Browsing by Author "Dai, S"

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    BIFURCATIONS IN A MODULATION EQUATION FOR ALTERNANS IN A CARDIAC FIBER
    (ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, 2010) Dai, S; Schaeffer, DG
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    Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation
    (2017-04-23) Dai, S; Li, B; Lu, J
    We study a phase-field variational model for the solvaiton of charged molecules with an implicit solvent. The solvation free-energy functional of all phase fields consists of the surface energy, solute excluded volume and solute-solvent van der Waals dispersion energy, and electrostatic free energy. The surface energy is defined by the van der Waals--Cahn--Hilliard functional with squared gradient and a double-well potential. The electrostatic part of free energy is defined through the electrostatic potential governed by the Poisson--Boltzmann equation in which the dielectric coefficient is defined through the underlying phase field. We prove the continuity of the electrostatics---its potential, free energy, and dielectric boundary force---with respect to the perturbation of dielectric boundary. We also prove the $\Gamma$-convergence of the phase-field free-energy functionals to their sharp-interface limit, and the equivalence of the convergence of total free energies to that of all individual parts of free energy. We finally prove the convergence of phase-field forces to their sharp-interface limit. Such forces are defined as the negative first variations of the free-energy functional; and arise from stress tensors. In particular, we obtain the force convergence for the van der Waals--Cahn--Hilliard functionals with minimal assumptions.
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    Spectrum of a linearized amplitude equation for alternans in a cardiac fiber
    (SIAM Journal on Applied Mathematics, 2008-12-01) Dai, S; Schaeffer, DG
    Under rapid periodic pacing, cardiac cells typically undergo a period-doubling bifurcation in which action potentials of short and long duration alternate with one another. If these action potentials propagate in a fiber, the short-long alternation may suffer reversals of phase at various points along the fiber, a phenomenon called (spatially) discordant alternans. Either stationary or moving patterns are possible. Using a weak approximation, Echebarria and Karma proposed an equation to describe the spatiotemporal dynamics of small-amplitude alternans in a class of simple cardiac models, and they showed that an instability in this equation predicts the spontaneous formation of discordant alternans. To study the bifurcation, they computed the spectrum of the relevant linearized operator numerically, supplemented with partial analytical results. In the present paper we calculate this spectrum with purely analytical methods in two cases where a small parameter may be exploited: (i) small dispersion or (ii) a long fiber. From this analysis we estimate the parameter ranges in which the phase reversals of discordant alternans are stationary or moving. © 2008 Society for Industrial and Applied Mathematics.