Variational Nonlinearities and Wave Selection at an $O(2)$-Hopf Bifurcation

Abstract

At an $O(2)$-equivariant Hopf bifurcation, the real parts $ζ_R$ and $ξ_R$ of the two cubic normal-form coefficients govern selection between traveling and standing waves. In a fourth-order regularized $p$-system, $ξ_R$ was observed to vanish and the cubic nonlinearity to shift only frequencies. We show that both observations are structural: variational nonlinearities enforce them even when the linear part is dissipative and non-Hamiltonian. We consider a class of two-component PDEs on the circle whose quadratic and cubic nonlinearities are generated by reflection-invariant local Hamiltonians of the fields and finitely many of their spatial derivatives, through a common constant-coefficient Poisson operator. Then $ξ_R=0$ for every admissible linear part, while $ζ_R$ depends only on the quadratic nonlinearity. For variational nonlinearities the direct cubic terms are purely imaginary. The mixed cascade term loses its real part because its second harmonic has zero temporal frequency and the resolvent keeps its parity, while the self-cascade at the doubled critical frequency keeps a real part. When $ζ_R\ne0$, the bifurcating traveling waves are saddles, while the standing waves are supercritical and orbitally asymptotically stable for $ζ_R<0$ and subcritical and unstable for $ζ_R>0$. When the nonlinearities are the Poisson operator applied to polynomials in the fields alone, and its symbol does not vanish at the critical and second-harmonic wavenumbers, cancellation for every admissible linear part conversely characterizes variationality. In a dissipatively regularized Boussinesq family, as dissipation tends to zero, $ζ_R$ vanishes linearly off resonance and diverges inversely at the 2:1 resonance. The Hamiltonian limit of the cubic coefficient is therefore singular.