3.3 Higher Order PDEs
109
*11. Classify the symmetries of nonlinear wave equation ) ( utt = c2 (u)ux x , and the linear wave equation utt = c2 (x)uxx (see Ames et al. [14] and Bluman and Kumei [5]).
3.3 HIGHER ORDER PDEs Previously, we constructed the symmetries of second-order PDEs. We now extend symmetries to higher order equations. Consider the general nth order PDE ) ( Δ t, x, u, ut , ux , utt , utx , uxx , ‌ , ut(n) , ¡ ¡ ¡ , ut(n−i)x(i) , ¡ ¡ ¡ ux(n) = 0, where for convenience, we have denoted ut(n−i)x(i) = đ?œ•tn−i đ?œ•xi u. We again introduce the infinitesimal operator Γ as đ?œ• đ?œ• đ?œ• +X +U , đ?œ•t đ?œ•x đ?œ•u where T = T(t, x, u), X = X (t, x, u) and U = U(t, x, u) are to be determined. We define the nth extension to the operator Γ as Γ(n) , given recursively by Γ=T
(n)
Γ
=Γ
(n−1)
+
n ∑
U[t(n−i)x(i)]
i=0
đ?œ•n . đ?œ•tn−i đ?œ•xi
Lie’s invariance condition becomes | = 0, Γ(n) Δ| |Δ=0 where the extended transformations are given as U[t] = Dt (U) − ut Dt (T) − ux Dt (X ), U[x] = Dx (U) − ut Dx (T) − ux Dx (X ), ⋎ U[t(n−i)x(i)] = Dt (U[t(n−i−1)x(i)] ) − u[t(n−i)x(i)] Dt (T) −u[t(n−i−1)x(i+1)] Dt (X ) or,
(3.84)