A Mini-course on Elliptic–Hyperbolic Equations - ICMS

Making the choice σ(t) = t 2 in the preceding discussion leads toequations of Boussinesq type, which arise in hydrodynamics. Weview these as first-order systems u t + B(u)u x = 0, takingu = (h, w) T , whereB(u) =(−hw1−w 21−h222−hwThe sub-domain of hyperbolic solutions, that is, the domain ofwave motion, coincides with that region of the domain on whichthe eigenvalues of B, that is, the numbers)√(1 − hλ ± B = −hw ± 2 ) (1 − w 2 ),2are real and the associated eigenvectors span R 2 . That correspondsto the region in which |h| and |w| are both less than 1. This is asquare, not in the xt-plane but in the “phase space” of thehw-plane – a square which cannot be determined without solvingthe system..