If there wasn't that minus sign, the answer would be a wave equation. http://uniquation.com/ was a bust. I asked wolframalpha, and it came back with an answer which looked just like the wave equation with an extra factor of $i$. Does the equation have a more general name?
Name for this wave-like differential equation, $u_{tt} = -u_{xx}$
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pde
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7It is a Laplace-type equation. – 2011-10-06
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0The question came up in a 4D context, specifically: u_tt + u_xx + u_yy + u_zz = 0, so that would be a 4D Laplace equation. The inhomogeneous case would be 4D Poisson. Thanks. – 2011-10-06
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1I'd like to add that the conic representation of this is an ellipse and the function is elliptic. – 2011-10-06