering is dictated by the partial derivatives and is the same implemented in the SortDerivatives command of the PDEtools Library programming routines.
 • The returned invariant equation is automatically simplified in size. To avoid that, pass the optional argument simplifier = none, or to use a different simplifier use simplifier = ....
 • By default, the equation returned by InvariantEquation is in function notation. You can change this by specifying the optional argument jetnotation = ... where the value can be any of the jet notations available.

Examples

 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$

Consider a PDE problem with two independent variables and one dependent variable, $u\left(x,t\right)$, and consider the list of infinitesimals of a symmetry group.

 > $S≔\left[{\mathrm{_ξ}}_{x}=x,{\mathrm{_ξ}}_{t}=1,{\mathrm{_η}}_{u}=u\right]$
 ${S}{≔}\left[{{\mathrm{_ξ}}}_{{x}}{=}{x}{,}{{\mathrm{_ξ}}}_{{t}}{=}{1}{,}{{\mathrm{_η}}}_{{u}}{=}{u}\right]$ (1)

In the input above you can also enter the symmetry $S$ without labels for the infinitesimals, as in $\left[x,1,u\right]$, or use the corresponding infinitesimal generator

 > $G≔\mathrm{InfinitesimalGenerator}\left(S,u\left(x,t\right)\right)$
 ${G}{≔}{f}{→}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{f}\right){+}\frac{{\partial }}{{\partial }{t}}{}{f}{+}{u}{}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right)$ (2)

The equation invariant under the symmetry transformation underlying the infinitesimal generator $S$ is (you can equally use G instead of S)

 > $\mathrm{PDE}≔\mathrm{InvariantEquation}\left(S,u\left(x,t\right),\mathrm{arbitraryfunctionname}=\mathrm{Λ}\right)$
 ${\mathrm{PDE}}{≔}{\mathrm{Λ}}{}\left({-}{\mathrm{ln}}{}\left({x}\right){+}{t}{,}\frac{{u}{}\left({x}{,}{t}\right)}{{x}}{,}\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right){,}\frac{\frac{{\partial }}{{\partial }{t}}{}{u}{}\left({x}{,}{t}\right)}{{x}}\right)$ (3)

This is the explicit form of the PDE system as explained in the Description, in terms of two arbitrary functions $\mathrm{_F1}$ and $\mathrm{_F2}$

 > $\mathrm{InvariantEquation}\left(S,u\left(x,t\right),\mathrm{explicit}\right)$
 $\left[\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right){=}{\mathrm{_F1}}{}\left({-}{\mathrm{ln}}{}\left({x}\right){+}{t}{,}\frac{{u}{}\left({x}{,}{t}\right)}{{x}}\right){,}\frac{{\partial }}{{\partial }{t}}{}{u}{}\left({x}{,}{t}\right){=}{\mathrm{_F2}}{}\left({-}{\mathrm{ln}}{}\left({x}\right){+}{t}{,}\frac{{u}{}\left({x}{,}{t}\right)}{{x}}\right){}{x}\right]$ (4)

The invariance of $\mathrm{PDE}$ can be verified in various ways, the simplest of which is perhaps to use SymmetryTest, which verifies that $S$ is a symmetry of $\mathrm{PDE}$.

 > $\mathrm{SymmetryTest}\left(S,\mathrm{PDE}\right)$
 $\left\{{0}\right\}$ (5)

A less abstract verification would be to explicitly construct the symmetry transformation related to $G$, for example, in terms of new variables $r,s,v\left(r,s\right)$.

 > $\mathrm{itr}≔\mathrm{SymmetryTransformation}\left(G,u\left(x,t\right),v\left(r,s\right)\right)$
 ${\mathrm{itr}}{≔}\left\{{r}{=}{x}{}{{ⅇ}}^{{\mathrm{_ε}}}{,}{s}{=}{\mathrm{_ε}}{+}{t}{,}{v}{}\left({r}{,}{s}\right){=}{{ⅇ}}^{{\mathrm{_ε}}}{}{u}{}\left({x}{,}{t}\right)\right\}$ (6)
 > $\mathrm{tr}≔\mathrm{solve}\left(\mathrm{itr},\left\{x,t,u\left(x,t\right)\right\}\right)$
 ${\mathrm{tr}}{≔}\left\{{t}{=}{s}{-}{\mathrm{_ε}}{,}{x}{=}\frac{{r}}{{{ⅇ}}^{{\mathrm{_ε}}}}{,}{u}{}\left({x}{,}{t}\right){=}\frac{{v}{}\left({r}{,}{s}\right)}{{{ⅇ}}^{{\mathrm{_ε}}}}\right\}$ (7)

Now, change the variables in $\mathrm{PDE}$ using $\mathrm{tr}$, and recall that the Lie group parameter $\mathrm{_ϵ}$ is real:

 > $\mathrm{dchange}\left(\mathrm{tr},\mathrm{PDE},\mathrm{known}=\mathrm{Λ},\left[r,s,v\left(r,s\right)\right]\right)$
 ${\mathrm{Λ}}{}\left({-}{\mathrm{ln}}{}\left(\frac{{r}}{{{ⅇ}}^{{\mathrm{_ε}}}}\right){+}{s}{-}{\mathrm{_ε}}{,}\frac{{v}{}\left({r}{,}{s}\right)}{{r}}{,}\frac{{\partial }}{{\partial }{r}}{}{v}{}\left({r}{,}{s}\right){,}\frac{\frac{{\partial }}{{\partial }{s}}{}{v}{}\left({r}{,}{s}\right)}{{r}}\right)$ (8)
 > $\mathrm{simplify}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{_ε}::\mathrm{real}$
 ${\mathrm{Λ}}{}\left({-}{\mathrm{ln}}{}\left({r}\right){+}{s}{,}\frac{{v}{}\left({r}{,}{s}\right)}{{r}}{,}\frac{{\partial }}{{\partial }{r}}{}{v}{}\left({r}{,}{s}\right){,}\frac{\frac{{\partial }}{{\partial }{s}}{}{v}{}\left({r}{,}{s}\right)}{{r}}\right)$ (9)

The equation above is identical to $\mathrm{PDE}$, that is the meaning of invariant in the context of symmetries. Alternatively, the following is the prolongation to order 1 of the infinitesimal generator $G$ (5.2) (that is, ready to act on functions depending on $x,t,u\left(x,t\right)$ and partial derivatives of $u\left(x,t\right)$ of order 1 at most).

 > $G≔\mathrm{InfinitesimalGenerator}\left(G,u\left(x,t\right),\mathrm{prolongation}=1,\mathrm{expanded}\right)$
 ${G}{≔}{f}{→}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{f}\right){+}\frac{{\partial }}{{\partial }{t}}{}{f}{+}{u}{}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right){+}{{u}}_{{t}}{}\left(\frac{{\partial }}{{\partial }{{u}}_{{t}}}{}{f}\right)$ (10)

If you apply this differential operator $G$ to $\mathrm{PDE}$, you obtain zero because of the invariance of the $\mathrm{PDE}$ under the transformation $\mathrm{tr}$ related to $G$. To apply $G$, an operator in jet notation, you need to rewrite $\mathrm{PDE}$ in the same notation

 > $\mathrm{jetPDE}≔\mathrm{ToJet}\left(\mathrm{PDE},u\left(x,t\right)\right)$
 ${\mathrm{jetPDE}}{≔}{\mathrm{Λ}}{}\left({-}{\mathrm{ln}}{}\left({x}\right){+}{t}{,}\frac{{u}}{{x}}{,}{{u}}_{{x}}{,}\frac{{{u}}_{{t}}}{{x}}\right)$ (11)
 > $\mathrm{normal}\left(G\left(\mathrm{jetPDE}\right)\right)$
 ${0}$ (12)

Underlying this zero, there is the way $\mathrm{PDE}$ is constructed: it is an arbitrary function of the Invariants associated to $G$, that is, the solutions of the differential operator (you can equally pass $S$ instead of $G$)

 > $\mathrm{Invariants}\left(G,u\left(x,t\right)\right)$
 $\frac{{u}}{{x}}{,}{-}{\mathrm{ln}}{}\left({x}\right){+}{t}{,}{{u}}_{{x}}{,}\frac{{{u}}_{{t}}}{{x}}$ (13)

InvariantEquation also handles dynamical symmetries, that is, symmetries that depend on the partial derivatives of the dependent variables of the problem. In these cases it is however of no use to directly compute the invariants, as in the example above, because they will all be of higher order. The approach used is then different. Consider for instance these infinitesimals depending on ${\left(\frac{\partial }{\partial x}u\left(x,t\right)\right)}^{\left(-1\right)}$

 > $\mathrm{dynamical_S}≔\left[{\mathrm{_ξ}}_{x}=1,{\mathrm{_ξ}}_{t}=0,{\mathrm{_η}}_{u}=\frac{1}{{u}_{x}}\right]$
 ${\mathrm{dynamical_S}}{≔}\left[{{\mathrm{_ξ}}}_{{x}}{=}{1}{,}{{\mathrm{_ξ}}}_{{t}}{=}{0}{,}{{\mathrm{_η}}}_{{u}}{=}\frac{{1}}{{{u}}_{{x}}}\right]$ (14)

As mentioned, the differential invariants, for instance, of order 1, automatically come depending on higher derivatives

 > $\mathrm{Invariants}\left(\mathrm{dynamical_S},u\left(x,t\right),\mathrm{order}=1\right)$
 ${t}{,}{{u}}_{{x}}^{{3}}{+}{3}{}{x}{}{{u}}_{{x}{,}{x}}{,}\frac{{1}}{{2}}{}\frac{{2}{}{u}{}{{u}}_{{x}{,}{x}}{+}{{u}}_{{x}}^{{2}}}{{{u}}_{{x}{,}{x}}}{,}\frac{{{u}}_{{t}}{}{{u}}_{{x}{,}{x}}{-}{{u}}_{{x}}{}{{u}}_{{x}{,}{t}}}{{{u}}_{{x}{,}{x}}}$ (15)

In fact, for order = n, and besides the invariant of order 0, all the other ones will always be of order $n+1$. To see the actual invariants of order 1 you would need to eliminate the higher derivatives using the related invariant equation, thus defeating the use of these invariants with the purposes of constructing invariant equations of order $n$. The approach used by InvariantEquation results instead in

 > $\mathrm{InvariantEquation}\left(\mathrm{dynamical_S},u\left(x,t\right),\mathrm{order}=2,\mathrm{name}=\mathrm{Λ}\right)=0$
 ${\mathrm{* Partial match of \text{'}name\text{'} against keyword \text{'}arbitraryfunctionname\text{'}}}$
 ${\mathrm{Λ}}{}\left({t}{,}\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right){,}\frac{{u}{}\left({x}{,}{t}\right){}{\left(\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)\right)}^{{2}}{-}{2}{}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)\right){+}{u}{}\left({x}{,}{t}\right)}{{\left(\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)\right)}^{{2}}{+}{1}}{,}\frac{{\partial }}{{\partial }{t}}{}{u}{}\left({x}{,}{t}\right){,}\frac{{\left(\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)\right)}^{{3}}{+}{2}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{u}{}\left({x}{,}{t}\right)\right){}{x}{+}\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)}{\left(\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{u}{}\left({x}{,}{t}\right)\right){}\left({\left(\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)\right)}^{{2}}{+}{1}\right)}{,}\frac{\left({\left(\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)\right)}^{{2}}{+}{1}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{u}{}\left({x}{,}{t}\right)\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{t}}{}{u}{}\left({x}{,}{t}\right)\right)}{\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{u}{}\left({x}{,}{t}\right)}{,}\frac{{\left(\frac{{{\partial }}^{{2}}}{{\partial }{x}{}{\partial }{t}}{}{u}{}\left({x}{,}{t}\right)\right)}^{{2}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{u}{}\left({x}{,}{t}\right)\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{t}}^{{2}}}{}{u}{}\left({x}{,}{t}\right)\right)}{\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{u}{}\left({x}{,}{t}\right)}\right){=}{0}$ (16)

Note the ${\mathrm{\phi }}_{j}$ depend on first and second derivatives of $u\left(x,t\right)$, as it should be. In explicit form (use declare to have a more compact and readable output with derivatives displayed in jet notation)

 > $\mathrm{declare}\left(u\left(x,t\right)\right)$
 ${u}{}\left({x}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{u}$ (17)
 > $\mathrm{InvariantEquation}\left(\mathrm{dynamical_S},u\left(x,t\right),\mathrm{order}=2,\mathrm{explicit}\right)$
 $\left[{{u}}_{{x}{,}{x}}{=}\frac{{{u}}_{{x}}{}\left({{u}}_{{x}}^{{2}}{+}{1}\right)}{{{u}}_{{x}}^{{3}}{}{\mathrm{_F1}}{}\left({t}{,}{{u}}_{{x}}{,}\frac{{u}{}{{u}}_{{x}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}}{+}{u}}{{{u}}_{{x}}^{{2}}{+}{1}}{,}{{u}}_{{t}}\right){+}{{u}}_{{x}}{}{\mathrm{_F1}}{}\left({t}{,}{{u}}_{{x}}{,}\frac{{u}{}{{u}}_{{x}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}}{+}{u}}{{{u}}_{{x}}^{{2}}{+}{1}}{,}{{u}}_{{t}}\right){-}{2}{}{x}}{,}{{u}}_{{t}{,}{x}}{=}\frac{{\mathrm{_F2}}{}\left({t}{,}{{u}}_{{x}}{,}{{u}}_{{t}}{,}\frac{{u}{}{{u}}_{{x}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}}{+}{u}}{{{u}}_{{x}}^{{2}}{+}{1}}\right)}{{{u}}_{{x}}^{{3}}{}{\mathrm{_F1}}{}\left({t}{,}{{u}}_{{x}}{,}\frac{{u}{}{{u}}_{{x}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}}{+}{u}}{{{u}}_{{x}}^{{2}}{+}{1}}{,}{{u}}_{{t}}\right){+}{{u}}_{{x}}{}{\mathrm{_F1}}{}\left({t}{,}{{u}}_{{x}}{,}\frac{{u}{}{{u}}_{{x}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}}{+}{u}}{{{u}}_{{x}}^{{2}}{+}{1}}{,}{{u}}_{{t}}\right){-}{2}{}{x}}{,}{{u}}_{{t}{,}{t}}{=}\frac{{{\mathrm{_F2}}{}\left({t}{,}{{u}}_{{x}}{,}{{u}}_{{t}}{,}\frac{{u}{}{{u}}_{{x}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}}{+}{u}}{{{u}}_{{x}}^{{2}}{+}{1}}\right)}^{{2}}}{{{u}}_{{x}}{}\left({{u}}_{{x}}^{{2}}{+}{1}\right){}\left(\left({{u}}_{{x}}^{{3}}{+}{{u}}_{{x}}\right){}{\mathrm{_F1}}{}\left({t}{,}{{u}}_{{x}}{,}\frac{{u}{}{{u}}_{{x}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}}{+}{u}}{{{u}}_{{x}}^{{2}}{+}{1}}{,}{{u}}_{{t}}\right){-}{2}{}{x}\right)}{+}{\mathrm{_F3}}{}\left({t}{,}{{u}}_{{x}}{,}\frac{{u}{}{{u}}_{{x}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}}{+}{u}}{{{u}}_{{x}}^{{2}}{+}{1}}{,}{{u}}_{{t}}\right)\right]$ (18)

where the PDE system returned contains as many differential equations as partial derivatives of order 2, that is, 3 equations, and the same number of arbitrary functions $\mathrm{_Fn}$.

Compatibility

 • The explicit option was introduced in Maple 15.