# American Institute of Mathematical Sciences

October  2012, 17(7): 2299-2311. doi: 10.3934/dcdsb.2012.17.2299

## A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies

 1 Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong 3 Australian Research Council Centre & Excellence for Mathematics & Statistics of Complex Systems, School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia

Received  December 2011 Revised  April 2012 Published  July 2012

In prior work, a series of two-point boundary value problems have been investigated for a steady state two-ion electro-diffusion model system in which the sum of the valencies $\nu_+$ and $\nu_-$ is zero. In that case, reduction is obtained to the canonical Painlevé II equation for the scaled electric field. Here, a physically important Neumann boundary value problem in the generic case when $\nu_+ + \nu_-\neq 0$ is investigated. The problem is novel in that the model equation for the electric field involves yet to be determined boundary values of the solution. A reduction of the Neumann boundary value problem in terms of elliptic functions is obtained for privileged valency ratios. A topological index argument is used to establish the existence of a solution in the general case, under the assumption $\nu_+ + \nu_- \leq 0$.
Citation: Pablo Amster, Man Kam Kwong, Colin Rogers. A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2299-2311. doi: 10.3934/dcdsb.2012.17.2299
##### References:
 [1] W. Nernst, Zur Kinetik der in Lösung befindlichen Körper: I Theorie der Diffusion, Z. Phys. Chem., 2 (1882), 613-637. [2] M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten, Ann. Phys. Chem., 39 (1890), 161-186. [3] K. S. Cole, "Membranes, Ions and Impulses," University of California Press, Berkeley, 1968. [4] T. L. Schwarz, "Biophysics and Physiology of Excitable Membranes" (ed. W. J. Adelman, Jr.), Van Rostrand, New York, 1971. [5] J. O'M Bokris and A. K. N. Reddy, "Modern Electrochemistry," Plenum, New York, 1971. [6] H. R. Leuchtag, A family of differential equations arising from multi-ion electrodiffusion, J. Mathematical Phys., 22 (1981), 1317-1320. [7] R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system, J. Phys. A, 40 (2007), F1031-F1040. doi: 10.1088/1751-8113/40/48/F01. [8] H. B. Thompson, Existence for two-point boundary value problems in two-ion electrodiffusion, J. Math. Anal. Appl, 184 (1994), 82-94. doi: 10.1006/jmaa.1994.1185. [9] B. M. Grafov and A. A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte, Dokl. Akad. Nauk. SSR, 146 (1962), 135-138. [10] L. Bass, Electrical structures of interfaces in steady electrolysis, Trans. Faraday Soc., 60 (1964), 1655-1663. doi: 10.1039/tf9646001656. [11] N. A. Kudryashov, The second Painlevé equation as a model for the electric field in a semiconductor, Phys. Lett. A, 233 (1997), 397-400. doi: 10.1016/S0375-9601(97)00545-8. [12] C. Rogers, A. Bassom and W. K. Schief, On a Painlevé II model in steady electrolysis: Application of a Bäcklund transformation, J. Math. Anal. Appl., 240 (1999), 367-381. doi: 10.1006/jmaa.1999.6589. [13] L. Bass, J. Nimmo, C. Rogers and W. K. Schief, Enhanced structures of interfaces: A Painlevé II model, Proc. Roy. Soc. London Ser. A Math. Phys. Eng. Sci., 466 (2010), 2117-2136. doi: 10.1098/rspa.2009.0620. [14] L. Bass, Irreversible interactions between metals and electrolytes, Proc. Roy. Soc. London A, 277 (1964), 125-136. doi: 10.1098/rspa.1964.0009. [15] P. Amster, M. K. Kwong and C. Rogers, On a Neumann boundary value problem for Painlevé II in two-ion electro-diffusion, Nonlinear Analysis, Theory, Methods and Applications, in press. [16] C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results, in "Nonlinear Analysis and Boundary Value Problems for ODEs," CISM Courses and Lectures, 371, Springer, Vienna, (1996), 1-78.

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##### References:
 [1] W. Nernst, Zur Kinetik der in Lösung befindlichen Körper: I Theorie der Diffusion, Z. Phys. Chem., 2 (1882), 613-637. [2] M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten, Ann. Phys. Chem., 39 (1890), 161-186. [3] K. S. Cole, "Membranes, Ions and Impulses," University of California Press, Berkeley, 1968. [4] T. L. Schwarz, "Biophysics and Physiology of Excitable Membranes" (ed. W. J. Adelman, Jr.), Van Rostrand, New York, 1971. [5] J. O'M Bokris and A. K. N. Reddy, "Modern Electrochemistry," Plenum, New York, 1971. [6] H. R. Leuchtag, A family of differential equations arising from multi-ion electrodiffusion, J. Mathematical Phys., 22 (1981), 1317-1320. [7] R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system, J. Phys. A, 40 (2007), F1031-F1040. doi: 10.1088/1751-8113/40/48/F01. [8] H. B. Thompson, Existence for two-point boundary value problems in two-ion electrodiffusion, J. Math. Anal. Appl, 184 (1994), 82-94. doi: 10.1006/jmaa.1994.1185. [9] B. M. Grafov and A. A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte, Dokl. Akad. Nauk. SSR, 146 (1962), 135-138. [10] L. Bass, Electrical structures of interfaces in steady electrolysis, Trans. Faraday Soc., 60 (1964), 1655-1663. doi: 10.1039/tf9646001656. [11] N. A. Kudryashov, The second Painlevé equation as a model for the electric field in a semiconductor, Phys. Lett. A, 233 (1997), 397-400. doi: 10.1016/S0375-9601(97)00545-8. [12] C. Rogers, A. Bassom and W. K. Schief, On a Painlevé II model in steady electrolysis: Application of a Bäcklund transformation, J. Math. Anal. Appl., 240 (1999), 367-381. doi: 10.1006/jmaa.1999.6589. [13] L. Bass, J. Nimmo, C. Rogers and W. K. Schief, Enhanced structures of interfaces: A Painlevé II model, Proc. Roy. Soc. London Ser. A Math. Phys. Eng. Sci., 466 (2010), 2117-2136. doi: 10.1098/rspa.2009.0620. [14] L. Bass, Irreversible interactions between metals and electrolytes, Proc. Roy. Soc. London A, 277 (1964), 125-136. doi: 10.1098/rspa.1964.0009. [15] P. Amster, M. K. Kwong and C. Rogers, On a Neumann boundary value problem for Painlevé II in two-ion electro-diffusion, Nonlinear Analysis, Theory, Methods and Applications, in press. [16] C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results, in "Nonlinear Analysis and Boundary Value Problems for ODEs," CISM Courses and Lectures, 371, Springer, Vienna, (1996), 1-78.
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