2012/05/31 by Alin Pogan, Pogan, Alin, Arnd Scheel +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Materials Science · Mathematics · #Analysis of PDEs (math.AP) #Dermatological and Skeletal Disorders #FOS: Mathematics #Magnetism in coordination complexes #Mathematical Biology Tumor Growth #Microbial metabolism and enzyme function #Microtubule and mitosis dynamics #math.AP
paper · pdf · doi:10.48550/arxiv.1205.7082
arxiv created 2012/05/31 · openalex publication_date 2012/05/31 · arxiv updated 2012/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∈ \RMr, and stability of solutions of a class of degenerate "quasi-gradient" systems dX/dt=-M(X)∇ 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 ∂ c/∂ ω∈ \Rr× r, where c(ω)=C(X^*ω)∈ \RMr denote the values of the imposed constraints and ω∈ \RMr 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.