Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns

Abstract

Extending the approach of Grillakis, Shatah, and Strauss, Bronski, Johnson, and Kapitula, and others for Hamiltonian systems, we explore relations between the constrained variational problem $$ \min_{X:C(X)=c_0} \mathcal{E}(X) , \ \ \ c_0\in \mathbb R^r , $$ and stability of solutions of a class of degenerate ``quasi-gradient'' systems $dX/dt=-M(X)\nabla \mathcal{E}(X)$ admitting constraints, including Cahn--Hilliard equations, one- and 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 $\partial c/\partial \omega \in \mathbb R^{r\times r}$, where $c(\omega)=C(X^*_\omega)\in \mathbb R ^r$ denote the values of the imposed constraints and $\omega\in \mathbb R^r$ the associated Lagrange multipliers at a critical point $X^*_\omega$, 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.