The default output of maxdimsystems (modulo *** below) consists of the set of cases having maximal dimension in maxdimvars, where each member has the form:

Solved is a list of the PDE which are in solved form with respect to their highest derivative (with respect to the given ranking of derivatives), Constraint is a list of the PDEs in the classvars (any dependent variables not in maxdimvars) which are nonlinear in their leading derivatives, Pivots is a list of the inequations in the classvars and/or maxdimvars, and dimension is the maximal degree of arbitrariness in maxdimvars (i.e. the number of arbitrary constants on which maxdimvars depends).

*** The result is valid provided that the Constraint and Pivots equations have a solution, and also that maxdimvars do not occur in the constraint equations. The user must check the first condition algebraically (e.g. see Groebner[Solve], and possibly DifferentialAlgebra). If the maxdimvars are present in the pivots, then the returned system will have lower dimension that indicated, and must be handled manually.

Each of the cases is in rifsimp form (a generalization of Gauss normal form to systems of PDE) and also contains a system of differential equations and inequations in the classvars characterizing the case.  In the same way that the Gauss normal form is determined by a ordering of variables, rifsimp form is determined by a ranking of derivatives.

The default ranking is determined by maxdimvars. It can be inferred by the application of checkrank, and always ranks the classvars and their derivatives lower than all maxdimvars.

The maximal cases can be determined in many applications where a complete analysis of all cases is not possible.  These cases are often the most interesting, with the richest properties, when compared to the relatively uninteresting (but often difficult to compute) lower dimensional cases.

The options for maxdimsystems consist of the output option (described below), and a subset of those for rifsimp, the main subroutine used by maxdimsystems. Of the subset of rifsimp options available, two that are often used are indep which specifies the independent variables, ezcriteria which controls the flow of equations from a fast (ultra-hog) setting (that can readily lead to memory explosion but sometimes faster calculations), to a conservative (one equation at a time) setting that can overcome some expression swell problems. The complete list is given by:

Many rifsimp options are not supported by maxdimsystems as their use could produce misleading results. For example, the rifsimp option $\mathrm{ctl}=\mathrm{time}$, places a time limit on rifsimp case computations, and hence could cause maxdimsystems to miss the maximal case if it takes too long to compute. Conversely, specification of other options are not supported as they must have a specific value for the maxdimsystems command to function properly (for example, casesplit must always be on). Other unsupported rifsimp options include itl, stl, nopiv, unsolved, store, storeall, spawn, spoly, and casecount.

output = type

The output type can either be set (default) or $\mathrm{rif}$. Note that set output is easier to read and manipulate, but cannot be used with the caseplot visualization tool. The $\mathrm{rif}$ output can be used with caseplot, but the output is somewhat more complicated, and is described in rifsimp[output]. Note that for the class of problems that maxdimsystems simplifies, many of the outputs from rifsimp described in the output page ( such as DiffConstraint and UnSolve) will never be present in the output.

The maxdimsystems command allows a subset of the rankings available for rifsimp. Specifically those rankings that are controlled by a partition of the dependent variables in maxdimvars, and those available by the order of the independent variables in indep. See rifsimp[ranking] and checkrank for an explanation of how to specify rankings based on these options.

The application of caseplot to the $\mathrm{rif}$-form output of maxdimsystems with the second argument given as the maxdimvars displays the tree of maximal cases and their dimensions, also indicating which branches of the tree have less than maximal dimension.

Note that maxdimsystems is essentially a simplified version of the rifsimp interface that also has the ability to automatically determine the maximum finite dimensional cases or all infinite dimensional cases for a classification ODE/PDE system. If the dimension of the solution space is known, multiple conditions on the dimension of the solution space are needed, or much finer control of the computation is desired, then rifsimp could be used with the mindim option described in rifsimp[cases].

References:

G.J. Reid and A.D. Wittkopf, "Determination of Maximal Symmetry Groups of Classes of Differential Equations", Proc. ISSAC 2000, pp.272-280

G.W. Bluman and S. Kumei, "Symmetries and Differential Equations", Springer-Verlag, vol. 81.

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

$\mathrm{maxdimsystems}\left(\left[u-av=0\,u-cv=0\right]\,\left[u\,v\right]\right)$

$\left\{\left[\mathrm{Solved}=\left[u=cv\,a=c\right]\,\mathrm{Constraint}=\left[\right]\,\mathrm{Pivots}=\left[\right]\,\mathrm{dimension}=1\right]\right\}$

$\mathrm{maxdimsystems}\left(\left[\left(a\left(x\right)-d\left(x\right)\right)\mathrm{diff}\left(u\left(x\right)\,x\,x\right)+b\left(x\right)\mathrm{diff}\left(u\left(x\right)\,x\right)+u\left(x\right)=0\right]\,\left[u\right]\right)$

$\left\{\left[\mathrm{Solved}=\left[\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)=\frac{-b\left(x\right)\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)-u\left(x\right)}{a\left(x\right)-d\left(x\right)}\right]\,\mathrm{Constraint}=\left[\right]\,\mathrm{Pivots}=\left[a\left(x\right)-d\left(x\right)\ne 0\right]\,\mathrm{dimension}=2\right]\right\}$

$\mathrm{maxdimsystems}\left(\left[\mathrm{diff}\left(u\left(x\,y\right)\,x\right)-f\left(x\,y\right)u\left(x\,y\right)=0\,\mathrm{diff}\left(u\left(x\,y\right)\,y\right)-g\left(x\,y\right)u\left(x\,y\right)=0\,{f\left(x\,y\right)}^3-1=0\right]\,\left[u\right]\right)$

$\left\{\left[\mathrm{Solved}=\left[\frac{\partial }{\partial x}u\left(x\,y\right)=f\left(x\,y\right)u\left(x\,y\right)\,\frac{\partial }{\partial y}u\left(x\,y\right)=g\left(x\,y\right)u\left(x\,y\right)\,\frac{\partial }{\partial x}f\left(x\,y\right)=0\,\frac{\partial }{\partial x}g\left(x\,y\right)=0\,\frac{\partial }{\partial y}f\left(x\,y\right)=0\right]\,\mathrm{Constraint}=\left[{f\left(x\,y\right)}^3-1=0\right]\,\mathrm{Pivots}=\left[\right]\,\mathrm{dimension}=1\right]\right\}$

$\mathrm{ans}≔\mathrm{rifsimp}\left(\left[\left(a\left(x\right)-d\left(x\right)\right)\mathrm{diff}\left(u\left(x\right)\,x\,x\right)+b\left(x\right)\mathrm{diff}\left(u\left(x\right)\,x\right)+u\left(x\right)=0\right]\,\left[u\right]\,\mathrm{casesplit}\right)$

$\mathrm{ans}≔table\left(\left[1=table\left(\left[\mathrm{Pivots}=\left[a\left(x\right)-d\left(x\right)\ne 0\right]\,\mathrm{Solved}=\left[\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)=\frac{-b\left(x\right)\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)-u\left(x\right)}{a\left(x\right)-d\left(x\right)}\right]\,\mathrm{Case}=\left[\left[a\left(x\right)-d\left(x\right)\ne 0\,\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right]\right]\right]\right)\,2=table\left(\left[\mathrm{Pivots}=\left[b\left(x\right)\ne 0\right]\,\mathrm{Solved}=\left[\frac{\ⅆ}{\ⅆx}u\left(x\right)=-\frac{u\left(x\right)}{b\left(x\right)}\,a\left(x\right)=d\left(x\right)\right]\,\mathrm{Case}=\left[\left[a\left(x\right)-d\left(x\right)=0\,\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right]\,\left[b\left(x\right)\ne 0\,\frac{\ⅆ}{\ⅆx}u\left(x\right)\right]\right]\right]\right)\,3=table\left(\left[\mathrm{Solved}=\left[u\left(x\right)=0\,a\left(x\right)=d\left(x\right)\,b\left(x\right)=0\right]\,\mathrm{Case}=\left[\left[a\left(x\right)-d\left(x\right)=0\,\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right]\,\left[b\left(x\right)=0\,\frac{\ⅆ}{\ⅆx}u\left(x\right)\right]\right]\right]\right)\,\mathrm{casecount}=3\right]\right)$

$\mathrm{caseplot}\left(\mathrm{ans}\,\left[u\right]\right)$

$\mathrm{ODE}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)=f\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=f\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{symeq}≔\left[\mathrm{DEtools}\left[\mathrm{odepde}\right]\left(\mathrm{ODE}\,\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\right]$

$\mathrm{symeq}≔\left[\left(-3\mathrm{\_y1}\left(\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right)-2\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)+\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\right)f\left(\mathrm{\_y1}\right)+\left(-\mathrm{\_y1}\left(\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\right)+{\mathrm{\_y1}}^2\left(\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\right)+\mathrm{\_y1}\left(\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)\right)-\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\_y1}}f\left(\mathrm{\_y1}\right)\right)+\frac{{\partial }^2}{\partial x^2}\mathrm{\eta }\left(x\,y\right)+\left(\frac{{\partial }^2}{\partial y^2}\mathrm{\eta }\left(x\,y\right)\right){\mathrm{\_y1}}^2-\left(\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)\right){\mathrm{\_y1}}^3-2\left(\frac{{\partial }^2}{\partial x\partial y}\mathrm{\xi }\left(x\,y\right)\right){\mathrm{\_y1}}^2+2\left(\frac{{\partial }^2}{\partial x\partial y}\mathrm{\eta }\left(x\,y\right)\right)\mathrm{\_y1}-\mathrm{\_y1}\left(\frac{{\partial }^2}{\partial x^2}\mathrm{\xi }\left(x\,y\right)\right)\right]$

$\mathrm{t1}≔\mathrm{time}\left(\right)\:$

$\mathrm{msys}≔\mathrm{maxdimsystems}\left(\left[\mathrm{op}\left(\mathrm{symeq}\right)\,f\left(\mathrm{\_y1}\right)\ne 0\right]\,\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)\:$

$\mathrm{time}\left(\right)-\mathrm{t1}$

$0.523$

$\mathrm{nops}\left(\mathrm{msys}\right)$

$1$

$\mathrm{length}\left(\mathrm{msys}\left[1\right]\right)$

$5362$

$\mathrm{msys}\left[1\right]\left[4\right]$

$\mathrm{dimension}=8$

$\mathrm{remove}\left(\mathrm{has}\,\mathrm{rhs}\left(\mathrm{msys}\left[1\right]\left[1\right]\right)\,\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\left[\frac{{\ⅆ}^4}{\ⅆ{\mathrm{\_y1}}^4}f\left(\mathrm{\_y1}\right)=0\right]$

$\mathrm{t2}≔\mathrm{time}\left(\right)\:$

$\mathrm{rsys}≔\mathrm{rifsimp}\left(\left[\mathrm{op}\left(\mathrm{symeq}\right)\,f\left(\mathrm{\_y1}\right)\ne 0\right]\,\left[\mathrm{\xi }\,\mathrm{\eta }\right]\,'\mathrm{casesplit}'\right)\:$

$\mathrm{time}\left(\right)-\mathrm{t2}$

$4.783$

$\mathrm{rsys}\left['\mathrm{casecount}'\right]$

$11$

$\mathrm{diff}\left(u\left(x\,t\right)\,t\right)=\mathrm{diff}\left(k\left(u\left(x\,t\right)\right)\mathrm{diff}\left(u\left(x\,t\right)\,x\right)\,x\right)$

$\frac{\partial }{\partial t}u\left(x\,t\right)=\mathrm{D}\left(k\right)\left(u\left(x\,t\right)\right){\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)}^2+k\left(u\left(x\,t\right)\right)\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right)$

$\mathrm{sys}≔\left[-2\mathrm{diff}\left(T\left(u\,t\,x\right)\,x\right)\,\mathrm{diff}\left(\mathrm{diff}\left(U\left(u\,t\,x\right)\,u\right)\,u\right)-2\mathrm{diff}\left(\mathrm{diff}\left(X\left(u\,t\,x\right)\,u\right)\,x\right)+\frac{U\left(u\,t\,x\right)\mathrm{diff}\left(\mathrm{diff}\left(k\left(u\right)\,u\right)\,u\right)}{k\left(u\right)}+\frac{1}{k\left(u\right)}\mathrm{diff}\left(k\left(u\right)\,u\right)\mathrm{diff}\left(U\left(u\,t\,x\right)\,u\right)-\frac{U\left(u\,t\,x\right)}{{k\left(u\right)}^2}{\mathrm{diff}\left(k\left(u\right)\,u\right)}^2\,\frac{1}{k\left(u\right)}\mathrm{diff}\left(X\left(u\,t\,x\right)\,u\right)\mathrm{diff}\left(k\left(u\right)\,u\right)-\mathrm{diff}\left(\mathrm{diff}\left(X\left(u\,t\,x\right)\,u\right)\,u\right)\,\mathrm{diff}\left(\mathrm{diff}\left(U\left(u\,t\,x\right)\,x\right)\,x\right)-\frac{1}{k\left(u\right)}\mathrm{diff}\left(U\left(u\,t\,x\right)\,t\right)\,\frac{1}{k\left(u\right)}\mathrm{diff}\left(T\left(u\,t\,x\right)\,t\right)+\frac{U\left(u\,t\,x\right)}{{k\left(u\right)}^2}\mathrm{diff}\left(k\left(u\right)\,u\right)-\mathrm{diff}\left(\mathrm{diff}\left(T\left(u\,t\,x\right)\,x\right)\,x\right)-\frac{2}{k\left(u\right)}\mathrm{diff}\left(X\left(u\,t\,x\right)\,x\right)\,-\mathrm{diff}\left(\mathrm{diff}\left(X\left(u\,t\,x\right)\,x\right)\,x\right)+2\mathrm{diff}\left(\mathrm{diff}\left(U\left(u\,t\,x\right)\,u\right)\,x\right)+\frac{1}{k\left(u\right)}\mathrm{diff}\left(X\left(u\,t\,x\right)\,t\right)+\frac{2\mathrm{diff}\left(k\left(u\right)\,u\right)}{k\left(u\right)}\mathrm{diff}\left(U\left(u\,t\,x\right)\,x\right)\,-\frac{2\mathrm{diff}\left(k\left(u\right)\,u\right)}{k\left(u\right)}\mathrm{diff}\left(T\left(u\,t\,x\right)\,x\right)-\frac{2}{k\left(u\right)}\mathrm{diff}\left(X\left(u\,t\,x\right)\,u\right)-2\mathrm{diff}\left(\mathrm{diff}\left(T\left(u\,t\,x\right)\,u\right)\,x\right)\,-\frac{1}{k\left(u\right)}\mathrm{diff}\left(k\left(u\right)\,u\right)\mathrm{diff}\left(T\left(u\,t\,x\right)\,u\right)-\mathrm{diff}\left(\mathrm{diff}\left(T\left(u\,t\,x\right)\,u\right)\,u\right)\,-2\mathrm{diff}\left(T\left(u\,t\,x\right)\,u\right)\right]\:$

$\mathrm{maxdimsystems}\left(\left[\mathrm{op}\left(\mathrm{sys}\right)\,\mathrm{diff}\left(k\left(u\right)\,u\right)\ne 0\right]\,\left[\left[U\right]\,\left[X\,T\right]\right]\right)$

$\left\{\left[\mathrm{Solved}=\left[U\left(u\,t\,x\right)=-\frac{k\left(u\right)\left(-2\frac{\partial }{\partial x}X\left(u\,t\,x\right)+\frac{\partial }{\partial t}T\left(u\,t\,x\right)\right)}{\frac{\ⅆ}{\ⅆu}k\left(u\right)}\,\frac{{\partial }^3}{\partial x^3}X\left(u\,t\,x\right)=0\,\frac{{\partial }^2}{\partial t^2}T\left(u\,t\,x\right)=0\,\frac{\partial }{\partial u}X\left(u\,t\,x\right)=0\,\frac{\partial }{\partial u}T\left(u\,t\,x\right)=0\,\frac{\partial }{\partial t}X\left(u\,t\,x\right)=0\,\frac{\partial }{\partial x}T\left(u\,t\,x\right)=0\,\frac{{\ⅆ}^2}{\ⅆu^2}k\left(u\right)=\frac{7{\left(\frac{\ⅆ}{\ⅆu}k\left(u\right)\right)}^2}{4k\left(u\right)}\right]\,\mathrm{Constraint}=\left[\right]\,\mathrm{Pivots}=\left[\frac{\ⅆ}{\ⅆu}k\left(u\right)\ne 0\right]\,\mathrm{dimension}=5\right]\right\}$

$\mathrm{maxdimsystems}\left(\left[\mathrm{op}\left(\mathrm{sys}\right)\,\mathrm{diff}\left(k\left(u\right)\,u\right)\ne 0\right]\,\left[\left[X\,T\right]\,\left[U\right]\right]\right)$

$\left\{\left[\mathrm{Solved}=\left[\frac{{\partial }^2}{\partial x^2}X\left(u\,t\,x\right)=\frac{\left(\frac{\ⅆ}{\ⅆu}k\left(u\right)\right)\left(\frac{\partial }{\partial x}U\left(u\,t\,x\right)\right)}{2k\left(u\right)}\,\frac{\partial }{\partial u}X\left(u\,t\,x\right)=0\,\frac{\partial }{\partial u}T\left(u\,t\,x\right)=0\,\frac{\partial }{\partial t}X\left(u\,t\,x\right)=0\,\frac{\partial }{\partial t}T\left(u\,t\,x\right)=\frac{2\left(\frac{\partial }{\partial x}X\left(u\,t\,x\right)\right)k\left(u\right)-U\left(u\,t\,x\right)\left(\frac{\ⅆ}{\ⅆu}k\left(u\right)\right)}{k\left(u\right)}\,\frac{\partial }{\partial x}T\left(u\,t\,x\right)=0\,\frac{{\partial }^2}{\partial x^2}U\left(u\,t\,x\right)=0\,\frac{\partial }{\partial u}U\left(u\,t\,x\right)=-\frac{3U\left(u\,t\,x\right)\left(\frac{\ⅆ}{\ⅆu}k\left(u\right)\right)}{4k\left(u\right)}\,\frac{\partial }{\partial t}U\left(u\,t\,x\right)=0\,\frac{{\ⅆ}^2}{\ⅆu^2}k\left(u\right)=\frac{7{\left(\frac{\ⅆ}{\ⅆu}k\left(u\right)\right)}^2}{4k\left(\right)}\right]\right]\right\}$