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.

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

$\mathrm{DE}\left[1\right]≔\left[\mathrm{diff}\left({u\left(x\right)}^2\,x\,x\,x\right)+{u\left(x\right)}^2=0\right]$

${\mathrm{DE}}_1≔\left[6\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)+2u\left(x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}u\left(x\right)\right)+{u\left(x\right)}^2=0\right]$

$\mathrm{MDE}\left[1\right]≔\mathrm{MapDE}\left(\left[\mathrm{DE}\left[1\right]\,\left[\left[x\right]\,\left[u\right]\right]\,\left[\left[\mathrm{\xi }\right]\,\left[\mathrm{\eta }\right]\right]\right]\,\mathrm{ToLinearDE}\,\left[\left[\mathrm{\psi }\right]\,\left[\mathrm{\phi }\right]\right]\right)$

${\mathrm{MDE}}_1≔table\left(\left[\mathrm{RedMapDE}=\left[\frac{{\partial }^2}{\partial u^2}\mathrm{\phi }\left(x\,u\right)=\frac{\frac{\partial }{\partial u}\mathrm{\phi }\left(x\,u\right)}{u}\,\frac{\partial }{\partial u}\mathrm{\psi }\left(x\,u\right)=0\right]\,\mathrm{MapDEPivs}=\left[\frac{\partial }{\partial u}\mathrm{\phi }\left(x\,u\right)\ne 0\,\frac{\partial }{\partial x}\mathrm{\psi }\left(x\,u\right)\ne 0\right]\,\mathrm{TargetDE}=\left[\frac{{\ⅆ}^3}{\ⅆ{\mathrm{xh}}^3}\mathrm{uh}\left(\mathrm{xh}\right)=-\mathrm{uh}\left(\mathrm{xh}\right)\right]\mathbf{where}\left[\mathrm{uh}\left(\mathrm{xh}\right)\ne 0\right]\,\mathrm{IntegratedMap}=\left[\mathrm{xh}=x\,\mathrm{uh}=u^2\right]\,\mathrm{MapExists}=\mathrm{true}\right]\right)$

$\mathrm{DE}\left[2\right]≔\left[u\left(x\right)\mathrm{diff}\left(\mathrm{diff}\left(\mathrm{diff}\left(u\left(x\right)\,x\right)\,x\right)\,x\right)+3\mathrm{diff}\left(u\left(x\right)\,x\right)\mathrm{diff}\left(\mathrm{diff}\left(u\left(x\right)\,x\right)\,x\right)-\frac{3{\left({\mathrm{diff}\left(u\left(x\right)\,x\right)}^2+u\left(x\right)\mathrm{diff}\left(\mathrm{diff}\left(u\left(x\right)\,x\right)\,x\right)+1\right)}^2}{u\left(x\right)\mathrm{diff}\left(u\left(x\right)\,x\right)+x}-\frac{8x{\left(u\left(x\right)\mathrm{diff}\left(u\left(x\right)\,x\right)+x\right)}^4\left(1+{u\left(x\right)}^2+x^2\right)}{{u\left(x\right)}^2+x^2}=0\right]$

${\mathrm{DE}}_2≔\left[u\left(x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}u\left(x\right)\right)+3\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)-\frac{3{\left({\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)}^2+u\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)\right)+1\right)}^2}{u\left(x\right)\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)+x}-\frac{8x{\left(u\left(x\right)\left(\frac{\ⅆ}{\ⅆx}u\left(x\right)\right)+x\right)}^4\left(1+{u\left(x\right)}^2+x^2\right)}{{u\left(x\right)}^2+x^2}=0\right]$

$\mathrm{MDE}\left[2\right]≔\mathrm{MapDE}\left(\left[\mathrm{DE}\left[2\right]\,\left[\left[x\right]\,\left[u\right]\right]\,\left[\left[\mathrm{\xi }\right]\,\left[\mathrm{\eta }\right]\right]\right]\,\left[\mathrm{ToLinearDE}\,\left[\left[\mathrm{xh}\right]\,\left[\mathrm{uh}\right]\right]\right]\,\left[\left[\mathrm{\psi }\right]\,\left[\mathrm{\phi }\right]\right]\right)$

${\mathrm{MDE}}_2≔table\left(\left[\mathrm{RedMapDE}=\left[\frac{{\partial }^2}{\partial x^2}\mathrm{\phi }\left(x\,u\right)=-\frac{-2\left(\frac{{\partial }^2}{\partial u\partial x}\mathrm{\phi }\left(x\,u\right)\right)u^{2}x+\left(\frac{{\partial }^2}{\partial u^2}\mathrm{\phi }\left(x\,u\right)\right)u{x}^2-\left(\frac{\partial }{\partial u}\mathrm{\phi }\left(x\,u\right)\right)u^2-\left(\frac{\partial }{\partial u}\mathrm{\phi }\left(x\,u\right)\right)x^2}{u^3}\,\frac{\partial }{\partial x}\mathrm{\psi }\left(x\,u\right)=\frac{\left(\frac{\partial }{\partial u}\mathrm{\psi }\left(x\,u\right)\right)x}{u}\right]\,\mathrm{MapDEPivs}=\left[\frac{\partial }{\partial u}\mathrm{\psi }\left(x\,u\right)\ne 0\,\left(\frac{\partial }{\partial x}\mathrm{\phi }\left(x\,u\right)\right)u-\left(\frac{\partial }{\partial u}\mathrm{\phi }\left(x\,u\right)\right)x\ne 0\right]\,\mathrm{TargetDE}=\left[\frac{{\ⅆ}^3}{\ⅆ{\mathrm{xh}}^3}\mathrm{uh}\left(\mathrm{xh}\right)=-\frac{\left(\mathrm{xh}+1\right)\mathrm{uh}\left(\mathrm{xh}\right)}{\mathrm{xh}}\right]\mathbf{where}\left[\mathrm{uh}\left(\mathrm{xh}\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{xh}}\mathrm{uh}\left(\mathrm{xh}\right)\right)\ne 0\,-{\mathrm{uh}\left(\mathrm{xh}\right)}^2+\mathrm{xh}\ne 0\,\mathrm{uh}\left(\mathrm{xh}\right)\ne 0\right]\,\mathrm{IntegratedMap}=\left[\mathrm{xh}=u^2+x^2\,\mathrm{uh}=-x\right]\,\mathrm{MapExists}=\mathrm{true}\right]\right)$

$\mathrm{DE}\left[3\right]≔\left[\mathrm{diff}\left(u\left(x\,t\right)\,x\,x\right)=\mathrm{diff}\left(u\left(x\,t\right)\,t\right)-u\left(x\,t\right)\mathrm{diff}\left(u\left(x\,t\right)\,x\right)\right]$

${\mathrm{DE}}_3≔\left[\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)=\frac{\partial }{\partial t}u\left(x\,t\right)-u\left(x\,t\right)\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)\right]$

$\mathrm{MDE}\left[3\right]≔\mathrm{MapDE}\left(\left[\mathrm{DE}\left[3\right]\,\left[\left[x\,t\right]\,\left[u\right]\right]\,\left[\left[\mathrm{\xi }\,\mathrm{\tau }\right]\,\left[\mathrm{\eta }\right]\right]\right]\,\left[\mathrm{ToLinearDE}\,\left[\left[x\,t\right]\,\left[u\right]\right]\right]\,\left[\left[\mathrm{\psi }\,\mathrm{\Upsilon }\right]\,\left[\mathrm{\phi }\right]\right]\right)$

${\mathrm{MDE}}_3≔table\left(\left[\mathrm{MapExists}=\mathrm{false}\right]\right)$

$\mathrm{DE}\left[4\right]≔\left[\mathrm{diff}\left(v\left(x\,t\right)\,x\right)=u\left(x\,t\right)\,\mathrm{diff}\left(v\left(x\,t\right)\,t\right)=\mathrm{diff}\left(u\left(x\,t\right)\,x\right)+\frac{{u\left(x\,t\right)}^2}{2}\right]$

${\mathrm{DE}}_4≔\left[\frac{\partial }{\partial x}v\left(x\,t\right)=u\left(x\,t\right)\,\frac{\partial }{\partial t}v\left(x\,t\right)=\frac{\partial }{\partial x}u\left(x\,t\right)+\frac{{u\left(x\,t\right)}^2}{2}\right]$

$\mathrm{MDE}\left[4\right]≔\mathrm{MapDE}\left(\left[\mathrm{DE}\left[4\right]\,\left[\left[x\,t\right]\,\left[u\,v\right]\right]\,\left[\left[\mathrm{\xi }\,\mathrm{\tau }\right]\,\left[\mathrm{\eta }\,\mathrm{\beta }\right]\right]\right]\,\left[\mathrm{ToLinearDE}\,\left[\left[\mathrm{xh}\,\mathrm{th}\right]\,\left[\mathrm{uh}\,\mathrm{vh}\right]\right]\right]\,\left[\left[\mathrm{\psi }\,\mathrm{\Upsilon }\right]\,\left[\mathrm{\varphi }\,\mathrm{\phi }\right]\right]\right)$

${\mathrm{MDE}}_4≔table\left(\left[\mathrm{RedMapDE}=\left[\frac{{\partial }^2}{\partial u^2}\mathrm{\phi }\left(x\,t\,u\,v\right)=0\,\frac{{\partial }^2}{\partial u^2}\mathrm{\varphi }\left(x\,t\,u\,v\right)=0\,\frac{{\partial }^2}{\partial u\partial v}\mathrm{\phi }\left(x\,t\,u\,v\right)=\frac{\frac{\partial }{\partial u}\mathrm{\phi }\left(x\,t\,u\,v\right)}{2}\,\frac{{\partial }^2}{\partial u\partial v}\mathrm{\varphi }\left(x\,t\,u\,v\right)=\frac{\frac{\partial }{\partial u}\mathrm{\varphi }\left(x\,t\,u\,v\right)}{2}\,\frac{{\partial }^2}{\partial v^2}\mathrm{\phi }\left(x\,t\,u\,v\right)=\frac{\frac{\partial }{\partial v}\mathrm{\phi }\left(x\,t\,u\,v\right)}{2}\,\frac{{\partial }^2}{\partial v^2}\mathrm{\varphi }\left(x\,t\,u\,v\right)=\frac{\frac{\partial }{\partial v}\mathrm{\varphi }\left(x\,t\,u\,v\right)}{2}\,\frac{\partial }{\partial u}\mathrm{\Upsilon }\left(x\,t\,u\,v\right)=0\,\frac{\partial }{\partial u}\mathrm{\psi }\left(x\,t\,u\,v\right)=0\,\frac{\partial }{\partial v}\mathrm{\Upsilon }\left(x\,t\,u\,v\right)=0\,\frac{\partial }{\partial v}\mathrm{\psi }\left(x\,t\,u\,v\right)=0\right]\,\mathrm{MapDEPivs}=\left[\left(\frac{\partial }{\partial v}\mathrm{\varphi }\left(x\,t\,u\,v\right)\right)\left(\frac{\partial }{\partial u}\mathrm{\phi }\left(x\,t\,u\,v\right)\right)-\left(\frac{\partial }{\partial u}\mathrm{\varphi }\left(x\,t\,u\,v\right)\right)\left(\frac{\partial }{\partial v}\mathrm{\phi }\left(x\,t\,u\,v\right)\right)\ne 0\,\left(\frac{\partial }{\partial x}\mathrm{\psi }\left(x\,t\,u\,v\right)\right)\left(\frac{\partial }{\partial t}\mathrm{\Upsilon }\left(x\,t\,u\,v\right)\right)-\left(\frac{\partial }{\partial t}\mathrm{\psi }\left(x\,t\,u\,v\right)\right)\left(\frac{\partial }{\partial x}\mathrm{\Upsilon }\left(x\,t\,u\,v\right)\right)\ne 0\,\frac{\partial }{\partial x}\mathrm{\psi }\left(x\,t\,u\,v\right)\ne 0\right]\,\mathrm{TargetDE}=\left[\frac{\partial }{\partial \mathrm{xh}}\mathrm{vh}\left(\mathrm{xh}\,\mathrm{th}\right)=\frac{\mathrm{uh}\left(\mathrm{xh}\,\mathrm{th}\right)}{2}\,\frac{\partial }{\partial \mathrm{th}}\mathrm{vh}\left(\mathrm{xh}\,\mathrm{th}\right)=\frac{\frac{\partial }{\partial \mathrm{xh}}\mathrm{uh}\left(\mathrm{xh}\,\mathrm{th}\right)}{2}\,\frac{{\partial }^2}{\partial {\mathrm{xh}}^2}\mathrm{uh}\left(\mathrm{xh}\,\mathrm{th}\right)=\frac{\partial }{\partial \mathrm{th}}\mathrm{uh}\left(\mathrm{xh}\,\mathrm{th}\right)\right]\mathbf{where}\left[\mathrm{vh}\left(\mathrm{xh}\,\mathrm{th}\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{MapExists}=\mathrm{true}\right]\right)$

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.

The LieAlgebrasOfVectorFields[MapDE] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.