Real eternal PDE solutions are not complex entire: a quadratic parabolic example

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