Secure Beamforming for MIMO Broadcasting With Wireless Information and Power Transfer

Abstract

We consider the Allen–Cahn equation Δu + u(1 − u2) = 0 in R3. We construct two classes of axially symmetric solutions u = u(|x |, x3) such that the (multiple) components of the zero set look for large |x | like catenoids, namely |x3| ∼ A log |x |. In Theorem 1, we find a solution which is even in x3, with Morse index one and a zero set with exactly two components, which are graphs. In Theorem 2, we construct a solution with a zero set with two or more nested catenoid-like components, whose Morse index become as large as we wish. While it is a common idea that nodal sets of the Allen–Cahn equation behave like minimal surfaces, these examples show that the nonlocal interaction between disjoint portions of the nodal set, governed in suitably asymptotic regimes by explicit systems of 2d PDE, induces richness and complex structure of the set of entire solutions, beyond the one in minimal surface theory.