ABSTRACT
We continue to introduce other types of shocks and singularities that are associated with NDE-3 (1).
uα(x, t) = (−t)αg(z), z = x/(−t)β , β = 1+α3 (α ∈ IR), (82) where g solves the ODE
A(g) ≡ (gg′)′′ = 1+α3 g′z − αg ≡ Cg in IR. (83)
and
In (82), we introduce an extra arbitrary parameter α ∈ IR. We next show that the behavior of the similarity profiles g(z) and, hence, of the corresponding solutions uα(x, t), essentially depend on whether α > 0 or α < 0. We first prove the following auxiliary result, explaining a key feature of
ODE (83):
Proposition 8.68 (i) Both nonlinearA and linearC operators in (83) admit the 4D linear invariant subspace
W4 = Span{1, z, z2, z3}. (84) (ii) ODE (83) possesses nontrivial solutions on W4 in two cases:
(I) α = αc = − 110 , with the solutions given by (85)
g(z) = C0 + C1z + 1 60 z
3, C0,1 ∈ IR are arbitrary, and (86) (II) α = −1, g(z) = 4003 C32 + 20C22z + C2z2 + 160 z3, C2 ∈ IR. (87)
Proof. (i) is straightforward, since, for any
g = C0 + C1z + C2z 2 + C3z
3 ∈ W4, (88) the following holds:
A(g) = 6(C1C2 + C0C3) + 12(C 2 2 + 2C1C3)z + 60C2C3z
2 + 60C23z 3 ∈ W4,
Cg = −αC0 + 1−2α3 C1z + 2−α3 C2z2 + C3x3 ∈ W4. (ii) According to ODE (83), equating the coefficients given in the expansion
above yields the algebraic system
⎧ ⎪⎨
⎪⎩
6(C1C2 + C0C3) = −αC0, 12(C22 + 2C1C3) =
60C2C3 = 2−α 3 C2,
60C23 = C3.
