ABSTRACT

In this short section, we touch on the problem of a formation of shocks for NDEs that are higher-order in time. Instead of studying the PDEs such as (we discussed these already)

utt = −(uux)xxxx, uttt = −(uux)xxxx, etc., (495) consider the fifth-order in time NDE (379), exhibiting certain simple and yet exceptional properties. Writing it for W = (u, v, w, g, h)T as

⎧ ⎪⎪⎪⎨

⎪⎪ ⎩

ut = vx, vt = wx, wt = gx, gt = hx, ht = uux,

or Wt = AWx, A =

⎢ ⎢ ⎢ ⎣

0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 u 0 0 0 0

⎥ ⎥ ⎥ ⎦ , (496)

(379) becomes a first-order system with the characteristic equation

−λ5 + u = 0. Hence, for any u = 0, there exist complex roots, so that advanced results on hyperbolic systems [50, 92] cannot be applied.