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Computation and Stability of Traveling Waves in Second Order Evolution Equations

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      Abstract

      The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a comoving frame in which the solution becomes stationary. In addition it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type.

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      Impulses and Physiological States in Theoretical Models of Nerve Membrane.

      Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model" has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a "physiological state diagram," with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.
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        Stability theory of solitary waves in the presence of symmetry, II

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          Exponential dichotomies and transversal homoclinic points

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            Author and article information

            Journal
            2016-06-28
            1606.08844

            http://arxiv.org/licenses/nonexclusive-distrib/1.0/

            Custom metadata
            65P40, 35L52, 47A25 (35B35, 35P30, 37C80)
            33 pages, 26 figures
            math.AP math.NA math.SP

            Analysis, Numerical & Computational mathematics, Functional analysis

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