# American Institute of Mathematical Sciences

March  2013, 8(1): 191-209. doi: 10.3934/nhm.2013.8.191

## Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems

 1 Institute of Mathematics for Industry, Kyusyu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395 2 Opinion Poll Research Center, The Asahi Shimbun Company, Tokyo 104-8011, Japan

Received  February 2012 Revised  January 2013 Published  April 2013

We consider pulse-like localized solutions for reaction-diffusion systems on a half line and impose various boundary conditions at one end of it. It is shown that the movement of a pulse solution with the homogeneous Neumann boundary condition is completely opposite from that with the Dirichlet boundary condition. As general cases, Robin type boundary conditions are also considered. Introducing one parameter connecting the Neumann and the Dirichlet boundary conditions, we clarify the transition of motions of solutions with respect to boundary conditions.
Citation: Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191
##### References:
 [1] J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t = epsilon^2 u_{x x} + f(u)$,, Comm. Pure Appl. Math., 42 (1989), 523.  doi: 10.1002/cpa.3160420502.  Google Scholar [2] A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model,, Physica D, 122 (1998), 1.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar [3] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynam. Differential Equations, 14 (2002), 85.  doi: 10.1023/A:1012980128575.  Google Scholar [4] P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335.   Google Scholar [5] G. Fusco and J. Hale, Slow motion manifold, dormant instability and singular perturbations,, J. Dynamics and Differential Equations, 1 (1989), 75.  doi: 10.1007/BF01048791.  Google Scholar [6] K. Kawasaki and T. Ohta, Kink dynamics in one-dimensional nonlinear systems,, Physica A, 116 (1982), 573.  doi: 10.1016/0378-4371(82)90178-9.  Google Scholar [7] J. M. Murray, "Mathematical Biology,", Springer-Verlag, (1989).   Google Scholar [8] Y. Nishiura, "Far-From-Equilibrium Dynamics,", (Translations of Mathematical Monographs), (2002).   Google Scholar

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##### References:
 [1] J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t = epsilon^2 u_{x x} + f(u)$,, Comm. Pure Appl. Math., 42 (1989), 523.  doi: 10.1002/cpa.3160420502.  Google Scholar [2] A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model,, Physica D, 122 (1998), 1.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar [3] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynam. Differential Equations, 14 (2002), 85.  doi: 10.1023/A:1012980128575.  Google Scholar [4] P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335.   Google Scholar [5] G. Fusco and J. Hale, Slow motion manifold, dormant instability and singular perturbations,, J. Dynamics and Differential Equations, 1 (1989), 75.  doi: 10.1007/BF01048791.  Google Scholar [6] K. Kawasaki and T. Ohta, Kink dynamics in one-dimensional nonlinear systems,, Physica A, 116 (1982), 573.  doi: 10.1016/0378-4371(82)90178-9.  Google Scholar [7] J. M. Murray, "Mathematical Biology,", Springer-Verlag, (1989).   Google Scholar [8] Y. Nishiura, "Far-From-Equilibrium Dynamics,", (Translations of Mathematical Monographs), (2002).   Google Scholar
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