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Various approaches to studying the stability of solutions of nonlinear PDEslead to explicit formulae determining the stability or instability of the wavefor a wide range of classes of equations. However, these are typicallyspecialized to a particular equation and checking the stability conditions maynot be not straightforward. We present results for a large class of problemsthat reduce the determination of spectral stability of a wave to a simple taskof locating zeros of explicitly constructed polynomials. We study spectralstability of small-amplitude periodic waves in scalar Hamiltonian problems as aperturbation of the zero-amplitude case. A necessary condition for stability ofthe wave is that the unperturbed spectrum is restricted to the imaginary axis.Instability can come about through a Hamiltonian-Hopf bifurcation, i.e., of acollision of purely imaginary eigenvalues of the Floquet spectrum of oppositeKrein signature. In recent work on the stability of small-amplitude waves thedispersion relation of the unperturbed problem was shown to play a centralrole. We demonstrate that the dispersion relation provides even more explicitinformation about wave stability: we construct a polynomial of half the degreeof the dispersion relation, and its roots directly characterize not onlycollisions of eigenvalues at zero-amplitude but also an agreement or adisagreement of their Krein signatures. Based on this explicit information itis possible to detect instabilities of non-zero amplitude waves. In ouranalysis we stay away from the possible instabilities at the origin of thespectral plane corresponding to modulation or Benjamin-Fair instability.Generalized KdV and its higher-order analogues are used as illustratingexamples.

Direct Characterization of Spectral Stability of Small Amplitude Periodic Waves in Scalar Hamiltonian Problems Via Dispersion Relation