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We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semi-strong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semi-strong limit the strongly localized... more
We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semi-strong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semi-strong limit the strongly localized activator pulses interact through the weakly localized inhibitor, the interaction is not tail-tail as in the weak interaction limit, and the pulses change amplitude and even stability as their separation distance evolves on algebraically slow time scales. The RG approach employed here validates the interaction laws of quasi-steady pulse patterns obtained formally in the literature, and establishes that the pulse dynamics reduce to a closed system of ordinary differential equations for the activator pulse locations. Moreover, we fully justify the reduction to the nonlocal eigenvalue problem (NLEP) showing the large difference between the quasi-steady NLEP operator and the operator arising from linearization about the pulse is controlled by the resolvent. 2...
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Research Interests: Mathematics, Applied Mathematics, Singular Perturbations, Biology, Numerical Analysis, and 15 moreConvergence, Algorithm, Fixed Point Theory, Approximation, Iteration, Mathematical Model, Manifold, Numerical Analysis and Computational Mathematics, Differential equation, Dynamic System, Projection Method, Singularly perturbed system, Iterative Algorithm, fixed point, and Iterative Method
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Research Interests: Mathematics, Applied Mathematics, Singular Perturbations, Biology, Numerical Analysis, and 15 moreConvergence, Algorithm, Fixed Point Theory, Approximation, Iteration, Mathematical Model, Manifold, Numerical Analysis and Computational Mathematics, Differential equation, Dynamic System, Projection Method, Singularly perturbed system, Iterative Algorithm, fixed point, and Iterative Method
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DOI: 10.1007/s00332-005-0731-4 J. Nonlinear Sci. Vol. 16: pp. 329349 (2006) ... A Geometric Construction of Traveling Waves in a Bioremediation Model ... M. Beck,1 A. Doelman,2 and TJ Kaper1 1 Department of Mathematics and Statistics,... more
DOI: 10.1007/s00332-005-0731-4 J. Nonlinear Sci. Vol. 16: pp. 329349 (2006) ... A Geometric Construction of Traveling Waves in a Bioremediation Model ... M. Beck,1 A. Doelman,2 and TJ Kaper1 1 Department of Mathematics and Statistics, Boston University, Boston, MA ...