A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV‐Like Eigenvalue Problems

Abstract

The Hamiltonian-Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem J Lu = λu, where J is skew-symmetric and L is self-adjoint. If J has a bounded inverse the index is well established, and it is given by the number of negative eigenvalues of the operator L constrained to act on some finite-codimensional subspace. There is an important class of problems-namely, those of KdV-type-for which J does not have a bounded inverse. In this paper, we overcome this difficulty and derive the index for eigenvalue problems of KdV-type. We use the index to discuss the spectral stability of homoclinic traveling waves for KdV-like problems and Benjamin-Bona-Mahony-type problems.

Gaussian solitary waves and compactons in Fermi–Pasta–Ulam lattices with Hertzian potentials

“…This result follows from the main theorems in [39,40] if the number of negative eigenvalues of L is equal to one and …”

“…Consistently with the above dynamical simulations, we prove in §3d the linear orbital stability of Gaussian solitary waves for the logKdV equation. Our analysis makes use of a suitable convex conserved Lyapunov function, but negative index techniques developed in recent works [39,40] for KdV-type equations would also apply.…”

Spectral Stability for Subsonic Traveling Pulses of the Boussinesq “abc" System

“…As we have mentioned above, this theory has been under development for some time, see [18], [14], [19], but we use a recent formulation due to Kapitula-Kevrekidis and Sandstede (KKS), [15] (see also [16]). In fact, even the (KKS) index count formula is not directly applicable 2 to the problem of (5), which is why Kapitula and Stefanov, [17] have found an approach, based on the KKS of the theory, which covers this situation. In order to simplify the exposition, we will restrict to a corollary of the main result in [17].…”

“…In fact, even the (KKS) index count formula is not directly applicable 2 to the problem of (5), which is why Kapitula and Stefanov, [17] have found an approach, based on the KKS of the theory, which covers this situation. In order to simplify the exposition, we will restrict to a corollary of the main result in [17]. More precisely, a the stability problem in the form is considered in the form…”

“…This would be achieved via the use of the instabilities indices counting formulas of Kapitula, Kevrekidis and Sandstede, [15], [16] and the subsequent refinement by Kapitula, Stefanov [17].…”