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LieAlgebrasOfVectorFields

 MapDE
 algorithmically checks if an input system of polynomially nonlinear ODE or PDE can be mapped into a target class of linear DEs by an invertible Map of the variables of the input system.

 Calling Sequence MapDE(InputDE, TargetClass, Map, Options) MapDE([DE,[x,u],[xi,eta]], ToLinearDE, [psi,phi], Options) MapDE([DE,[x,u],[xi,eta]], [ToLinearDE, [xh,uh]], [psi,phi], Options)

Parameters

 InputDE - A list of lists [DE, vars, infinitesimals] for an input (source) DE system. DE is a list of polynomially nonlinear partial or ordinary differential equations and inequations, and vars is a list $\left[x,u\right]$ where $x=\left[{x}_{1},{x}_{2},\dots ,{x}_{n}\right]$ is a list of independent variable names and $u=\left[{u}_{1},{u}_{2},\dots ,{u}_{m}\right]$ is a list of dependent variable names for the DE. infinitesimals is a list $\left[\mathrm{\xi },\mathrm{\eta }\right]$ where $\mathrm{\xi }=\left[{\mathrm{\xi }}_{1},{\mathrm{\xi }}_{2},\dots ,{\mathrm{\xi }}_{n}\right]$ and $\mathrm{\eta }=\left[{\mathrm{\eta }}_{1},\dots ,{\mathrm{\eta }}_{m}\right]$ are lists of the components of vectorfields of the Lie algebra corresponding to continuous Lie point symmetry transformations of the DE in the space of independent and dependent variables of the DE. TargetClass - Properties of the class of target DE. Currently TargetClass is either ToLinearDE or has form [ToLinearDE, $\left[\mathrm{xh},\mathrm{uh}\right]$] where $\mathrm{xh}=\left[{\mathrm{xh}}_{1},\dots ,{\mathrm{xh}}_{n}\right]$ is a list of independent variable names and $\mathrm{uh}=\left[{\mathrm{uh}}_{1},\dots ,{\mathrm{uh}}_{m}\right]$ is a list of dependent variable names for the target DE. Map - $\left[\mathrm{\psi },\mathrm{\phi }\right]$ where $\mathrm{\psi }$ and $\mathrm{\phi }$ are lists of mapping functions corresponding to the independent variables $\mathrm{xh}$ and dependent variables $\mathrm{uh}$. Thus Map transforms the InputDE to a member of the TargetClass where $\left(\mathrm{xh},\mathrm{uh}\right)=\mathrm{Map}\left(x,u\right)=\left(\mathrm{\psi }\left(x,u\right),\mathrm{\phi }\left(x,u\right)\right)$ at points $\left(x,u\right)$ where the Jacobian of Map is nonzero. Options - OutputDetails when included in Options returns additional details of the computation (including structure and dimensions of various objects computed in intermediate computations).

Description

 • The command MapDE determines the invertible linearizability of nonlinear differential polynomial systems and attempts to construct the map if it exists. Thus it determines the existence of an invertible mapping $\left(\mathrm{xh},\mathrm{uh}\right)=\left(\mathrm{\psi }\left(x,u\right),\mathrm{\phi }\left(x,u\right)\right)$ on points $\left(x,u\right)$ in the space of independent and dependent variables of the InputDE to points $\left(\mathrm{xh},\mathrm{uh}\right)$ in the space of independent and dependent variables of a target linear DE.
 • The output of MapDE is a table. The output table includes the label MapExists = true or MapExists = false, if a mapping of the InputDE to a linear DE exists or does not exist. Additional outputs include the reduced Map DE (RedMapDE) which are differential equations satisfied by the functions $\left(\mathrm{\psi },\mathrm{\phi }\right)$ of the mapping, together with inequality (pivot) conditions on $\left(\mathrm{\psi },\mathrm{\phi }\right)$ given by the output table entry labeled by MapDEPivs. If pdsolve is successful in explicitly integrating the RedMapDE system, then the output also includes the integrated form of the transformations labeled by IntegratedMap, and an explicit form of the target DE, labeled by TargetDE.
 • A user can get more information about the mapping system, including the structure and dimensions of various objects (solution spaces, derived algebra, Lie algebra structure, etc) by including OutputDetails or by assigning infolevel[MapDE] to a non-negative integer value.

Examples

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

DE[1] below is obviously linearizable by the transformation $\mathrm{uh}={u}^{2},\mathrm{xh}=x$:

 > $\mathrm{DE}\left[1\right]≔\left[\mathrm{diff}\left({u\left(x\right)}^{2},x,x,x\right)+{u\left(x\right)}^{2}=0\right]$
 ${{\mathrm{DE}}}_{{1}}{≔}\left[{6}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{2}{}{u}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{{u}{}\left({x}\right)}^{{2}}{=}{0}\right]$ (1)
 > $\mathrm{MDE}\left[1\right]≔\mathrm{MapDE}\left(\left[\mathrm{DE}\left[1\right],\left[\left[x\right],\left[u\right]\right],\left[\left[\mathrm{\xi }\right],\left[\mathrm{\eta }\right]\right]\right],\mathrm{ToLinearDE},\left[\left[\mathrm{\psi }\right],\left[\mathrm{\phi }\right]\right]\right)$
 ${{\mathrm{MDE}}}_{{1}}{≔}{table}{}\left(\left[{\mathrm{MapDEPivs}}{=}\left[\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right){\ne }{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{u}\right){\ne }{0}\right]{,}{\mathrm{IntegratedMap}}{=}\left[{\mathrm{xh}}{=}{x}{,}{\mathrm{uh}}{=}{{u}}^{{2}}\right]{,}{\mathrm{RedMapDE}}{=}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right){=}\frac{\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right)}{{u}}{,}\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{u}\right){=}{0}\right]{,}{\mathrm{MapExists}}{=}{\mathrm{true}}{,}{\mathrm{TargetDE}}{=}\left[\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{\mathrm{xh}}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{uh}}{}\left({\mathrm{xh}}\right){=}{-}{\mathrm{uh}}{}\left({\mathrm{xh}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[{\mathrm{uh}}{}\left({\mathrm{xh}}\right){\ne }{0}\right]\right]\right)$ (2)

 > $\mathrm{DE}\left[2\right]≔\left[u\left(x\right)\mathrm{diff}\left(\mathrm{diff}\left(\mathrm{diff}\left(u\left(x\right),x\right),x\right),x\right)+3\mathrm{diff}\left(u\left(x\right),x\right)\mathrm{diff}\left(\mathrm{diff}\left(u\left(x\right),x\right),x\right)-\frac{3{\left({\mathrm{diff}\left(u\left(x\right),x\right)}^{2}+u\left(x\right)\mathrm{diff}\left(\mathrm{diff}\left(u\left(x\right),x\right),x\right)+1\right)}^{2}}{u\left(x\right)\mathrm{diff}\left(u\left(x\right),x\right)+x}-\frac{8x{\left(u\left(x\right)\mathrm{diff}\left(u\left(x\right),x\right)+x\right)}^{4}\left(1+{u\left(x\right)}^{2}+{x}^{2}\right)}{{u\left(x\right)}^{2}+{x}^{2}}=0\right]$
 ${{\mathrm{DE}}}_{{2}}{≔}\left[{u}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{3}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){-}\frac{{3}{}{\left({\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right)}^{{2}}{+}{u}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{1}\right)}^{{2}}}{{u}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{x}}{-}\frac{{8}{}{x}{}{\left({u}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}\right)\right){+}{x}\right)}^{{4}}{}\left({1}{+}{{u}{}\left({x}\right)}^{{2}}{+}{{x}}^{{2}}\right)}{{{u}{}\left({x}\right)}^{{2}}{+}{{x}}^{{2}}}{=}{0}\right]$ (3)
 > $\mathrm{MDE}\left[2\right]≔\mathrm{MapDE}\left(\left[\mathrm{DE}\left[2\right],\left[\left[x\right],\left[u\right]\right],\left[\left[\mathrm{\xi }\right],\left[\mathrm{\eta }\right]\right]\right],\left[\mathrm{ToLinearDE},\left[\left[\mathrm{xh}\right],\left[\mathrm{uh}\right]\right]\right],\left[\left[\mathrm{\psi }\right],\left[\mathrm{\phi }\right]\right]\right)$
 ${{\mathrm{MDE}}}_{{2}}{≔}{table}{}\left(\left[{\mathrm{MapDEPivs}}{=}\left[\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{u}\right){\ne }{0}{,}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right)\right){}{x}{-}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right)\right){}{u}{\ne }{0}\right]{,}{\mathrm{IntegratedMap}}{=}\left[{\mathrm{xh}}{=}{{u}}^{{2}}{+}{{x}}^{{2}}{,}{\mathrm{uh}}{=}{-}{x}\right]{,}{\mathrm{RedMapDE}}{=}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right){=}{-}\frac{\left(\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right)\right){}{u}{}{{x}}^{{2}}{-}{2}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right)\right){}{{u}}^{{2}}{}{x}{-}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right)\right){}{{u}}^{{2}}{-}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{u}\right)\right){}{{x}}^{{2}}}{{{u}}^{{3}}}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{u}\right){=}\frac{\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{u}\right)\right){}{x}}{{u}}\right]{,}{\mathrm{MapExists}}{=}{\mathrm{true}}{,}{\mathrm{TargetDE}}{=}\left[\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{\mathrm{xh}}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{uh}}{}\left({\mathrm{xh}}\right){=}{-}\frac{\left({\mathrm{xh}}{+}{1}\right){}{\mathrm{uh}}{}\left({\mathrm{xh}}\right)}{{\mathrm{xh}}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[{\mathrm{uh}}{}\left({\mathrm{xh}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{xh}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{uh}}{}\left({\mathrm{xh}}\right)\right){\ne }{0}{,}{-}{{\mathrm{uh}}{}\left({\mathrm{xh}}\right)}^{{2}}{+}{\mathrm{xh}}{\ne }{0}{,}{\mathrm{uh}}{}\left({\mathrm{xh}}\right){\ne }{0}\right]\right]\right)$ (4)

For the example above, the transformation $\left[\mathrm{xh}={u}^{2}+{x}^{2},\mathrm{uh}=-x\right]$ maps DE[2] to the given TargetLinearDE, which can be independently checked by applying PDEtools[dchange].

Burgers' PDE below is the simplest nonlinear model combining convection and diffusion:

 > $\mathrm{DE}\left[3\right]≔\left[\mathrm{diff}\left(u\left(x,t\right),x,x\right)=\mathrm{diff}\left(u\left(x,t\right),t\right)-u\left(x,t\right)\mathrm{diff}\left(u\left(x,t\right),x\right)\right]$
 ${{\mathrm{DE}}}_{{3}}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right){=}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right){-}{u}{}\left({x}{,}{t}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right)\right)\right]$ (5)
 > $\mathrm{MDE}\left[3\right]≔\mathrm{MapDE}\left(\left[\mathrm{DE}\left[3\right],\left[\left[x,t\right],\left[u\right]\right],\left[\left[\mathrm{\xi },\mathrm{\tau }\right],\left[\mathrm{\eta }\right]\right]\right],\left[\mathrm{ToLinearDE},\left[\left[x,t\right],\left[u\right]\right]\right],\left[\left[\mathrm{\psi },\mathrm{\Upsilon }\right],\left[\mathrm{\phi }\right]\right]\right)$
 ${{\mathrm{MDE}}}_{{3}}{≔}{table}{}\left(\left[{\mathrm{MapExists}}{=}{\mathrm{false}}\right]\right)$ (6)

MapDE proves above that there is no invertible change of variables in $\left(x,t,u\right)$-space linearizing Burgers' equation. Bluman et al (2010) nonlocally embed Burgers' PDE in the system below

 > $\mathrm{DE}\left[4\right]≔\left[\mathrm{diff}\left(v\left(x,t\right),x\right)=u\left(x,t\right),\mathrm{diff}\left(v\left(x,t\right),t\right)=\mathrm{diff}\left(u\left(x,t\right),x\right)+\frac{{u\left(x,t\right)}^{2}}{2}\right]$
 ${{\mathrm{DE}}}_{{4}}{≔}\left[\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}{,}{t}\right){=}{u}{}\left({x}{,}{t}\right){,}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}{,}{t}\right){=}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right){+}\frac{{{u}{}\left({x}{,}{t}\right)}^{{2}}}{{2}}\right]$ (7)

The reader can check that Burgers' PDE results from computing cross partials. Application of MapDE below reveals that Burgers' DE[4] is linearizable in (x,t,u,v)-space and recovers the famous Hopf-Cole linearizing transformation (also see Bluman et al (2010)).

 > $\mathrm{MDE}\left[4\right]≔\mathrm{MapDE}\left(\left[\mathrm{DE}\left[4\right],\left[\left[x,t\right],\left[u,v\right]\right],\left[\left[\mathrm{\xi },\mathrm{\tau }\right],\left[\mathrm{\eta },\mathrm{\beta }\right]\right]\right],\left[\mathrm{ToLinearDE},\left[\left[\mathrm{xh},\mathrm{th}\right],\left[\mathrm{uh},\mathrm{vh}\right]\right]\right],\left[\left[\mathrm{\psi },\mathrm{\Upsilon }\right],\left[\mathrm{\varphi },\mathrm{\phi }\right]\right]\right)$
 ${{\mathrm{MDE}}}_{{4}}{≔}{table}{}\left(\left[{\mathrm{MapDEPivs}}{=}\left[\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\varphi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\varphi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){\ne }{0}{,}{-}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Upsilon }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}\left(\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){+}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){}\left(\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Upsilon }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)\right){\ne }{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){\ne }{0}\right]{,}{\mathrm{IntegratedMap}}{=}\left[{\mathrm{xh}}{=}{x}{,}{\mathrm{th}}{=}{t}{,}{\mathrm{uh}}{=}{u}{}{{ⅇ}}^{\frac{{v}}{{2}}}{,}{\mathrm{vh}}{=}{{ⅇ}}^{\frac{{v}}{{2}}}\right]{,}{\mathrm{RedMapDE}}{=}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{u}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\varphi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}\frac{\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)}{{2}}{,}\frac{{{\partial }}^{{2}}}{{\partial }{u}{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\varphi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}\frac{\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\varphi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)}{{2}}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{v}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}\frac{\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)}{{2}}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{v}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\varphi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}\frac{\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\varphi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right)}{{2}}{,}\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Upsilon }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}{0}{,}\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}{0}{,}\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Upsilon }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}{0}{,}\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({x}{,}{t}{,}{u}{,}{v}\right){=}{0}\right]{,}{\mathrm{MapExists}}{=}{\mathrm{true}}{,}{\mathrm{TargetDE}}{=}\left[\frac{{\partial }}{{\partial }{\mathrm{xh}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{vh}}{}\left({\mathrm{xh}}{,}{\mathrm{th}}\right){=}\frac{{\mathrm{uh}}{}\left({\mathrm{xh}}{,}{\mathrm{th}}\right)}{{2}}{,}\frac{{\partial }}{{\partial }{\mathrm{th}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{vh}}{}\left({\mathrm{xh}}{,}{\mathrm{th}}\right){=}\frac{\frac{{\partial }}{{\partial }{\mathrm{xh}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{uh}}{}\left({\mathrm{xh}}{,}{\mathrm{th}}\right)}{{2}}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{\mathrm{xh}}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{uh}}{}\left({\mathrm{xh}}{,}{\mathrm{th}}\right){=}\frac{{\partial }}{{\partial }{\mathrm{th}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{uh}}{}\left({\mathrm{xh}}{,}{\mathrm{th}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[{\mathrm{vh}}{}\left({\mathrm{xh}}{,}{\mathrm{th}}\right){\ne }{0}\right]\right]\right)$ (8)

The above apparently paradoxical result is resolved, by noting that the two spaces $\left(x,t,u\right)$-space and $\left(x,t,u,v\right)$-space are noninvertibly related, since non-unique $v$ yields $u$ by integration. Systematically exploring such nonlocal embeddings is discussed in Bluman et al (2010).

References

 Z. Mohammadi, G. Reid, and S.-L.T Huang. "Introduction of the MapDE algorithm for determination of mappings relating differential equations." Proceedings of ISSAC 2019, pp.331-338. ACM Press, 2019.
 Z. Mohammadi, G. Reid, and S.-L.T. Huang. "Symmetry-based algorithms for invertible mappings of polynomially nonlinear PDE to linear PDE". Mathematics in Computer Science, 1–24, 2020.
 G. W. Bluman, A. F. Cheviakov, and S. C. Anco. Applications of Symmetry Methods to Partial Differential Equations, Springer, Chapter 2, 2010.
 D. Lyakhov, V. Gerdt, and D. Michels. "Algorithmic verification of linearizability for ordinary differential equations." Proceedings of ISSAC 2017, pp. 285–292. ACM Press, 2017.

Compatibility

 • The LieAlgebrasOfVectorFields[MapDE] command was introduced in Maple 2021.
 • For more information on Maple 2021 changes, see Updates in Maple 2021.

 See Also