Eta_k - Maple Help

PDEtools

 Eta_k
 construct a table-procedure that computes, on request, any prolongation of the infinitesimals of a symmetry generator

 Calling Sequence Eta_k(S, DepVars, options=value)

Parameters

 S - list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator DepVars - function or list of functions indicating the dependent variables of the problem expanded = ... - (optional) can be true or false (default); to expand or not the prolongation of the infinitesimals as opposed to returning just the table procedure that computes it jetnotation = ... - (optional) can be true (default, the notation found in S), false, jetvariables, jetvariableswithbrackets, jetnumbers or jetODE; to respectively return or not using the different jet notations available prolongation = ... - (optional) list with two operands: a positive integer identifying the dependent variable, and a list of them where the different numbers identify different independent variables and their amount indicates the prolongation order (see notation of D); default is 'all'

Description

 • Given a list of infinitesimals of a symmetry generator, or the corresponding infinitesimal generator differential operator, the Eta_k command returns a table-procedure $\mathrm{\eta }$ that can compute any particular prolongation of the symmetry. By table-procedure it is meant a Maple table that however acts as a procedure: it takes $n$ indices and works with them as a function of $n$ parameter does - see the examples and also indexing functions for table. Eta_k also works with anticommutative variables set using the Physics package.
 • The key idea in the implementation of Eta_k is that constructing a procedure optimized for a given symmetry is faster and more economical in computational resources than actually computing one or many prolongations from a single generic procedure. The table-procedure returned by Eta_k can in turn compute any prolongation, and will do so without repeating operations - that is the meaning of optimized in the previous sentence.
 • The table procedure $\mathrm{\eta }$ returned by Eta_k can be indexed using independent and dependent variables in jet notation, or their respective numeric positions in the lists of independent and dependent variables (see GetIndepVars and GetDepVars), and correspondingly, the output will be in jetvariables or jetnumbers notation. For example, if $\left[u\left(x,t\right),v\left(x,t\right)\right]$ are the dependent variables, you can index the table $\mathrm{\eta }$ returned by Eta_k as in ${\mathrm{\eta }}_{v,\left[x,x,t\right]}$, to receive the value of this prolongation in jetvariables notation. If you use jetnumbers, so the pair [1, 2] representing $\left[x,t\right]$ as well as $\left[u,v\right]$, as in ${\mathrm{\eta }}_{2,\left[1,1,2\right]}$, the output will be in jetnumbers notation, equivalent to apply ToJet with option jetnotation = jetnumbers over the output of ${\mathrm{\eta }}_{u,\left[x,x,t\right]}$. By using the option jetnotation - ..., where the right-hand-side could be jetvariables, jetvariableswithbrackets or jetODE, when indexing $\mathrm{\eta }$ with independent or dependent variables the output will appear in the jetnotation indicated.
 • Independent of this ability of returning a "prolongation procedure" without having in mind any particular prolongation, you can also specify to Eta_k the prolongation order using the optional argument $\mathrm{prolongation}=\left[m,\left[i,j,\mathrm{...}\right]\right]$, where $m$ is a positive integer identifying a dependent variable and $\left[i,j,\mathrm{...}\right]$ is a list with the positions of independent variables, in which case that particular prolongation is returned, unexpanded; evaluating it further (for instance with eval) will automatically expand it. Or, if in addition to specifying the prolongation it is passed the optional argument expanded, then instead of the unexpanded form of the prolongation specified, the expanded form is returned.
 • To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

Examples

 > $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{declare},\mathrm{Eta_k},\mathrm{InfinitesimalGenerator},\mathrm{ToJet}\right):$

Consider the generic form of a list of infinitesimals of a PDE problem in - say - two independent and two dependent variables u(x, t), v(x, t): there are then two $\mathrm{\xi }$ infinitesimals associated to each the independent variables and two $\mathrm{\eta }$ infinitesimals associated to each of the dependent variables, as in

 > $\mathrm{DepVars}≔\left[u,v\right]\left(x,t\right)$
 ${\mathrm{DepVars}}{≔}\left[{u}{}\left({x}{,}{t}\right){,}{v}{}\left({x}{,}{t}\right)\right]$ (1)
 > $S≔\left[\mathrm{seq}\left({\mathrm{ξ}}_{j}\left(x,t,u,v\right),j=\left[x,t\right]\right),\mathrm{seq}\left({\mathrm{η}}_{j}\left(x,t,u,v\right),j=\left[u,v\right]\right)\right]$
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{{\mathrm{\eta }}}_{{v}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right]$ (2)

For illustration purposes consider also the infinitesimal generator operator corresponding to this list of infinitesimals

 > $G≔\mathrm{InfinitesimalGenerator}\left(S,\mathrm{DepVars}\right)$
 ${G}{≔}{f}{→}{{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{f}\right){+}{{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){}\left(\frac{{\partial }}{{\partial }{t}}{}{f}\right){+}{{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right){+}{{\mathrm{η}}}_{{v}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){}\left(\frac{{\partial }}{{\partial }{v}}{}{f}\right)$ (3)

The table-procedure returned by Eta_k, able to compute any prolongation, is constructed instantly and without consuming any computational resources, equally from $S$ or $G$

 > $\mathrm{Η}≔\mathrm{Eta_k}\left(G,\mathrm{DepVars}\right)$
 ${\mathrm{Η}}{≔}{\mathrm{\eta }}$ (4)

This table output is always displayed as $\mathrm{\eta }$. The prolongation of order "zero" of the first dependent infinitesimal, that is, the $\mathrm{\eta }$ infinitesimal associated to the dependent variable $v\left(x,t\right)$ is obtained by entering ${\mathrm{Η}}_{v}$ or ${\mathrm{Η}}_{1}$ and is just equal to ${\mathrm{\eta }}_{v}\left(x,t,u,v\right)$ respectively written in jetvariables or jetnumbers notation

 > ${\mathrm{Η}}_{u}$
 ${{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)$ (5)
 > ${\mathrm{Η}}_{1}$
 ${{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)$ (6)

The difference between jetvariables and jetnumbers is more evident when actually computing prolongations, for example for the first prolongation of ${\mathrm{\eta }}_{v}$ with respect to $x$

 > ${\mathrm{Η}}_{v,\left[x\right]}$
 ${-}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{v}}_{{x}}^{{2}}{-}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{v}}_{{x}}{}{{v}}_{{t}}{+}{{u}}_{{x}}{}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{v}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){+}{{v}}_{{x}}{}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{v}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){-}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{v}}_{{x}}{-}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{v}}_{{t}}{+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{v}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)$ (7)
 > ${\mathrm{Η}}_{2,\left[1\right]}$
 ${-}\left(\frac{{\partial }}{{\partial }{u}\left[\right]}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)\right){}{{u}}_{{1}}{}{{v}}_{{1}}{-}\left(\frac{{\partial }}{{\partial }{v}\left[\right]}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)\right){}{{v}}_{{1}}^{{2}}{-}\left(\frac{{\partial }}{{\partial }{u}\left[\right]}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)\right){}{{u}}_{{1}}{}{{v}}_{{2}}{-}\left(\frac{{\partial }}{{\partial }{v}\left[\right]}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)\right){}{{v}}_{{1}}{}{{v}}_{{2}}{+}{{u}}_{{1}}{}\left(\frac{{\partial }}{{\partial }{u}\left[\right]}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{v}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)\right){+}{{v}}_{{1}}{}\left(\frac{{\partial }}{{\partial }{v}\left[\right]}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{v}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)\right){-}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)\right){}{{v}}_{{1}}{-}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)\right){}{{v}}_{{2}}{+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{v}}{}\left({x}{,}{t}{,}{u}\left[\right]{,}{v}\left[\right]\right)$ (8)

In this output (4.10) expressed using jetnumbers $u\left[\right]$ represents $u\left(x,t\right)$, ${u}_{1}$ represents the derivative with respect to $x$, ${u}_{1,1}$ the second derivative with respect to $x$ and so on; the convention is the same one used by the Maple D differentiation operator.

On the other hand, in the output (4.9) expressed in jetvariables, $u$ represents $u\left(x,t\right)$, ${u}_{x}$ represents the derivative with respect to $x$, ${u}_{x,x}$ the second derivative with respect to $x$ and so on; this convention is also used in all the symmetry commands of PDEtools, the declare command in the mathematical display of derivatives, and by the DifferentialAlgebra package.

To avoid redundant display in the following output use the declare facility, which will also make the derivatives be displayed compactly, indexed

 > $\mathrm{declare}\left(\left(\mathrm{ξ},\mathrm{η}\right)\left(x,t,u,v\right)\right)$
 ${\mathrm{\xi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){}{\mathrm{will now be displayed as}}{}{\mathrm{\xi }}$
 ${\mathrm{\eta }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){}{\mathrm{will now be displayed as}}{}{\mathrm{\eta }}$ (9)

For example, the now compact display for ${\mathrm{\eta }}_{v,\left[x\right]}$ lines above is

 > ${\mathrm{Η}}_{v,\left[x\right]}$
 ${-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}}^{{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}}{}{{v}}_{{t}}{+}{{u}}_{{x}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{v}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){+}{{v}}_{{x}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{v}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{v}}_{{t}}{+}{\mathrm{diff}}{}\left({{\mathrm{η}}}_{{v}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)$ (10)

There are three possible prolongations of order two for each of the ${\mathrm{\eta }}_{u}$ and ${\mathrm{\eta }}_{v}$ infinitesimals, these are: two times with respect to the first independent variable, ditto with respect to the second independent variable, and the mixed prolongation; for ${\mathrm{\eta }}_{u}$ these are

 > ${\mathrm{Η}}_{u,\left[x,x\right]}$
 ${\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right){,}{x}\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{3}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{2}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}{,}{x}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}{,}{x}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}{,}{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}}^{{2}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}{,}{x}}{+}{2}{}{{u}}_{{x}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){+}{2}{}{{v}}_{{x}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right){,}{x}\right)\right){}{{u}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right){,}{x}\right)\right){}{{u}}_{{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){}{{u}}_{{x}}^{{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{x}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{x}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}^{{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}^{{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}{,}{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}{,}{x}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}^{{2}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{2}}{}{{u}}_{{t}}{+}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{-}{3}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{x}{,}{x}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{x}}{}{{v}}_{{x}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{x}}$ (11)
 > ${\mathrm{Η}}_{u,\left[t,t\right]}$
 ${\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{t}\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{t}}^{{2}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right)\right){}{{u}}_{{x}{,}{t}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right)\right){}{{u}}_{{t}{,}{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{v}}_{{t}}^{{2}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{v}}_{{t}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{t}}^{{2}}{+}{2}{}{{u}}_{{t}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){+}{2}{}{{v}}_{{t}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{t}\right)\right){}{{u}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{t}\right)\right){}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{t}}^{{3}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{t}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}^{{2}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{t}}^{{2}}{}{{v}}_{{t}}{+}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}{,}{t}}{-}{3}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{t}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}{,}{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}^{{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{t}}^{{2}}{-}{2}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{t}}$ (12)
 > ${\mathrm{Η}}_{u,\left[x,t\right]}$
 ${\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{2}}{}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}^{{2}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}^{{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{x}}{}{{v}}_{{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{x}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{t}}^{{2}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}{,}{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{2}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right)\right){}{{u}}_{{x}{,}{x}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right)\right){}{{u}}_{{x}{,}{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{u}}_{{t}{,}{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){}{{u}}_{{t}}^{{2}}{+}{{u}}_{{t}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){+}{{v}}_{{t}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right)\right){}{{u}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right)\right){}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{}{{v}}_{{t}}$ (13)

Note the compact display in the output above, due to the use of declare. To see the contents behind this compact display use show

 > $\mathrm{show}$
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}{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{+}\left(\frac{{{\partial }}^{{2}}}{{\partial }{t}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{v}}_{{x}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}^{{2}}{+}{{u}}_{{t}}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{v}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{x}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{v}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{x}}{-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}{,}{x}}{}{{v}}_{{t}}{-}{2}{}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{x}{,}{t}}{-}{2}{}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{t}}{-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{t}}{-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}{,}{t}}{-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}{,}{t}}{-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{x}}{-}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}{,}{t}}{-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}{,}{t}}{}{{v}}_{{x}}{-}\left(\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}{,}{x}}{+}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{-}\left(\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{+}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{v}}_{{x}{,}{t}}{-}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{-}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}{,}{t}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{v}{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{v}{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{v}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{t}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{+}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{t}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{t}}^{{2}}{}{{v}}_{{x}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}^{{2}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{t}{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\eta }}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}^{{2}}{}{{u}}_{{t}}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\xi }}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}{{u}}_{{x}}^{{2}}{}{{v}}_{{t}}$ (14)

It is sometimes possible to compactify these expressions a bit more by using, for instance, simplify,size or convert,horner.

You can also request a particular prolongation directly, either using jetvariables or jetnumbers notation, with or without the option expanded

 > $\mathrm{Eta_k}\left(S,\mathrm{DepVars},\mathrm{prolongation}=\left[u,\left[x,t\right]\right]\right)$
 ${{\mathrm{\eta }}}_{{u}{,}\left[{x}{,}{t}\right]}$ (15)
 > 
 ${\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{2}}{}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}^{{2}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}^{{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{x}}{}{{v}}_{{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{x}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{t}}^{{2}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}{,}{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{2}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right)\right){}{{u}}_{{x}{,}{x}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right)\right){}{{u}}_{{x}{,}{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{u}}_{{t}{,}{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){}{{u}}_{{t}}^{{2}}{+}{{u}}_{{t}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){+}{{v}}_{{t}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right)\right){}{{u}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right)\right){}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{}{{v}}_{{t}}$ (16)
 > $\mathrm{Eta_k}\left(S,\mathrm{DepVars},\mathrm{prolongation}=\left[u,\left[x,t\right]\right],\mathrm{expanded}\right)$
 ${\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{2}}{}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}^{{2}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}^{{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{x}}{}{{v}}_{{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{x}{,}{t}}{-}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{t}}{}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{t}}^{{2}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}{,}{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}{,}{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}^{{2}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{u}\right)\right){}{{u}}_{{x}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right)\right){}{{u}}_{{x}{,}{x}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right)\right){}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right)\right){}{{u}}_{{x}{,}{t}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{v}\right)\right){}{{v}}_{{x}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right)\right){}{{v}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{u}}_{{x}{,}{t}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{x}\right)\right){}{{u}}_{{t}{,}{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){}{{u}}_{{t}}^{{2}}{+}{{u}}_{{t}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{x}\right)\right){+}{{v}}_{{t}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{x}\right)\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right)\right){}{{u}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{t}\right){,}{x}\right)\right){}{{u}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{t}}{}{{v}}_{{x}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{t}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{u}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{u}}_{{t}}{}{{v}}_{{x}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{u}{,}{v}\right){,}{v}\right){,}{v}\right)\right){}{{u}}_{{x}}{}{{v}}_{{x}}{}{{v}}_{{t}}$ (17)
 > $-$
 ${0}$ (18)
 > $\mathrm{Eta_k}\left(S,\mathrm{DepVars},\mathrm{prolongation}=\left[1,\left[1,2\right]\right],\mathrm{expanded}\right)$
 ${\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{t}\right){,}{x}\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{t}\right){,}{{u}}_{{[}{]}}\right)\right){}{{u}}_{{1}}^{{2}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{t}\right){,}{{u}}_{{[}{]}}\right)\right){}{{u}}_{{1}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{t}\right)\right){}{{u}}_{{1}{,}{1}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{u}}_{{[}{]}}\right)\right){}{{u}}_{{1}{,}{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{t}\right)\right){}{{u}}_{{1}{,}{2}}{+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{t}\right){,}{{v}}_{{[}{]}}\right)\right){}{{v}}_{{1}}{+}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right)\right){}{{v}}_{{1}{,}{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{x}\right)\right){}{{u}}_{{1}{,}{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{x}\right)\right){}{{u}}_{{2}{,}{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{x}\right){,}{{u}}_{{[}{]}}\right)\right){}{{u}}_{{2}}^{{2}}{+}{{u}}_{{2}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{x}\right){,}{{u}}_{{[}{]}}\right)\right){+}{{v}}_{{2}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{η}}}_{{u}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{x}\right){,}{{v}}_{{[}{]}}\right)\right){-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{t}\right){,}{x}\right)\right){}{{u}}_{{1}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{t}\right){,}{x}\right)\right){}{{u}}_{{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{u}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right)\right){}{{u}}_{{1}}{}{{u}}_{{2}}{}{{v}}_{{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{u}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right)\right){}{{u}}_{{1}}{}{{u}}_{{2}}{}{{v}}_{{1}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right)\right){}{{u}}_{{1}}{}{{v}}_{{1}}{}{{v}}_{{2}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right)\right){}{{u}}_{{2}}{}{{v}}_{{1}}{}{{v}}_{{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right)\right){}{{u}}_{{2}}{}{{v}}_{{1}{,}{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right)\right){}{{u}}_{{1}{,}{2}}{}{{v}}_{{1}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{u}}_{{[}{]}}\right)\right){}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}{-}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{t}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{v}}_{{[}{]}}\right)\right){}{{u}}_{{2}{,}{2}}{}{{v}}_{{1}}{-}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{x}}{}\left({x}{,}{t}{,}{{u}}_{{[}{]}}{,}{{v}}_{{[}{]}}\right){,}{{u}}_{{[}{]}}\right){,}{{u}}_{{[}{]}}\right)\right){}{{u}}_{{1}}^{{2}}{}{{u}}_{{2}}{-}\left({\mathrm{diff}}{}\right)$