Mariana Haragus, Jin Li, and Dmitry E. Pelinovsky

Counting unstable eigenvalues in Hamiltonian spectral problems via commuting operators

Commun. Math. Phys. 354, 247-268 (2017)

We present a general counting result for the unstable eigenvalues of linear operators of the form JL in which J and L are skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator K such that the operators JL and JK commute, we prove that the number of unstable eigenvalues of JL is bounded by the number of nonpositive eigenvalues of K. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev–Petviashvili) equation. We show that these one-dimensional periodic waves are transversely spectrally stable with respect to general two-dimensional bounded perturbations, including periodic and localized perturbations in either the longitudinal or the transverse direction, and that they are transversely linearly stable with respect to doubly periodic perturbations.

Hamiltonian systems; spectral stability; linearized stability; periodic waves; negative index theory; Kadomtsev-Petviashvili equation; transverse stability;