$\mathrm{restart}\;$

$\mathrm{with}\left(\mathrm{DifferentialThomas}\right)$

$\left[\mathrm{ComplementOfDecomposition}\,\mathrm{Display}\,\mathrm{Equations}\,\mathrm{Inequations}\,\mathrm{IntersectDecompositions}\,\mathrm{LinearCombination}\,\mathrm{NormalForm}\,\mathrm{PowerSeriesSolution}\,\mathrm{Ranking}\,\mathrm{ReducedForm}\,\mathrm{ThomasDecomposition}\,\mathrm{Tools}\right]$

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

$\mathrm{Ranking}\left(\left[x\,y\right]\,\left[u\right]\right)\:$

$\mathrm{Eq1} ≔ \left[\left(\mathrm{diff}\left(u\left(x\, y\right)\, y\right)\right) \left(\mathrm{diff}\left(u\left(x\, y\right)\, x\right)\right)\+\mathrm{diff}\left(u\left(x\, y\right)\, x\right)\+1\=0\, \left(\mathrm{diff}\left(u\left(x\, y\right)\, x\, x\right)\right) u\left(x\, y\right)-{\left(\mathrm{diff}\left(u\left(x\, y\right)\, y\right)\right)}^2\+u\left(x\, y\right)\=0\right]$

$\mathrm{Eq1}≔\left[\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)\left(\frac{\partial }{\partial x}u\left(x\,y\right)\right)+\frac{\partial }{\partial x}u\left(x\,y\right)+1=0\,\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\right)\right)u\left(x\,y\right)-{\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)}^2+u\left(x\,y\right)=0\right]$

$\mathrm{TD}≔\mathrm{ThomasDecomposition}\left(\mathrm{Eq1}\right)$

$\mathrm{TD}≔\left[\right]$

$\mathrm{Eq2} ≔ \left[\left(\mathrm{diff}\left(u\left(x\, y\right)\, y\right)\right) \left(\mathrm{diff}\left(u\left(x\, y\right)\, x\right)\right)\+\mathrm{diff}\left(u\left(x\, y\right)\, x\right)\+1\, \left(\mathrm{diff}\left(u\left(x\, y\right)\, x\, x\right)\right) u\left(x\, y\right)-{\left(\mathrm{diff}\left(u\left(x\, y\right)\, y\right)\right)}^2-\left(\mathrm{diff}\left(u\left(x\, y\right)\, x\right)\right)\right]$

$\mathrm{Eq2}≔\left[\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)\left(\frac{\partial }{\partial x}u\left(x\,y\right)\right)+\frac{\partial }{\partial x}u\left(x\,y\right)+1\,\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\right)\right)u\left(x\,y\right)-{\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)}^2-\frac{\partial }{\partial x}u\left(x\,y\right)\right]$

$\mathrm{TD}≔\mathrm{ThomasDecomposition}\left(\mathrm{Eq2}\right)$

$\mathrm{TD}≔\left[\mathrm{DifferentialSystem}\right]$

$\mathrm{Equations}\left(\mathrm{TD}\right)$

$\left[\left[\frac{\partial }{\partial x}u\left(x\,y\right)=-\frac{1}{\frac{\partial }{\partial y}u\left(x\,y\right)+1}\,{\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)}^3=-{\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)}^2+1\right]\right]$

$\mathrm{Eq3} ≔ \left[\left(\mathrm{diff}\left(u\left(x\, y\right)\, y\right)\right) \left(\mathrm{diff}\left(u\left(x\, y\right)\, x\right)\right)\+\mathrm{diff}\left(u\left(x\, y\right)\, x\right)\+1\=0\, a\left(x\, y\right) u\left(x\, y\right)\+\mathrm{diff}\left(u\left(x\, y\right)\, x\, x\right) u\left(x\, y\right)-\mathrm{diff}{\left(u\left(x\, y\right)\, y\right)}^2-\mathrm{diff}\left(u\left(x\, y\right)\, x\right)\=0\, \mathrm{diff}\left(a\left(x\, y\right)\, x\right)\=0\, \mathrm{diff}\left(a\left(x\, y\right)\, y\right)\=0\right]$

$\mathrm{Eq3}≔\left[\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)\left(\frac{\partial }{\partial x}u\left(x\,y\right)\right)+\frac{\partial }{\partial x}u\left(x\,y\right)+1=0\,a\left(x\,y\right)u\left(x\,y\right)+\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,y\right)\right)u\left(x\,y\right)-{\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)}^2-\frac{\partial }{\partial x}u\left(x\,y\right)=0\,\frac{\partial }{\partial x}a\left(x\,y\right)=0\,\frac{\partial }{\partial y}a\left(x\,y\right)=0\right]$

$\mathrm{Ranking}\left(\left[x\,y\right]\,\left[\left[u\right]\,\left[a\right]\right]\right)$

$\mathrm{ranking}$

$\mathrm{TD}≔\mathrm{ThomasDecomposition}\left(\mathrm{Eq3}\right)$

$\mathrm{TD}≔\left[\mathrm{DifferentialSystem}\right]$

$\mathrm{Equations}\left(\mathrm{TD}\left[1\right]\right)$

$\left[{\left(\frac{\partial }{\partial x}u\left(x\,y\right)\right)}^3=-{\left(\frac{\partial }{\partial x}u\left(x\,y\right)\right)}^2-2\frac{\partial }{\partial x}u\left(x\,y\right)-1\,\frac{\partial }{\partial x}u\left(x\,y\right)=-\frac{1}{\frac{\partial }{\partial y}u\left(x\,y\right)+1}\,a\left(x\,y\right)=0\right]$

$\mathrm{Inequations}\left(\mathrm{TD}\left[1\right]\right)$

$\left[u\left(x\,y\right)\ne 0\right]$

$\mathrm{Display}\left(\mathrm{TD}\left[1\right]\right)$

$\left[-{\left(\frac{\partial }{\partial x}u\left(x\,y\right)\right)}^3-{\left(\frac{\partial }{\partial x}u\left(x\,y\right)\right)}^2-2\frac{\partial }{\partial x}u\left(x\,y\right)-1=0\,\left(\frac{\partial }{\partial y}u\left(x\,y\right)\right)\left(\frac{\partial }{\partial x}u\left(x\,y\right)\right)+\frac{\partial }{\partial x}u\left(x\,y\right)+1=0\,a\left(x\,y\right)=0\,u\left(x\,y\right)\ne 0\right]$

$\mathrm{ivar}≔\left[\left[t\right]\,\left[x\right]\right]$

$\mathrm{ivar}≔\left[\left[t\right]\,\left[x\right]\right]$

$\mathrm{dvar}≔\left[\left[u\,v\,w\right]\right]$

$\mathrm{dvar}≔\left[\left[u\,v\,w\right]\right]$

$\mathrm{Ranking}\left(\mathrm{ivar}\,\mathrm{dvar}\right)$

$\mathrm{ranking}$

$\mathrm{Ranking}\left(\right)$

The current Ranking for the independent variables [t, x] and the dependent variables [u, v, w] is:
Matrixordering with Matrix(5, 5, [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,1,1],[0,0,0,0,-1],[0,0,0,-1,0]])

$\mathrm{ranking}$

$L ≔ \frac{1}{2} {\left(\mathrm{diff}\left(u\left(t\, x\right)\, t\right)\right)}^2\+\mathrm{diff}{\left(v\left(t\, x\right)\, t\right)}^2 u\left(t\, x\right) \mathrm{diff}\left(w\left(t\, x\right)\, x\right)\+\mathrm{diff}\left(w\left(t\, x\right)\, t\right) \left(\mathrm{diff}\left(u\left(t\, x\right)\, t\right)\+\mathrm{diff}\left(v\left(t\, x\right)\, x\right)\right)$

$L≔\frac{{\left(\frac{\partial }{\partial t}u\left(t\,x\right)\right)}^2}{2}+{\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)}^{2}u\left(t\,x\right)\left(\frac{\partial }{\partial x}w\left(t\,x\right)\right)+\left(\frac{\partial }{\partial t}w\left(t\,x\right)\right)\left(\frac{\partial }{\partial t}u\left(t\,x\right)+\frac{\partial }{\partial x}v\left(t\,x\right)\right)$

$$\begin{gathered} \mathrm{EL} ≔ \left[-\mathrm{diff}{\left(v\left(t\, x\right)\, t\right)}^2 \mathrm{diff}\left(w\left(t\, x\right)\, x\right)\+\mathrm{diff}\left(u\left(t\, x\right)\, t\, t\right)\+\mathrm{diff}\left(w\left(t\, x\right)\, t\, t\right) \= 0\, \\ 2 \mathrm{diff}\left(v\left(t\, x\right)\, t\right) u\left(t\, x\right) \mathrm{diff}\left(w\left(t\, x\right)\, t\, x\right)\+2 \mathrm{diff}\left(v\left(t\, x\right)\, t\, t\right) u\left(t\, x\right) \mathrm{diff}\left(w\left(t\, x\right)\, x\right)\+2 \mathrm{diff}\left(v\left(t\, x\right)\, t\right) \mathrm{diff}\left(u\left(t\, x\right)\, t\right) \mathrm{diff}\left(w\left(t\, x\right)\, x\right)\+\mathrm{diff}\left(w\left(t\, x\right)\, t\, x\right)\=0\, \\ 2 \mathrm{diff}\left(v\left(t\, x\right)\, t\, x\right) \mathrm{diff}\left(v\left(t\, x\right)\, t\right) u\left(t\, x\right)\+\mathrm{diff}{\left(v\left(t\, x\right)\, t\right)}^2 \mathrm{diff}\left(u\left(t\, x\right)\, x\right)\+\mathrm{diff}\left(u\left(t\, x\right)\, t\, t\right)\+\mathrm{diff}\left(v\left(t\, x\right)\, t\, x\right)\=0\right] \end{gathered}$$

$\mathrm{EL}≔\left[-{\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)}^2\left(\frac{\partial }{\partial x}w\left(t\,x\right)\right)+\frac{{\partial }^2}{\partial t^2}u\left(t\,x\right)+\frac{{\partial }^2}{\partial t^2}w\left(t\,x\right)=0\,2\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)u\left(t\,x\right)\left(\frac{{\partial }^2}{\partial t\partial x}w\left(t\,x\right)\right)+2\left(\frac{{\partial }^2}{\partial t^2}v\left(t\,x\right)\right)u\left(t\,x\right)\left(\frac{\partial }{\partial x}w\left(t\,x\right)\right)+2\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)\left(\frac{\partial }{\partial t}u\left(t\,x\right)\right)\left(\frac{\partial }{\partial x}w\left(t\,x\right)\right)+\frac{{\partial }^2}{\partial t\partial x}w\left(t\,x\right)=0\,2\left(\frac{{\partial }^2}{\partial t\partial x}v\left(t\,x\right)\right)\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)u\left(t\,x\right)+{\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)}^2\left(\frac{\partial }{\partial x}u\left(t\,x\right)\right)+\frac{{\partial }^2}{\partial t^2}u\left(t\,x\right)+\frac{{\partial }^2}{\partial t\partial x}v\left(t\,x\right)=0\right]$

$\mathrm{TD}≔\mathrm{ThomasDecomposition}\left(\mathrm{EL}\right)$

$\mathrm{TD}≔\left[\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\right]$

$\mathrm{Display}\left(\mathrm{TD}\left[1\right]\right)$

$\left[-2\left(\frac{{\partial }^2}{\partial t\partial x}v\left(t\,x\right)\right)\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)u\left(t\,x\right)-{\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)}^2\left(\frac{\partial }{\partial x}u\left(t\,x\right)\right)-\frac{{\partial }^2}{\partial t^2}u\left(t\,x\right)-\frac{{\partial }^2}{\partial t\partial x}v\left(t\,x\right)=0\,2\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)u\left(t\,x\right)\left(\frac{{\partial }^2}{\partial t\partial x}w\left(t\,x\right)\right)+2\left(\frac{{\partial }^2}{\partial t^2}v\left(t\,x\right)\right)u\left(t\,x\right)\left(\frac{\partial }{\partial x}w\left(t\,x\right)\right)+2\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)\left(\frac{\partial }{\partial t}u\left(t\,x\right)\right)\left(\frac{\partial }{\partial x}w\left(t\,x\right)\right)+\frac{{\partial }^2}{\partial t\partial x}w\left(t\,x\right)=0\,2\left(\frac{{\partial }^2}{\partial t\partial x}v\left(t\,x\right)\right)\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)u\left(t\,x\right)+{\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)}^2\left(\frac{\partial }{\partial x}u\left(t\,x\right)\right)+{\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)}^2\left(\frac{\partial }{\partial x}w\left(t\,x\right)\right)+\frac{{\partial }^2}{\partial t\partial x}v\left(t\,x\right)-\frac{{\partial }^2}{\partial t^2}w\left(t\,x\right)=0\,\frac{\partial }{\partial x}w\left(t\,x\right)\ne 0\,u\left(t\,x\right)\ne 0\right]$

$\mathrm{Display}\left(\mathrm{TD}\left[3\right]\right)$

$\left[u\left(t\,x\right)=0\,\frac{{\partial }^2}{\partial t\partial x}v\left(t\,x\right)=0\,-\frac{{\partial }^2}{\partial t^2}w\left(t\,x\right)=0\,\frac{{\partial }^2}{\partial t\partial x}w\left(t\,x\right)=0\,\frac{\partial }{\partial x}w\left(t\,x\right)=0\right]$

$\mathrm{Display}\left(\mathrm{TD}\left[4\right]\right)$

$\left[-\frac{{\partial }^2}{\partial t^2}u\left(t\,x\right)=0\,\frac{{\partial }^2}{\partial t\partial x}u\left(t\,x\right)=0\,\frac{\partial }{\partial x}u\left(t\,x\right)=0\,2\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)u\left(t\,x\right)+1=0\,-\frac{{\partial }^2}{\partial t^2}w\left(t\,x\right)=0\,\frac{{\partial }^2}{\partial t\partial x}w\left(t\,x\right)=0\,\frac{\partial }{\partial x}w\left(t\,x\right)=0\,u\left(t\,x\right)\ne 0\right]$

$\mathrm{Display}\left(\mathrm{TD}\left[5\right]\right)$

$\left[u\left(t\,x\right)=0\,\frac{\partial }{\partial t}v\left(t\,x\right)=0\,-\frac{{\partial }^2}{\partial t^2}w\left(t\,x\right)=0\,\frac{{\partial }^2}{\partial t\partial x}w\left(t\,x\right)=0\,\frac{\partial }{\partial x}w\left(t\,x\right)\ne 0\right]$

$\mathrm{Display}\left(\mathrm{TD}\left[6\right]\right)$

$\left[u\left(t\,x\right)=0\,\frac{{\partial }^2}{\partial t\partial x}v\left(t\,x\right)=0\,{\left(\frac{\partial }{\partial t}v\left(t\,x\right)\right)}^2\left(\frac{\partial }{\partial x}w\left(t\,x\right)\right)-\frac{{\partial }^2}{\partial t^2}w\left(t\,x\right)=0\,\frac{{\partial }^2}{\partial t\partial x}w\left(t\,x\right)=0\,\frac{{\partial }^2}{\partial x^2}w\left(t\,x\right)=0\,\frac{\partial }{\partial x}w\left(t\,x\right)\ne 0\,\frac{\partial }{\partial t}v\left(t\,x\right)\ne 0\right]$

$\mathrm{Ranking}\left(\left[t\right]\,\left[\mathrm{q1}\,\mathrm{q2}\right]\right)$

$\mathrm{ranking}$

$L ≔ \mathrm{q1}{\left(t\right)}^2 \mathrm{diff}{\left(\mathrm{q2}\left(t\right)\, t\right)}^2\+2 \mathrm{q1}\left(t\right) \mathrm{q2}\left(t\right) \mathrm{diff}\left(\mathrm{q1}\left(t\right)\, t\right) \mathrm{diff}\left(\mathrm{q2}\left(t\right)\, t\right)\+\mathrm{q2}{\left(t\right)}^2 \mathrm{diff}{\left(\mathrm{q1}\left(t\right)\, t\right)}^2\+\mathrm{q1}{\left(t\right)}^2\+\mathrm{q2}{\left(t\right)}^2$

$L≔{\mathrm{q1}\left(t\right)}^2{\left(\frac{\ⅆ}{\ⅆt}\mathrm{q2}\left(t\right)\right)}^2+2\mathrm{q1}\left(t\right)\mathrm{q2}\left(t\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{q1}\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{q2}\left(t\right)\right)+{\mathrm{q2}\left(t\right)}^2{\left(\frac{\ⅆ}{\ⅆt}\mathrm{q1}\left(t\right)\right)}^2+{\mathrm{q1}\left(t\right)}^2+{\mathrm{q2}\left(t\right)}^2$

$$\begin{gathered} \mathrm{EL} ≔ \left[2 \mathrm{q1}\left(t\right) \mathrm{q2}\left(t\right) \mathrm{diff}\left(\mathrm{q2}\left(t\right)\, t\, t\right)\+4 \mathrm{diff}\left(\mathrm{q1}\left(t\right)\, t\right) \mathrm{q2}\left(t\right) \mathrm{diff}\left(\mathrm{q2}\left(t\right)\, t\right)\+2 \mathrm{diff}\left(\mathrm{q1}\left(t\right)\, t\, t\right) \mathrm{q2}{\left(t\right)}^2-2 \mathrm{q1}\left(t\right)\=0\, \\ 2 \mathrm{q1}{\left(t\right)}^2 \mathrm{diff}\left(\mathrm{q2}\left(t\right)\, t\, t\right)\+4 \mathrm{diff}\left(\mathrm{q1}\left(t\right)\, t\right) \mathrm{q1}\left(t\right) \mathrm{diff}\left(\mathrm{q2}\left(t\right)\, t\right)\+2 \mathrm{q1}\left(t\right) \mathrm{q2}\left(t\right) \mathrm{diff}\left(\mathrm{q1}\left(t\right)\, t\, t\right)-2 \mathrm{q2}\left(t\right)\=0\right] \end{gathered}$$

$\mathrm{EL}≔\left[2\mathrm{q1}\left(t\right)\mathrm{q2}\left(t\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{q2}\left(t\right)\right)+4\left(\frac{\ⅆ}{\ⅆt}\mathrm{q1}\left(t\right)\right)\mathrm{q2}\left(t\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{q2}\left(t\right)\right)+2\left(\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{q1}\left(t\right)\right){\mathrm{q2}\left(t\right)}^2-2\mathrm{q1}\left(t\right)=0\,2{\mathrm{q1}\left(t\right)}^2\left(\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{q2}\left(t\right)\right)+4\left(\frac{\ⅆ}{\ⅆt}\mathrm{q1}\left(t\right)\right)\mathrm{q1}\left(t\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{q2}\left(t\right)\right)+2\mathrm{q1}\left(t\right)\mathrm{q2}\left(t\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{q1}\left(t\right)\right)-2\mathrm{q2}\left(t\right)=0\right]$

$\mathrm{TD}≔\mathrm{ThomasDecomposition}\left(\mathrm{EL}\right)$

$\mathrm{TD}≔\left[\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\,\mathrm{DifferentialSystem}\right]$

$\mathrm{Display}\left(\mathrm{TD}\right)$

$\left[\left[\mathrm{q1}\left(t\right)+\mathrm{q2}\left(t\right)=0\,2\mathrm{q2}\left(t\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{q2}\left(t\right)\right)+2{\left(\frac{\ⅆ}{\ⅆt}\mathrm{q2}\left(t\right)\right)}^2-1=0\,\mathrm{q2}\left(t\right)\ne 0\right]\,\left[\mathrm{q1}\left(t\right)-\mathrm{q2}\left(t\right)=0\,2\mathrm{q2}\left(t\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{q2}\left(t\right)\right)+2{\left(\frac{\ⅆ}{\ⅆt}\mathrm{q2}\left(t\right)\right)}^2-1=0\,\mathrm{q2}\left(t\right)\ne 0\right]\,\left[\mathrm{q1}\left(t\right)=0\,\mathrm{q2}\left(t\right)=0\right]\right]$

$\mathrm{Burgers}≔\mathrm{diff}\left(u\left(x\,t\right)\,t\right)\+u\left(x\,t\right) \mathrm{diff}\left(u\left(x\,t\right)\,x\right)-\mathrm{diff}\left(u\left(x\,t\right)\,x\,x\right)\;$

$\mathrm{Burgers}≔\frac{\partial }{\partial t}u\left(x\,t\right)+u\left(x\,t\right)\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)-\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)$

$\mathrm{WTC\_Bur}≔\mathrm{subs}\left(u\left(x\,t\right)=\frac{\mathrm{add}\left(v\Vert j\left(x\,t\right){f\left(x\,t\right)}^j\,j=0..3\right)}{f\left(x\,t\right)}\,\mathrm{Burgers}\right)$

$\mathrm{WTC\_Bur}≔\frac{\partial }{\partial t}\left(\frac{\mathrm{v0}\left(x\,t\right)+\mathrm{v1}\left(x\,t\right)f\left(x\,t\right)+\mathrm{v2}\left(x\,t\right){f\left(x\,t\right)}^2+\mathrm{v3}\left(x\,t\right){f\left(x\,t\right)}^3}{f\left(x\,t\right)}\right)+\frac{\left(\mathrm{v0}\left(x\,t\right)+\mathrm{v1}\left(x\,t\right)f\left(x\,t\right)+\mathrm{v2}\left(x\,t\right){f\left(x\,t\right)}^2+\mathrm{v3}\left(x\,t\right){f\left(x\,t\right)}^3\right)\left(\frac{\partial }{\partial x}\left(\frac{\mathrm{v0}\left(x\,t\right)+\mathrm{v1}\left(x\,t\right)f\left(x\,t\right)+\mathrm{v2}\left(x\,t\right){f\left(x\,t\right)}^2+\mathrm{v3}\left(x\,t\right){f\left(x\,t\right)}^3}{f\left(x\,t\right)}\right)\right)}{f\left(x\,t\right)}-\frac{{\partial }^2}{\partial x^2}\left(\frac{\mathrm{v0}\left(x\,t\right)+\mathrm{v1}\left(x\,t\right)f\left(x\,t\right)+\mathrm{v2}\left(x\,t\right){f\left(x\,t\right)}^2+\mathrm{v3}\left(x\,t\right){f\left(x\,t\right)}^3}{f\left(x\,t\right)}\right)$

$\mathrm{Num}≔\mathrm{numer}\left(\mathrm{WTC\_Bur}\right)$

$\mathrm{Num}≔{f\left(x\,t\right)}^2\mathrm{v1}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v0}\left(x\,t\right)\right)+{f\left(x\,t\right)}^2\mathrm{v0}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v1}\left(x\,t\right)\right)-f\left(x\,t\right)\left(\frac{\partial }{\partial t}f\left(x\,t\right)\right)\mathrm{v0}\left(x\,t\right)+f\left(x\,t\right)\mathrm{v0}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v0}\left(x\,t\right)\right)+2f\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v0}\left(x\,t\right)\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+f\left(x\,t\right)\mathrm{v0}\left(x\,t\right)\left(\frac{{\partial }^2}{\partial x^2}f\left(x\,t\right)\right)-2{f\left(x\,t\right)}^4\mathrm{v3}\left(x\,t\right)\left(\frac{{\partial }^2}{\partial x^2}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^7\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)+2{f\left(x\,t\right)}^6{\mathrm{v3}\left(x\,t\right)}^2\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^6\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)+{f\left(x\,t\right)}^6\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)+{f\left(x\,t\right)}^5\mathrm{v1}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)+{f\left(x\,t\right)}^5\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)+{f\left(x\,t\right)}^5\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v1}\left(x\,t\right)\right)+{f\left(x\,t\right)}^4{\mathrm{v2}\left(x\,t\right)}^2\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^4\mathrm{v1}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)+2{f\left(x\,t\right)}^4\left(\frac{\partial }{\partial t}f\left(x\,t\right)\right)\mathrm{v3}\left(x\,t\right)+{f\left(x\,t\right)}^4\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v1}\left(x\,t\right)\right)+{f\left(x\,t\right)}^4\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v0}\left(x\,t\right)\right)+{f\left(x\,t\right)}^4\mathrm{v0}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)-4{f\left(x\,t\right)}^4\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)-2{f\left(x\,t\right)}^3\mathrm{v3}\left(x\,t\right){\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)}^2+{f\left(x\,t\right)}^3\mathrm{v1}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v1}\left(x\,t\right)\right)+{f\left(x\,t\right)}^3\left(\frac{\partial }{\partial t}f\left(x\,t\right)\right)\mathrm{v2}\left(x\,t\right)+{f\left(x\,t\right)}^3\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v0}\left(x\,t\right)\right)+{f\left(x\,t\right)}^3\mathrm{v0}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)-2{f\left(x\,t\right)}^3\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)-{f\left(x\,t\right)}^3\mathrm{v2}\left(x\,t\right)\left(\frac{{\partial }^2}{\partial x^2}f\left(x\,t\right)\right)+3{f\left(x\,t\right)}^5\mathrm{v2}\left(x\,t\right)\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+2{f\left(x\,t\right)}^4\mathrm{v1}\left(x\,t\right)\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^3\mathrm{v1}\left(x\,t\right)\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^3\mathrm{v3}\left(x\,t\right)\mathrm{v0}\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)-f\left(x\,t\right)\mathrm{v1}\left(x\,t\right)\mathrm{v0}\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^5\left(\frac{\partial }{\partial t}\mathrm{v3}\left(x\,t\right)\right)+{f\left(x\,t\right)}^4\left(\frac{\partial }{\partial t}\mathrm{v2}\left(x\,t\right)\right)+\left(\frac{\partial }{\partial t}\mathrm{v1}\left(x\,t\right)\right){f\left(x\,t\right)}^3+{f\left(x\,t\right)}^2\left(\frac{\partial }{\partial t}\mathrm{v0}\left(x\,t\right)\right)-{\mathrm{v0}\left(x\,t\right)}^2\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)-2\mathrm{v0}\left(x\,t\right){\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)}^2-{f\left(x\,t\right)}^5\left(\frac{{\partial }^2}{\partial x^2}\mathrm{v3}\left(x\,t\right)\right)-{f\left(x\,t\right)}^4\left(\frac{{\partial }^2}{\partial x^2}\mathrm{v2}\left(x\,t\right)\right)-{f\left(x\,t\right)}^3\left(\frac{{\partial }^2}{\partial x^2}\mathrm{v1}\left(x\,t\right)\right)-{f\left(x\,t\right)}^2\left(\frac{{\partial }^2}{\partial x^2}\mathrm{v0}\left(x\,t\right)\right)$

$P≔\mathrm{convert}\left(\mathrm{Num}\,\mathrm{D}\right)\:$

$\mathrm{subs}\left(f\left(x\,t\right)=0\,P\right)$

$-{\mathrm{v0}\left(x\,t\right)}^2{\mathrm{D}}_1\left(f\right)\left(x\,t\right)-2\mathrm{v0}\left(x\,t\right){{\mathrm{D}}_1\left(f\right)\left(x\,t\right)}^2$

$\mathrm{sol\_v0}≔\mathrm{solve}\left(\left\{\mathrm{v0}\left(x\,t\right)\ne 0\,=0\right\}\,\mathrm{v0}\left(x\,t\right)\right)$

$\mathrm{sol\_v0}≔\left\{\mathrm{v0}\left(x\,t\right)=-2{\mathrm{D}}_1\left(f\right)\left(x\,t\right)\right\}$

$\mathrm{convert}\left(\mathrm{op}\left(\mathrm{sol\_v0}\right)\,\mathrm{diff}\right)$

$\mathrm{v0}\left(x\,t\right)=-2\frac{\partial }{\partial x}f\left(x\,t\right)$

$P≔\mathrm{expand}\left(\mathrm{subs}\left(\mathrm{convert}\left(\mathrm{op}\left(\mathrm{sol\_v0}\right)\,\mathrm{diff}\right)\,\mathrm{Num}\right)\right)$

$P≔2{f\left(x\,t\right)}^2\left(\frac{{\partial }^3}{\partial x^3}f\left(x\,t\right)\right)-2{f\left(x\,t\right)}^2\left(\frac{{\partial }^2}{\partial t\partial x}f\left(x\,t\right)\right)-4{f\left(x\,t\right)}^4\mathrm{v3}\left(x\,t\right)\left(\frac{{\partial }^2}{\partial x^2}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^7\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)+2{f\left(x\,t\right)}^6{\mathrm{v3}\left(x\,t\right)}^2\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^6\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)+{f\left(x\,t\right)}^6\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)+{f\left(x\,t\right)}^5\mathrm{v1}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)+{f\left(x\,t\right)}^5\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)+{f\left(x\,t\right)}^5\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v1}\left(x\,t\right)\right)+{f\left(x\,t\right)}^4{\mathrm{v2}\left(x\,t\right)}^2\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^4\mathrm{v1}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)+2{f\left(x\,t\right)}^4\left(\frac{\partial }{\partial t}f\left(x\,t\right)\right)\mathrm{v3}\left(x\,t\right)+{f\left(x\,t\right)}^4\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v1}\left(x\,t\right)\right)-6{f\left(x\,t\right)}^4\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)\left(\frac{\partial }{\partial x}\mathrm{v3}\left(x\,t\right)\right)-4{f\left(x\,t\right)}^3\mathrm{v3}\left(x\,t\right){\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)}^2+{f\left(x\,t\right)}^3\mathrm{v1}\left(x\,t\right)\left(\frac{\partial }{\partial x}\mathrm{v1}\left(x\,t\right)\right)+{f\left(x\,t\right)}^3\left(\frac{\partial }{\partial t}f\left(x\,t\right)\right)\mathrm{v2}\left(x\,t\right)-4{f\left(x\,t\right)}^3\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)\left(\frac{\partial }{\partial x}\mathrm{v2}\left(x\,t\right)\right)-3{f\left(x\,t\right)}^3\mathrm{v2}\left(x\,t\right)\left(\frac{{\partial }^2}{\partial x^2}f\left(x\,t\right)\right)-2{f\left(x\,t\right)}^2\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)\left(\frac{\partial }{\partial x}\mathrm{v1}\left(x\,t\right)\right)+2f\left(x\,t\right)\left(\frac{\partial }{\partial t}f\left(x\,t\right)\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+2f\left(x\,t\right)\mathrm{v1}\left(x\,t\right){\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)}^2-2f\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)\left(\frac{{\partial }^2}{\partial x^2}f\left(x\,t\right)\right)-2{f\left(x\,t\right)}^2\mathrm{v1}\left(x\,t\right)\left(\frac{{\partial }^2}{\partial x^2}f\left(x\,t\right)\right)+3{f\left(x\,t\right)}^5\mathrm{v2}\left(x\,t\right)\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+2{f\left(x\,t\right)}^4\mathrm{v1}\left(x\,t\right)\mathrm{v3}\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^3\mathrm{v1}\left(x\,t\right)\mathrm{v2}\left(x\,t\right)\left(\frac{\partial }{\partial x}f\left(x\,t\right)\right)+{f\left(x\,t\right)}^5\left(\frac{\partial }{\partial t}\mathrm{v3}\left(x\,t\right)\right)+{f\left(x\,t\right)}^4\left(\frac{\partial }{\partial t}\mathrm{v2}\left(x\,t\right)\right)+\left(\frac{\partial }{\partial t}\mathrm{v1}\left(x\,t\right)\right){f\left(x\,t\right)}^3-{f\left(x\,t\right)}^5\left(\frac{{\partial }^2}{\partial x^2}\mathrm{v3}\left(x\,t\right)\right)-{f\left(x\,t\right)}^4\left(\frac{{\partial }^2}{\partial x^2}\mathrm{v2}\left(x\,t\right)\right)-{f\left(x\,t\right)}^3\left(\frac{{\partial }^2}{\partial x^2}\mathrm{v1}\left(x\,t\right)\right)$