Krein signature for instability of $\mathcal{PT}$-symmetric states

Abstract: Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the $\mathcal{PT}$-symmetric nonlinear Schr\"{o}dinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature.