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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.nonrwa.2013.09.011.  Google Scholar

[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.  doi: 10.3934/dcdsb.2012.17.2299.  Google Scholar

[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.  doi: 10.1016/j.na.2010.06.063.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2003.09.075.  Google Scholar

[5]

P. Amster and C. Rogers, On boundary value problems in three-ion electrodiffusion,, J. Math. Anal. Appl., 333 (2007), 42.  doi: 10.1016/j.jmaa.2007.03.067.  Google Scholar

[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.  doi: 10.1016/j.na.2008.10.036.  Google Scholar

[7]

L. Bass, Irreversible interactions between metals and electrolytes,, Proc. Roy. Soc. Lond. A, 277 (1964), 125.  doi: 10.1098/rspa.1964.0009.  Google Scholar

[8]

L. Bass, Electric structures of interfaces in steady electrolysis,, Trans. Faraday Soc., 60 (1964), 1656.  doi: 10.1039/tf9646001656.  Google Scholar

[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.  doi: 10.1098/rspa.2009.0620.  Google Scholar

[10]

P. C. T. de Boer and G. S. S. Ludford, Spherical electric probe in a continuum gas,, Plasma Phys., 17 (1975), 29.  doi: 10.1088/0032-1028/17/1/004.  Google Scholar

[11]

J. O'M. Bokris and A. K. N. Reddy, Modern Electrochemistry,, Plenum, (1970).   Google Scholar

[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).  doi: 10.1088/1751-8113/45/10/105204.  Google Scholar

[13]

K. S. Cole, Membranes, Ions and Impulses,, University of California Press, (1968).   Google Scholar

[14]

R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system,, J. Phys. A: Math. Theor., 40 (2007).  doi: 10.1088/1751-8113/40/48/F01.  Google Scholar

[15]

C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions,, Mathematics in Science and Engineering, 205 (2006).   Google Scholar

[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.  doi: 10.1007/BF01867851.  Google Scholar

[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.  doi: 10.1007/BF00283254.  Google Scholar

[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.   Google Scholar

[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, (1999), 687.   Google Scholar

[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.  doi: 10.1017/S0956792598003428.  Google Scholar

[21]

P. Holmes and D. Spence, On a Painlevé-type boundary value problem,, Quart. J. Mech. Appl. Math., 37 (1984), 525.  doi: 10.1093/qjmam/37.4.525.  Google Scholar

[22]

N. A. Kudryashov, The second Painlevé equation as a model for the electric field in a semiconductor,, Phys. Lett. A, 233 (1997), 397.  doi: 10.1016/S0375-9601(97)00545-8.  Google Scholar

[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.  doi: 10.1017/S0022377806004648.  Google Scholar

[24]

H. R. Leuchtag, A family of differential equations arising from multi-ion electro-diffusion,, J. Math. Phys., 22 (1981), 1317.  doi: 10.1063/1.525026.  Google Scholar

[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.  doi: 10.1016/S0006-3495(77)85625-7.  Google Scholar

[26]

M. C. Mackey, Admittance properties of electrodiffusion membrane models,, Math. Biosci., 25 (1975), 67.  doi: 10.1016/0025-5564(75)90052-8.  Google Scholar

[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.  doi: 10.1006/jmaa.2001.7675.  Google Scholar

[28]

W. Nernst, Zur Kinetik der in Lösung befindlichen Körper. Erste Abhandlung. Theorie der Diffusion,, Z. Phys. Chem., 2 (1888), 613.   Google Scholar

[29]

O. K. Pashaev and J. H. Lee, Resonance solitons as black holes in Madelung fluid,, Mod. Phys. Lett. A, 17 (2002), 1601.  doi: 10.1142/S0217732302007995.  Google Scholar

[30]

M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten,, Ann. Phys. Chem., 39 (1890), 161.   Google Scholar

[31]

J. R. Ray, Nonlinear superposition law for generalised Ermakov systems,, Phys. Lett. A, 78 (1980), 4.  doi: 10.1016/0375-9601(80)90789-6.  Google Scholar

[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.  doi: 10.1063/1.524625.  Google Scholar

[33]

C. Rogers, A novel Ermakov-Painlevé II system. $N+1$-dimensional coupled NLS and elastodynamic reductions,, Stud. Appl. Math., 133 (2014), 214.  doi: 10.1111/sapm.12039.  Google Scholar

[34]

C. Rogers and H. An, Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory,, Stud. Appl. Math., 125 (2010), 275.  doi: 10.1111/j.1467-9590.2010.00488.x.  Google Scholar

[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.  doi: 10.1006/jmaa.1999.6589.  Google Scholar

[36]

C. Rogers, C. Hoenselaers and J. R. Ray, On $2+1$-dimensional Ermakov systems,, J. Phys. A: Math. Gen., 26 (1993), 2625.  doi: 10.1088/0305-4470/26/11/012.  Google Scholar

[37]

C. Rogers, B. Malomed and H. An, Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics,, Stud. Appl. Math., 129 (2012), 389.  doi: 10.1111/j.1467-9590.2012.00557.x.  Google Scholar

[38]

C. Rogers and W. K. Schief, Multi-component Ermakov systems: Structure and linearization,, J. Math. Anal. Appl., 198 (1996), 194.  doi: 10.1006/jmaa.1996.0076.  Google Scholar

[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.  doi: 10.1006/jmaa.1997.5674.  Google Scholar

[40]

J. Sandblom, Anomalous reactances in electrodiffusion systems,, Biophys. J., 12 (1972), 1118.   Google Scholar

[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.  doi: 10.1088/0305-4470/29/4/017.  Google Scholar

[42]

R. Schlögl, Elektrodiffusion in freier Lösung und geladenen Membranen,, Z. Physik. Chem. Neue Folge, 1 (1954), 305.  doi: 10.1524/zpch.1954.1.5_6.305.  Google Scholar

[43]

T. L. Schwarz, in Biophysics and Physiology of Excitable Membranes,, W.J. Adelman Jr. (Ed.), (1971).   Google Scholar

[44]

H. B. Thompson, Existence of solutions for a two point boundary value problem arising in electro-diffusion,, Acta. Math. Sci., 8 (1988), 373.   Google Scholar

[45]

H. B. Thompson, Existence for Two-Point Boundary Value Problems in Two Ion Electrodiffusion,, Journal of Mathematical Analysis and Applications, 184 (1994), 82.  doi: 10.1006/jmaa.1994.1185.  Google Scholar

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.  doi: 10.1016/j.nonrwa.2013.09.011.  Google Scholar

[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.  doi: 10.3934/dcdsb.2012.17.2299.  Google Scholar

[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.  doi: 10.1016/j.na.2010.06.063.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2003.09.075.  Google Scholar

[5]

P. Amster and C. Rogers, On boundary value problems in three-ion electrodiffusion,, J. Math. Anal. Appl., 333 (2007), 42.  doi: 10.1016/j.jmaa.2007.03.067.  Google Scholar

[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.  doi: 10.1016/j.na.2008.10.036.  Google Scholar

[7]

L. Bass, Irreversible interactions between metals and electrolytes,, Proc. Roy. Soc. Lond. A, 277 (1964), 125.  doi: 10.1098/rspa.1964.0009.  Google Scholar

[8]

L. Bass, Electric structures of interfaces in steady electrolysis,, Trans. Faraday Soc., 60 (1964), 1656.  doi: 10.1039/tf9646001656.  Google Scholar

[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.  doi: 10.1098/rspa.2009.0620.  Google Scholar

[10]

P. C. T. de Boer and G. S. S. Ludford, Spherical electric probe in a continuum gas,, Plasma Phys., 17 (1975), 29.  doi: 10.1088/0032-1028/17/1/004.  Google Scholar

[11]

J. O'M. Bokris and A. K. N. Reddy, Modern Electrochemistry,, Plenum, (1970).   Google Scholar

[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).  doi: 10.1088/1751-8113/45/10/105204.  Google Scholar

[13]

K. S. Cole, Membranes, Ions and Impulses,, University of California Press, (1968).   Google Scholar

[14]

R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system,, J. Phys. A: Math. Theor., 40 (2007).  doi: 10.1088/1751-8113/40/48/F01.  Google Scholar

[15]

C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions,, Mathematics in Science and Engineering, 205 (2006).   Google Scholar

[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.  doi: 10.1007/BF01867851.  Google Scholar

[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.  doi: 10.1007/BF00283254.  Google Scholar

[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.   Google Scholar

[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, (1999), 687.   Google Scholar

[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.  doi: 10.1017/S0956792598003428.  Google Scholar

[21]

P. Holmes and D. Spence, On a Painlevé-type boundary value problem,, Quart. J. Mech. Appl. Math., 37 (1984), 525.  doi: 10.1093/qjmam/37.4.525.  Google Scholar

[22]

N. A. Kudryashov, The second Painlevé equation as a model for the electric field in a semiconductor,, Phys. Lett. A, 233 (1997), 397.  doi: 10.1016/S0375-9601(97)00545-8.  Google Scholar

[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.  doi: 10.1017/S0022377806004648.  Google Scholar

[24]

H. R. Leuchtag, A family of differential equations arising from multi-ion electro-diffusion,, J. Math. Phys., 22 (1981), 1317.  doi: 10.1063/1.525026.  Google Scholar

[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.  doi: 10.1016/S0006-3495(77)85625-7.  Google Scholar

[26]

M. C. Mackey, Admittance properties of electrodiffusion membrane models,, Math. Biosci., 25 (1975), 67.  doi: 10.1016/0025-5564(75)90052-8.  Google Scholar

[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.  doi: 10.1006/jmaa.2001.7675.  Google Scholar

[28]

W. Nernst, Zur Kinetik der in Lösung befindlichen Körper. Erste Abhandlung. Theorie der Diffusion,, Z. Phys. Chem., 2 (1888), 613.   Google Scholar

[29]

O. K. Pashaev and J. H. Lee, Resonance solitons as black holes in Madelung fluid,, Mod. Phys. Lett. A, 17 (2002), 1601.  doi: 10.1142/S0217732302007995.  Google Scholar

[30]

M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten,, Ann. Phys. Chem., 39 (1890), 161.   Google Scholar

[31]

J. R. Ray, Nonlinear superposition law for generalised Ermakov systems,, Phys. Lett. A, 78 (1980), 4.  doi: 10.1016/0375-9601(80)90789-6.  Google Scholar

[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.  doi: 10.1063/1.524625.  Google Scholar

[33]

C. Rogers, A novel Ermakov-Painlevé II system. $N+1$-dimensional coupled NLS and elastodynamic reductions,, Stud. Appl. Math., 133 (2014), 214.  doi: 10.1111/sapm.12039.  Google Scholar

[34]

C. Rogers and H. An, Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory,, Stud. Appl. Math., 125 (2010), 275.  doi: 10.1111/j.1467-9590.2010.00488.x.  Google Scholar

[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.  doi: 10.1006/jmaa.1999.6589.  Google Scholar

[36]

C. Rogers, C. Hoenselaers and J. R. Ray, On $2+1$-dimensional Ermakov systems,, J. Phys. A: Math. Gen., 26 (1993), 2625.  doi: 10.1088/0305-4470/26/11/012.  Google Scholar

[37]

C. Rogers, B. Malomed and H. An, Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics,, Stud. Appl. Math., 129 (2012), 389.  doi: 10.1111/j.1467-9590.2012.00557.x.  Google Scholar

[38]

C. Rogers and W. K. Schief, Multi-component Ermakov systems: Structure and linearization,, J. Math. Anal. Appl., 198 (1996), 194.  doi: 10.1006/jmaa.1996.0076.  Google Scholar

[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.  doi: 10.1006/jmaa.1997.5674.  Google Scholar

[40]

J. Sandblom, Anomalous reactances in electrodiffusion systems,, Biophys. J., 12 (1972), 1118.   Google Scholar

[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.  doi: 10.1088/0305-4470/29/4/017.  Google Scholar

[42]

R. Schlögl, Elektrodiffusion in freier Lösung und geladenen Membranen,, Z. Physik. Chem. Neue Folge, 1 (1954), 305.  doi: 10.1524/zpch.1954.1.5_6.305.  Google Scholar

[43]

T. L. Schwarz, in Biophysics and Physiology of Excitable Membranes,, W.J. Adelman Jr. (Ed.), (1971).   Google Scholar

[44]

H. B. Thompson, Existence of solutions for a two point boundary value problem arising in electro-diffusion,, Acta. Math. Sci., 8 (1988), 373.   Google Scholar

[45]

H. B. Thompson, Existence for Two-Point Boundary Value Problems in Two Ion Electrodiffusion,, Journal of Mathematical Analysis and Applications, 184 (1994), 82.  doi: 10.1006/jmaa.1994.1185.  Google Scholar

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