Quasi-gradient Systems, Modulational Dichotomies, and Stability of Spatially Periodic Patterns

Extending the approach of Grillakis–Shatah–Strauss, Bronski–Johnson–Kapitula, and others for Hamiltonian systems, we explore relations between the constrained variational problem minX:C(X)=c0 E(X), c0 ∈ R , and stability of solutions of a class of degenerate “quasi-gradient” systems dX/dt = −M∇E admitting constraints, including Cahn-Hilliard equations, oneand multi-dimensional viscoelasticity, and coupled conservation law-reaction diffusion systems arising in chemotaxis and related settings. Using the relation between variational stability and the signature of ∂c/∂ω ∈ Rr×r, where c(ω) = C(X∗ ω) ∈ R denote the values of the imposed constraints and ω ∈ R the associated Lagrange multipliers at a critical point X∗ ω, we obtain as in the Hamiltonian case a general criterion for co-periodic stability of periodic waves, illuminating and extending a number of previous results obtained by direct Evans function techniques. More interestingly, comparing the form of the Jacobian arising in the co-periodic theory to Jacobians arising in the formal Whitham equations associated with modulation, we recover and substantially generalize a previously mysterious “modulational dichotomy” observed in special cases by Oh–Zumbrun and Howard, showing that co-periodic and sideband stability are incompatible. In particular, we both illuminate and extend to general viscosity/strain-gradient effects and multidimensional deformations the result of Oh-Zumbrun of universal modulational instability of periodic solutions of the equations of viscoelasticity with strain-gradient effects, considered as functions on the whole line. Likewise, we generalize to multi-dimensions corresponding results of Howard on periodic solutions of Cahn–Hilliard equations.