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My problem is as follows: I have two matrices $P$ and $Q$ which are made up of algebraic expressions of a function $q(x,t)$ which solves a somewhat complicated pde, and its derivatives, for example, the lower left coordinate of $P$ is given by $z^2+\frac12zq-\frac18q_{xx}+\frac38q^2$ where $z$ is a complex parameter.

All I want to do is verify that $P$ and $Q$ solve the following PDE,

$P_x-Q_t+QP-PQ=0$

Given that $q$ satisfies the second PDE.

Any help is greatly appreciated, I have very little experience with mathematica.

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  • $\begingroup$ Look up D. For further details, we to see the precise definition of P and Q in copyable Mathematica code. $\endgroup$ – Henrik Schumacher Apr 28 at 18:10
  • $\begingroup$ @HenrikSchumacher that's fair, I didn't want to just post my problem with the expectation that others do it. Do you have any tips on how to use the "Assume" function to let Mathematica know $q$ solves a differential equation? thank you for your response. I know how to compute with $D$ if that helps $\endgroup$ – qbert Apr 28 at 18:13
  • $\begingroup$ Hm. You could try to replace D[q,x,x] by minus the rest of the PDE. Then the remaining terms may cancel out. $\endgroup$ – Henrik Schumacher Apr 28 at 18:16
  • $\begingroup$ @HenrikSchumacher ah, that is a nice idea but unfortunately the pde is 5th order in space and all the terms in $P$ and $Q$ are lower order in space. I can add details $\endgroup$ – qbert Apr 28 at 18:20
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    $\begingroup$ You're welcome. I hope it works out for you. $\endgroup$ – Henrik Schumacher Apr 28 at 18:34
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Turning my comments into an answer.

You could try something along the lines of

Expand[ 
 D[P, x] - D[Q, t] + Q.P - P.Q 
 /. 
 {"one term of the PDE for q" -> -"rest of the PDE for q"} 
 ]

where the strings have to be replaced by appropriate expressions.

For that, is is important that the dependencies of q on x and t are explicit, i.e., you have to write

z^2 + 1/2 z q[x,t] - 1/8 D[q[x,t],x,x] +3/8 q[x,t]^2

and so on.

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