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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
L. Bass, Irreversible interactions between metals and electrolytes,, Proc. Roy. Soc. Lond. A, 277 (1964), 125.
doi: 10.1098/rspa.1964.0009. |
[8] |
L. Bass, Electric structures of interfaces in steady electrolysis,, Trans. Faraday Soc., 60 (1964), 1656.
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.
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.
doi: 10.1088/0032-1028/17/1/004. |
[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. |
[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. |
[15] |
C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions,, Mathematics in Science and Engineering, 205 (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.
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.
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. 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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
M. C. Mackey, Admittance properties of electrodiffusion membrane models,, Math. Biosci., 25 (1975), 67.
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.
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. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
L. Bass, Irreversible interactions between metals and electrolytes,, Proc. Roy. Soc. Lond. A, 277 (1964), 125.
doi: 10.1098/rspa.1964.0009. |
[8] |
L. Bass, Electric structures of interfaces in steady electrolysis,, Trans. Faraday Soc., 60 (1964), 1656.
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.
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.
doi: 10.1088/0032-1028/17/1/004. |
[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. |
[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. |
[15] |
C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions,, Mathematics in Science and Engineering, 205 (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.
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.
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. 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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
M. C. Mackey, Admittance properties of electrodiffusion membrane models,, Math. Biosci., 25 (1975), 67.
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.
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. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
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