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

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.
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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