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

Abstract

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-\lambda, \tag{*} \end{equation*} for $0<x<\tfrac{1}{2}$, under Neumann boundary conditions. By its gradient-like structure, all real eternal non-equilibrium orbits $\Gamma(r)$ 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 $\Gamma(r+\mathrm{i}s)$, of analytic real heteroclinic orbits towards $W_0=-\sqrt{\lambda/6}$, are not complex entire. For example, consider the time-reversible complex-valued solution $\psi(s)$ of the nonlinear and nonconservative quadratic Schr\"odinger equation \begin{equation*} \label{**} \mathrm{i}\psi_s=\psi_{xx}+6\psi^2-\lambda \tag{**} \end{equation*} with real initial condition $\psi_0=\Gamma(r_0)$. Then there exist $r_0$ such that $\psi(s)$ blows up at some finite real times $\pm s^*$. Abstractly, our results are formulated in the setting of analytic semigroups. They are based on Poincar\'e 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 $n\leq22$, or to fast unstable manifolds of dimensions $d<1+\tfrac{1}{\sqrt{2}}n$.