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 ∈ R are often called PDE entire or eternal. For a nonlinear example, consider the quadratic parabolic PDE wt = wxx + 6w2 − λ, (*) for 0 < x < 1 2 , under Neumann boundary conditions. By its gradient-like structure, all real eternal non-equilibrium orbits (r ) of (*) are heteroclinic among equilibria w = Wn(x). For parameters λ > 0, the trivial homogeneous equilibria are locally asymptotically stable W0 = − √ λ/6, and W∞ = + √ λ/6 of unstable dimension (Morse index) i (W∞) = 1, 2, 3, . . ., depending on λ. All nontrivial real Wn are rescaled and properly translated real-valuedWeierstrass elliptic functions with Morse index i (Wn) = n.We show that the complex time extensions (r+is), of analytic real heteroclinic orbits (r ) towards W0, are not complex entire. For example, consider the time-reversible complex-valued solution ψ(s) = (r0 − is) of the nonlinear and nonconservative quadratic Schrödinger equation iψs = ψxx + 6ψ2 − λ with real initial conditionψ0 = (r0). Then there exist real r0 such that ψ(s) blows up at some finite real times±s∗ = 0. Abstractly, our results are formulated in the setting of analytic semigroups. They are based on Poincaré non-resonance of unstable eigenvalues at equilibriaWn, near pitchfork bifurcation. Technically,we have to except discrete sets of parameters λ, and are currently limited to unstable dimensions i (Wn) ≤ 22, or to fast unstable manifolds of dimensions d < 1 + √1 2 i (Wn).