Strange DSolve behavior for PDE solution - Bug?

• To: mathgroup at smc.vnet.net
• Subject: [mg102353] Strange DSolve behavior for PDE solution - Bug?
• From: Tom <th.meurer at gmail.com>
• Date: Thu, 6 Aug 2009 06:32:39 -0400 (EDT)

```Hello,

I am trying to solve the following system of 1st order PDEs

dfun = {2*D[x1[x, y], x] - 2*D[x1[x, y], y] + a*x2[x, y] +
b*x4[x, y] == 0,
2*D[x2[x, y], x] - 2*D[x2[x, y], y] + a*x1[x, y] - b*x3[x, y] == 0,
2*D[x3[x, y], x] + 2*D[x3[x, y], y] - b*x2[x, y] - a*x4[x, y] == 0,
2*D[x4[x, y], x] + 2*D[x4[x, y], y] + b*x1[x, y] - a*x3[x, y] == 0}

running

DSolve[dfun, {x1, x2, x3, x4}, {x, y}]

Interestingly Mathematica provides a set of symbolic solutions. However,
substituting these solutions back into the PDE yields for example

FullSimplify[dfun /. %[[1]]]

{a C[7] + b C[13] +
b C[23] Sech[(x - y) C[2] - C[3]] Tanh[(x - y) C[2] - C[3]] == 0,
a C[4] == b C[10],
b C[7] + a C[13] +
a C[23] Sech[(x - y) C[2] - C[3]] Tanh[(x - y) C[2] - C[3]] == 0,
b C[4] == a C[10]}

which obviously implies that the determined solutions do not satisfy
the PDEs. Once replacing the values of a and b in the PDEs by numbers,
DSolve does not return a "solution". Any clues or is this a bug? When
running the DSolve command above I am in particularly concerned about
the rather large number of integration constants that are introduced.

Thanks, Thomas

dfun = {2*D[x1[x, y], x] -
2*D[x1[x, y], y] + \[Alpha]*x2[x, y] + \[Gamma]*x4[x, y] == 0,
2*D[x2[x, y], x] -
2*D[x2[x, y], y] + \[Alpha]*x1[x, y] - \[Gamma]*x3[x, y] == 0,
2*D[x3[x, y], x] +
2*D[x3[x, y], y] - \[Gamma]*x2[x, y] - \[Alpha]*x4[x, y] == 0,
2*D[x4[x, y], x] +
2*D[x4[x, y], y] + \[Gamma]*x1[x, y] - \[Alpha]*x3[x, y] ==
0} /. {\[Alpha] -> AA, \[Gamma] -> BB};

```