ABSTRACT
We are already familiar with our basic higher-order NDEs, which we are going to study more systematically, together with some new equations. Namely, these new models (we called such equations NDE-(k, l)) are ordered by numbers of derivatives inside (k) and outside (l) the quadratic differential operators involved on the right-hand sides:
ut = −uuxxxxx ( NDE-(5, 0)
) , (370)
ut = −(uuxxxx)x ( NDE-(4, 1)
) , (371)
ut = −(uuxxx)xx ( NDE-(3, 2)
) , (372)
ut = −(uuxx)xxx ( NDE-(2, 3)
) , (373)
ut = −(uux)xxxx ( NDE-(1, 4)
) . (374)
The only fully divergence operator is in the last NDE-(1, 4):
ut = −(uux)xxxx ≡ − 12 (u2)xxxxx ( NDE-(1, 4) = NDE-(0, 5)
) . (375)
This is also the NDE-(0, 5), or simply the NDE-5. This completes the list of such quasilinear degenerate PDEs under consideration.
