On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem

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August 2015, 35(8): 3277-3292. doi: 10.3934/dcds.2015.35.3277

On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem

Pablo Amster ^1, and Colin Rogers ^2,

1.	Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires
2.	Australian Research Council Centre of Excellence for Mathematics & Statistics of Complex Systems, School of Mathematics, The University of New South Wales, Sydney, NSW2052

Received April 2014 Revised January 2015 Published February 2015

Abstract
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Two-point boundary value problems of Dirichlet type are investigated for a Ermakov-Painlevé II equation which arises out of a reduction of a three-ion electrodiffusion Nernst-Planck model system. In addition, it is shown how Ermakov invariants may be employed to solve a hybrid Ermakov-Painlevé II triad in terms of a solution of the single component integrable Ermakov-Painlevé II reduction. The latter is related to the classical Painlevé II equation.

Keywords: Ermakov-Painlevé reduction, three-ion electro-diffusion, boundary value problem..

Mathematics Subject Classification: Primary: 34B15; Secondary: 34B6.

Citation: Pablo Amster, Colin Rogers. On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3277-3292. doi: 10.3934/dcds.2015.35.3277

References:

[1]	P. Amster, M. K. Kwong and C. Rogers, A Painlevé II model in two-ion electrodiffusion with radiation boundary conditions, Nonlinear Analysis: Real World Applications, 16 (2014), 120-131. doi: 10.1016/j.nonrwa.2013.09.011.
[2]	P. Amster, M. K. Kwong and C. Rogers, A Neumann boundary value problem in two-ion electro-diffusion with unequal valencies, Discrete and Continuous Dynamical Systems, 17 (2012), 2299-2311. doi: 10.3934/dcdsb.2012.17.2299.
[3]	P. Amster, M. K. Kwong and C. Rogers, On a Neumann boundary value problem for the Painlevé II equation in two-ion electro-diffusion, Nonlinear Analysis, 74 (2011), 2897-2907. doi: 10.1016/j.na.2010.06.063.
[4]	P. Amster, M. C. Mariani, C. Rogers and C. C. Tisdell, On two-point boundary value problems in multi-ion electrodiffusion, J. Math. Anal. Appl., 289 (2004), 712-721. doi: 10.1016/j.jmaa.2003.09.075.
[5]	P. Amster and C. Rogers, On boundary value problems in three-ion electrodiffusion, J. Math. Anal. Appl., 333 (2007), 42-51. doi: 10.1016/j.jmaa.2007.03.067.
[6]	P. Amster, L. Vicchi and C. Rogers, Boundary value problems on the half-line for a generalised Painlevé II equation, Nonlinear Analysis, 71 (2009), 149-154. doi: 10.1016/j.na.2008.10.036.
[7]	L. Bass, Irreversible interactions between metals and electrolytes, Proc. Roy. Soc. Lond. A, 277 (1964), 125-136. doi: 10.1098/rspa.1964.0009.
[8]	L. Bass, Electric structures of interfaces in steady electrolysis, Trans. Faraday Soc., 60 (1964), 1656-1663. doi: 10.1039/tf9646001656.
[9]	L. Bass, J. Nimmo, C. Rogers and W. K. Schief, Electrical structures of interfaces. A Painlevé II model, Proc. Roy. Soc. London A, 466 (2010), 2117-2136. doi: 10.1098/rspa.2009.0620.
[10]	P. C. T. de Boer and G. S. S. Ludford, Spherical electric probe in a continuum gas, Plasma Phys., 17 (1975), 29-43. doi: 10.1088/0032-1028/17/1/004.
[11]	J. O'M. Bokris and A. K. N. Reddy, Modern Electrochemistry, Plenum, New York, 1970.
[12]	A. J. Bracken, L. Bass and C. Rogers, Bäcklund flux-quantization in a model of electrodiffusion based on Painlevé II, J. Phys. A: Math. & Theor., 45 (2012), 105204, 20pp. doi: 10.1088/1751-8113/45/10/105204.
[13]	K. S. Cole, Membranes, Ions and Impulses, University of California Press, Berkeley, 1968.
[14]	R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system, J. Phys. A: Math. Theor., 40 (2007), F1031-F1040. doi: 10.1088/1751-8113/40/48/F01.
[15]	C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering, 205 Elsevier, Amsterdam, 2006.
[16]	J. V. Hägglund, Single-ion electrodiffusion models of the late sodium and potassium currents in the giant axon of the squid, J. Membrane Biol, 10 (1972), 153-170. doi: 10.1007/BF01867851.
[17]	S. P. Hastings and J. B. Mcleod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rat. Mech. Anal., 73 (1980), 31-51. doi: 10.1007/BF00283254.
[18]	B. M. Grafov and A. A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte, Dokl. Akad. Navk. SSR, 146 (1962), 135-138.
[19]	V. I. Gromak, Bäcklund transformations of Painlevé equations and their applications, in The Painlevé Property. One Century Later, (Ed. R. Conte), CRM Series in Mathematical Physics, Springer Verlag, New York, (1999), 687-734.
[20]	B. Heiffer and F. B. Weissler, On a family of solutions of the second Painlevé equation related to superconductivity, Eur. J. Appl. Math., 9 (1998), 223-243. doi: 10.1017/S0956792598003428.
[21]	P. Holmes and D. Spence, On a Painlevé-type boundary value problem, Quart. J. Mech. Appl. Math., 37 (1984), 525-538. doi: 10.1093/qjmam/37.4.525.
[22]	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.
[23]	J. H. Lee, O. K. Pashaev, C. Rogers and W. K. Schief, The resonant nonlinear Schrödinger equation in cold plasma physics. Application of Bäcklund-Darboux transformations and superposition principles, J. Plasma Phys., 73 (2007), 257-272. doi: 10.1017/S0022377806004648.
[24]	H. R. Leuchtag, A family of differential equations arising from multi-ion electro-diffusion, J. Math. Phys., 22 (1981), 1317-1320. doi: 10.1063/1.525026.
[25]	H. R. Leuchtag and J. C. Swihart, Steady state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane, Biophys. J., 17 (1977), 27-46. doi: 10.1016/S0006-3495(77)85625-7.
[26]	M. C. Mackey, Admittance properties of electrodiffusion membrane models, Math. Biosci., 25 (1975), 67-80. doi: 10.1016/0025-5564(75)90052-8.
[27]	M. C. Mariani, P. Amster and C. Rogers, Dirichlet and periodic-type boundary value problems for Painlevé II, J. Math. Anal. Appl., 265 (2002), 1-11. doi: 10.1006/jmaa.2001.7675.
[28]	W. Nernst, Zur Kinetik der in Lösung befindlichen Körper. Erste Abhandlung. Theorie der Diffusion, Z. Phys. Chem., 2 (1888), 613-637.
[29]	O. K. Pashaev and J. H. Lee, Resonance solitons as black holes in Madelung fluid, Mod. Phys. Lett. A, 17 (2002), 1601-1619. doi: 10.1142/S0217732302007995.
[30]	M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten, Ann. Phys. Chem., 39 (1890), 161-186.
[31]	J. R. Ray, Nonlinear superposition law for generalised Ermakov systems, Phys. Lett. A, 78 (1980), 4-6. doi: 10.1016/0375-9601(80)90789-6.
[32]	J. L. Reid and J. R. Ray, Ermakov systems, nonlinear superposition and solution of nonlinear equations of motion, J. Math. Phys., 21 (1980), 1583-1587. doi: 10.1063/1.524625.
[33]	C. Rogers, A novel Ermakov-Painlevé II system. $N+1$-dimensional coupled NLS and elastodynamic reductions, Stud. Appl. Math., 133 (2014), 214-231. doi: 10.1111/sapm.12039.
[34]	C. Rogers and H. An, Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory, Stud. Appl. Math., 125 (2010), 275-299. doi: 10.1111/j.1467-9590.2010.00488.x.
[35]	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.
[36]	C. Rogers, C. Hoenselaers and J. R. Ray, On $2+1$-dimensional Ermakov systems, J. Phys. A: Math. Gen., 26 (1993), 2625-2633. doi: 10.1088/0305-4470/26/11/012.
[37]	C. Rogers, B. Malomed and H. An, Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics, Stud. Appl. Math., 129 (2012), 389-413. doi: 10.1111/j.1467-9590.2012.00557.x.
[38]	C. Rogers and W. K. Schief, Multi-component Ermakov systems: Structure and linearization, J. Math. Anal. Appl., 198 (1996), 194-220. doi: 10.1006/jmaa.1996.0076.
[39]	C. Rogers, W. K. Schief and P. Winternitz, Lie theoretical generalization and discretisation of the Pinney equation, J. Math. Anal. Appl., 216 (1997), 246-264. doi: 10.1006/jmaa.1997.5674.
[40]	J. Sandblom, Anomalous reactances in electrodiffusion systems, Biophys. J., 12 (1972), 1118-1130.
[41]	W. K. Schief, C. Rogers and A. Bassom, Ermakov systems with arbitrary order and dimension. Structure and linearization, J. Phys. A: Math. Gen., 29 (1996), 903-911. doi: 10.1088/0305-4470/29/4/017.
[42]	R. Schlögl, Elektrodiffusion in freier Lösung und geladenen Membranen, Z. Physik. Chem. Neue Folge, 1 (1954), 305-339. doi: 10.1524/zpch.1954.1.5_6.305.
[43]	T. L. Schwarz, in Biophysics and Physiology of Excitable Membranes, W.J. Adelman Jr. (Ed.), Van Rostrand, New York, 1971.
[44]	H. B. Thompson, Existence of solutions for a two point boundary value problem arising in electro-diffusion, Acta. Math. Sci., 8 (1988), 373-387.
[45]	H. B. Thompson, Existence for Two-Point Boundary Value Problems in Two Ion Electrodiffusion, Journal of Mathematical Analysis and Applications, 184 (1994), 82-94. doi: 10.1006/jmaa.1994.1185.

show all references

References:

[1]	P. Amster, M. K. Kwong and C. Rogers, A Painlevé II model in two-ion electrodiffusion with radiation boundary conditions, Nonlinear Analysis: Real World Applications, 16 (2014), 120-131. doi: 10.1016/j.nonrwa.2013.09.011.
[2]	P. Amster, M. K. Kwong and C. Rogers, A Neumann boundary value problem in two-ion electro-diffusion with unequal valencies, Discrete and Continuous Dynamical Systems, 17 (2012), 2299-2311. doi: 10.3934/dcdsb.2012.17.2299.
[3]	P. Amster, M. K. Kwong and C. Rogers, On a Neumann boundary value problem for the Painlevé II equation in two-ion electro-diffusion, Nonlinear Analysis, 74 (2011), 2897-2907. doi: 10.1016/j.na.2010.06.063.
[4]	P. Amster, M. C. Mariani, C. Rogers and C. C. Tisdell, On two-point boundary value problems in multi-ion electrodiffusion, J. Math. Anal. Appl., 289 (2004), 712-721. doi: 10.1016/j.jmaa.2003.09.075.
[5]	P. Amster and C. Rogers, On boundary value problems in three-ion electrodiffusion, J. Math. Anal. Appl., 333 (2007), 42-51. doi: 10.1016/j.jmaa.2007.03.067.
[6]	P. Amster, L. Vicchi and C. Rogers, Boundary value problems on the half-line for a generalised Painlevé II equation, Nonlinear Analysis, 71 (2009), 149-154. doi: 10.1016/j.na.2008.10.036.
[7]	L. Bass, Irreversible interactions between metals and electrolytes, Proc. Roy. Soc. Lond. A, 277 (1964), 125-136. doi: 10.1098/rspa.1964.0009.
[8]	L. Bass, Electric structures of interfaces in steady electrolysis, Trans. Faraday Soc., 60 (1964), 1656-1663. doi: 10.1039/tf9646001656.
[9]	L. Bass, J. Nimmo, C. Rogers and W. K. Schief, Electrical structures of interfaces. A Painlevé II model, Proc. Roy. Soc. London A, 466 (2010), 2117-2136. doi: 10.1098/rspa.2009.0620.
[10]	P. C. T. de Boer and G. S. S. Ludford, Spherical electric probe in a continuum gas, Plasma Phys., 17 (1975), 29-43. doi: 10.1088/0032-1028/17/1/004.
[11]	J. O'M. Bokris and A. K. N. Reddy, Modern Electrochemistry, Plenum, New York, 1970.
[12]	A. J. Bracken, L. Bass and C. Rogers, Bäcklund flux-quantization in a model of electrodiffusion based on Painlevé II, J. Phys. A: Math. & Theor., 45 (2012), 105204, 20pp. doi: 10.1088/1751-8113/45/10/105204.
[13]	K. S. Cole, Membranes, Ions and Impulses, University of California Press, Berkeley, 1968.
[14]	R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system, J. Phys. A: Math. Theor., 40 (2007), F1031-F1040. doi: 10.1088/1751-8113/40/48/F01.
[15]	C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering, 205 Elsevier, Amsterdam, 2006.
[16]	J. V. Hägglund, Single-ion electrodiffusion models of the late sodium and potassium currents in the giant axon of the squid, J. Membrane Biol, 10 (1972), 153-170. doi: 10.1007/BF01867851.
[17]	S. P. Hastings and J. B. Mcleod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rat. Mech. Anal., 73 (1980), 31-51. doi: 10.1007/BF00283254.
[18]	B. M. Grafov and A. A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte, Dokl. Akad. Navk. SSR, 146 (1962), 135-138.
[19]	V. I. Gromak, Bäcklund transformations of Painlevé equations and their applications, in The Painlevé Property. One Century Later, (Ed. R. Conte), CRM Series in Mathematical Physics, Springer Verlag, New York, (1999), 687-734.
[20]	B. Heiffer and F. B. Weissler, On a family of solutions of the second Painlevé equation related to superconductivity, Eur. J. Appl. Math., 9 (1998), 223-243. doi: 10.1017/S0956792598003428.
[21]	P. Holmes and D. Spence, On a Painlevé-type boundary value problem, Quart. J. Mech. Appl. Math., 37 (1984), 525-538. doi: 10.1093/qjmam/37.4.525.
[22]	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.
[23]	J. H. Lee, O. K. Pashaev, C. Rogers and W. K. Schief, The resonant nonlinear Schrödinger equation in cold plasma physics. Application of Bäcklund-Darboux transformations and superposition principles, J. Plasma Phys., 73 (2007), 257-272. doi: 10.1017/S0022377806004648.
[24]	H. R. Leuchtag, A family of differential equations arising from multi-ion electro-diffusion, J. Math. Phys., 22 (1981), 1317-1320. doi: 10.1063/1.525026.
[25]	H. R. Leuchtag and J. C. Swihart, Steady state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane, Biophys. J., 17 (1977), 27-46. doi: 10.1016/S0006-3495(77)85625-7.
[26]	M. C. Mackey, Admittance properties of electrodiffusion membrane models, Math. Biosci., 25 (1975), 67-80. doi: 10.1016/0025-5564(75)90052-8.
[27]	M. C. Mariani, P. Amster and C. Rogers, Dirichlet and periodic-type boundary value problems for Painlevé II, J. Math. Anal. Appl., 265 (2002), 1-11. doi: 10.1006/jmaa.2001.7675.
[28]	W. Nernst, Zur Kinetik der in Lösung befindlichen Körper. Erste Abhandlung. Theorie der Diffusion, Z. Phys. Chem., 2 (1888), 613-637.
[29]	O. K. Pashaev and J. H. Lee, Resonance solitons as black holes in Madelung fluid, Mod. Phys. Lett. A, 17 (2002), 1601-1619. doi: 10.1142/S0217732302007995.
[30]	M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten, Ann. Phys. Chem., 39 (1890), 161-186.
[31]	J. R. Ray, Nonlinear superposition law for generalised Ermakov systems, Phys. Lett. A, 78 (1980), 4-6. doi: 10.1016/0375-9601(80)90789-6.
[32]	J. L. Reid and J. R. Ray, Ermakov systems, nonlinear superposition and solution of nonlinear equations of motion, J. Math. Phys., 21 (1980), 1583-1587. doi: 10.1063/1.524625.
[33]	C. Rogers, A novel Ermakov-Painlevé II system. $N+1$-dimensional coupled NLS and elastodynamic reductions, Stud. Appl. Math., 133 (2014), 214-231. doi: 10.1111/sapm.12039.
[34]	C. Rogers and H. An, Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory, Stud. Appl. Math., 125 (2010), 275-299. doi: 10.1111/j.1467-9590.2010.00488.x.
[35]	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.
[36]	C. Rogers, C. Hoenselaers and J. R. Ray, On $2+1$-dimensional Ermakov systems, J. Phys. A: Math. Gen., 26 (1993), 2625-2633. doi: 10.1088/0305-4470/26/11/012.
[37]	C. Rogers, B. Malomed and H. An, Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics, Stud. Appl. Math., 129 (2012), 389-413. doi: 10.1111/j.1467-9590.2012.00557.x.
[38]	C. Rogers and W. K. Schief, Multi-component Ermakov systems: Structure and linearization, J. Math. Anal. Appl., 198 (1996), 194-220. doi: 10.1006/jmaa.1996.0076.
[39]	C. Rogers, W. K. Schief and P. Winternitz, Lie theoretical generalization and discretisation of the Pinney equation, J. Math. Anal. Appl., 216 (1997), 246-264. doi: 10.1006/jmaa.1997.5674.
[40]	J. Sandblom, Anomalous reactances in electrodiffusion systems, Biophys. J., 12 (1972), 1118-1130.
[41]	W. K. Schief, C. Rogers and A. Bassom, Ermakov systems with arbitrary order and dimension. Structure and linearization, J. Phys. A: Math. Gen., 29 (1996), 903-911. doi: 10.1088/0305-4470/29/4/017.
[42]	R. Schlögl, Elektrodiffusion in freier Lösung und geladenen Membranen, Z. Physik. Chem. Neue Folge, 1 (1954), 305-339. doi: 10.1524/zpch.1954.1.5_6.305.
[43]	T. L. Schwarz, in Biophysics and Physiology of Excitable Membranes, W.J. Adelman Jr. (Ed.), Van Rostrand, New York, 1971.
[44]	H. B. Thompson, Existence of solutions for a two point boundary value problem arising in electro-diffusion, Acta. Math. Sci., 8 (1988), 373-387.
[45]	H. B. Thompson, Existence for Two-Point Boundary Value Problems in Two Ion Electrodiffusion, Journal of Mathematical Analysis and Applications, 184 (1994), 82-94. doi: 10.1006/jmaa.1994.1185.

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