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

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.

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

$S≔\left[\mathrm{\_\ξ}\left[x\right]=x\,\mathrm{\_\ξ}\left[t\right]=1\,\mathrm{\_\η}\left[u\right]=u\right]$

$S≔\left[{\mathrm{\_\ξ}}_x=x\,{\mathrm{\_\ξ}}_t=1\,{\mathrm{\_\η}}_u=u\right]$

$G≔\mathrm{InfinitesimalGenerator}\left(S\,u\left(x\,t\right)\right)$

$G≔f\→x\left(\frac{\partial }{\partial x}f\right)\+\frac{\partial }{\partial t}f\+u\left(\frac{\partial }{\partial u}f\right)$

$\mathrm{PDE}≔\mathrm{InvariantEquation}\left(S\,u\left(x\,t\right)\,\mathrm{arbitraryfunctionname}=\mathrm{\Lambda }\right)$

$\mathrm{PDE}≔\mathrm{\Lambda }\left(-\ln \left(x\right)+t\,\frac{u\left(x\,t\right)}{x}\,\frac{\partial }{\partial x}u\left(x\,t\right)\,\frac{\frac{\partial }{\partial t}u\left(x\,t\right)}{x}\right)$

$\mathrm{InvariantEquation}\left(S\,u\left(x\,t\right)\,\mathrm{explicit}\right)$

$\left[\frac{\partial }{\partial x}u\left(x\,t\right)=\mathrm{f\_\_1}\left(-\ln \left(x\right)+t\,\frac{u\left(x\,t\right)}{x}\right)\,\frac{\partial }{\partial t}u\left(x\,t\right)=\mathrm{f\_\_2}\left(-\ln \left(x\right)+t\,\frac{u\left(x\,t\right)}{x}\right)x\right]$

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

$\left\{0\right\}$

$\mathrm{itr}≔\mathrm{SymmetryTransformation}\left(G\,u\left(x\,t\right)\,v\left(r\,s\right)\right)$

$\mathrm{itr}≔\left\{r=x{\ⅇ}^{\mathrm{\_\ε}}\,s=\mathrm{\_\ε}+t\,v\left(r\,s\right)={\ⅇ}^{\mathrm{\_\ε}}u\left(x\,t\right)\right\}$

$\mathrm{tr}≔\mathrm{solve}\left(\mathrm{itr}\,\left\{t\,x\,u\left(x\,t\right)\right\}\right)$

$\mathrm{tr}≔\left\{t=s-\mathrm{\_\ε}\,x=\frac{r}{{\ⅇ}^{\mathrm{\_\ε}}}\,u\left(x\,t\right)=\frac{v\left(r\,s\right)}{{\ⅇ}^{\mathrm{\_\ε}}}\right\}$

$\mathrm{dchange}\left(\mathrm{tr}\,\mathrm{PDE}\,\mathrm{known}=\mathrm{\Lambda }\,\left[r\,s\,v\left(r\,s\right)\right]\right)$

$\mathrm{\Lambda }\left(-\ln \left(\frac{r}{{\ⅇ}^{\mathrm{\_\ε}}}\right)+s-\mathrm{\_\ε}\,\frac{v\left(r\,s\right)}{r}\,\frac{\partial }{\partial r}v\left(r\,s\right)\,\frac{\frac{\partial }{\partial s}v\left(r\,s\right)}{r}\right)$

$\mathrm{simplify}\left(\right)\mathbf{assuming}\mathrm{\_\ε}::\mathrm{real}$

$\mathrm{\Lambda }\left(-\ln \left(r\right)+s\,\frac{v\left(r\,s\right)}{r}\,\frac{\partial }{\partial r}v\left(r\,s\right)\,\frac{\frac{\partial }{\partial s}v\left(r\,s\right)}{r}\right)$

$G≔\mathrm{InfinitesimalGenerator}\left(G\,u\left(x\,t\right)\,\mathrm{prolongation}=1\,\mathrm{expanded}\right)$

$G≔f\→x\left(\frac{\partial }{\partial x}f\right)\+\frac{\partial }{\partial t}f\+u\left(\frac{\partial }{\partial u}f\right)\+u_t\left(\frac{\partial }{\partial u_t}f\right)$

$\mathrm{jetPDE}≔\mathrm{ToJet}\left(\mathrm{PDE}\,u\left(x\,t\right)\right)$

$\mathrm{jetPDE}≔\mathrm{\Lambda }\left(-\ln \left(x\right)+t\,\frac{u}{x}\,u_x\,\frac{u_t}{x}\right)$

$\mathrm{normal}\left(G\left(\mathrm{jetPDE}\right)\right)$

$0$

$\mathrm{Invariants}\left(G\,u\left(x\,t\right)\right)$

$\frac{u}{x},-\ln \left(x\right)+t,u_x,\frac{u_t}{x}$

$\mathrm{dynamical\_S}≔\left[\mathrm{\_\ξ}\left[x\right]=1\,\mathrm{\_\ξ}\left[t\right]=0\,\mathrm{\_\η}\left[u\right]=\frac{1}{u\left[x\right]}\right]$

$\mathrm{dynamical\_S}≔\left[{\mathrm{\_\ξ}}_x=1\,{\mathrm{\_\ξ}}_t=0\,{\mathrm{\_\η}}_u=\frac{1}{u_x}\right]$

$\mathrm{Invariants}\left(\mathrm{dynamical\_S}\,u\left(x\,t\right)\,\mathrm{order}=1\right)$

$t,u_x^3+3x{u}_{x,x},\frac{2u{u}_{x,x}+u_x^2}{2u_{x,x}},\frac{u_t{u}_{x,x}-u_x{u}_{x,t}}{u_{x,x}}$

$\mathrm{InvariantEquation}\left(\mathrm{dynamical\_S}\,u\left(x\,t\right)\,\mathrm{order}=2\,\mathrm{name}=\mathrm{\Lambda }\right)=0$

$\mathrm{*\; Partial\; match\; of\; \text{'}name\text{'}\; against\; keyword\; \text{'}arbitraryfunctionname\text{'}}$

$\mathrm{\Lambda }\left(t\,\frac{\partial }{\partial x}u\left(x\,t\right)\,\frac{u\left(x\,t\right){\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)}^2-2x\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)+u\left(x\,t\right)}{{\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)}^2+1}\,\frac{\partial }{\partial t}u\left(x\,t\right)\,\frac{{\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)}^3+2\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right)x+\frac{\partial }{\partial x}u\left(x\,t\right)}{\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right)\left({\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)}^2+1\right)}\,\frac{\left({\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)}^2+1\right)\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)\left(\frac{{\partial }^2}{\partial t\partial x}u\left(x\,t\right)\right)}{\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)}\,\frac{\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right)\left(\frac{{\partial }^2}{\partial t^2}u\left(x\,t\right)\right)-{\left(\frac{{\partial }^2}{\partial t\partial x}u\left(x\,t\right)\right)}^2}{\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)}\right)=0$

$\mathrm{declare}\left(u\left(x\,t\right)\right)$

$u\left(x\,t\right)\mathrm{will\; now\; be\; displayed\; as}u$

$\mathrm{InvariantEquation}\left(\mathrm{dynamical\_S}\,u\left(x\,t\right)\,\mathrm{order}=2\,\mathrm{explicit}\right)$

$\left[u_{x,x}=\frac{u_x\left({u_x}^2+1\right)}{{u_x}^3\mathrm{f\_\_1}\left(t\,u_x\,\frac{u{u_x}^2-2x{u}_x+u}{{u_x}^2+1}\,u_t\right)+u_x\mathrm{f\_\_1}\left(t\,u_x\,\frac{u{u_x}^2-2x{u}_x+u}{{u_x}^2+1}\,u_t\right)-2x}\,u_{t,x}=\frac{\mathrm{f\_\_2}\left(t\,u_x\,u_t\,\frac{u{u_x}^2-2x{u}_x+u}{{u_x}^2+1}\right)}{{u_x}^3\mathrm{f\_\_1}\left(t\,u_x\,\frac{u{u_x}^2-2x{u}_x+u}{{u_x}^2+1}\,u_t\right)+u_x\mathrm{f\_\_1}\left(t\,u_x\,\frac{u{u_x}^2-2x{u}_x+u}{{u_x}^2+1}\,u_t\right)-2x}\,u_{t,t}=-\frac{{\mathrm{f\_\_2}\left(t\,u_x\,u_t\,\frac{u{u_x}^2-2x{u}_x+u}{{u_x}^2+1}\right)}^2}{-{u_x}^2{\left({u_x}^2+1\right)}^2\mathrm{f\_\_1}\left(t\,u_x\,\frac{u{u_x}^2-2x{u}_x+u}{{u_x}^2+1}\,u_t\right)+2u_{x}x\left({u_x}^2+1\right)}+\mathrm{f\_\_3}\left(t\,u_x\,\frac{u{u_x}^2-2x{u}_x+u}{{u_x}^2+1}\,u_t\right)\right]$

The explicit option was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.