In parabolic or hyperbolic PDEs, solutions which remain uniformly bounded for all real times
$$t=r\in \mathbb {R}$$
t
=
r
∈
R
are often called PDE entire or eternal. For a nonlinear example, consider the quadratic parabolic PDE
$$\begin{aligned} w_t=w_{xx}+6w^2-\lambda , \end{aligned}$$
w
t
=
w
xx
+
6
w
2
-
λ
,
for
$$0<x<\tfrac{1}{2}$$
0
<
x
<
1
2
, under Neumann boundary conditions. By its gradient-like structure, all real eternal non-equilibrium orbits
$$\Gamma (r)$$
Γ
(
r
)
of (*) are heteroclinic among equilibria
$$w=W_n(x)$$
w
=
W
n
(
x
)
. For parameters
$$\lambda >0$$
λ
>
0
, the trivial homogeneous equilibria are locally asymptotically stable
$$W_0=-\sqrt{\lambda /6}$$
W
0
=
-
λ
/
6
, and
$$W_\infty =+\sqrt{\lambda /6}$$
W
∞
=
+
λ
/
6
of unstable dimension (Morse index)
$$i(W_\infty )=1,2,3,\ldots $$
i
(
W
∞
)
=
1
,
2
,
3
,
…
, depending on
$$\lambda $$
λ
. All nontrivial real
$$W_n$$
W
n
are rescaled and properly translated real-valued Weierstrass elliptic functions with Morse index
$$i(W_n)=n$$
i
(
W
n
)
=
n
. We show that the complex time extensions
$$\Gamma (r+\textrm{i}s)$$
Γ
(
r
+
i
s
)
, of analytic real heteroclinic orbits
$$\Gamma (r)$$
Γ
(
r
)
towards
$$W_0$$
W
0
, are not complex entire. For example, consider the time-reversible complex-valued solution
$$\psi (s)=\Gamma (r_0-\textrm{i}s)$$
ψ
(
s
)
=
Γ
(
r
0
-
i
s
)
of the nonlinear and nonconservative quadratic Schrödinger equation
$$\begin{aligned} \textrm{i}\psi _s=\psi _{xx}+6\psi ^2-\lambda \end{aligned}$$
i
ψ
s
=
ψ
xx
+
6
ψ
2
-
λ
with real initial condition
$$\psi _0=\Gamma (r_0)$$
ψ
0
=
Γ
(
r
0
)
. Then there exist real
$$r_0$$
r
0
such that
$$\psi (s)$$
ψ
(
s
)
blows up at some finite real times
$$\pm s^*\ne 0$$
±
s
∗
≠
0
. Abstractly, our results are formulated in the setting of analytic semigroups. They are based on Poincaré non-resonance of unstable eigenvalues at equilibria
$$W_n$$
W
n
, near pitchfork bifurcation. Technically, we have to except discrete sets of parameters
$$\lambda $$
λ
, and are currently limited to unstable dimensions
$$i(W_n)\le 22$$
i
(
W
n
)
≤
22
, or to fast unstable manifolds of dimensions
$$d<1+\tfrac{1}{\sqrt{2}}i(W_n)$$
d
<
1
+
1
2
i
(
W
n
)
.
