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Solutions of Painlevé II on real intervals: novel approximating sequences. (English) Zbl 1483.34122

An approximation method is proposed for a boundary value problem for the second Painlevé equation with Neumann boundary conditions \[ y^{\prime\prime}(z) =2y(z)^3 +z y(z) +C, \quad y'(a)=0, \quad y'(b)=0, \quad (a<z<b). \] In the first step, a generalization of the second Painlevé equation is considered. The generalized equation on \(E(x)\) contains \(E(0), E(1)\) as nonlinear terms. In the expansion series \( E(x)=E_1(x) +E_2(x)+\cdots \), each \(E_n\) satisfies a nonhomogeneous Airy equation. By a suitable change of variables, a curious approximation series \(y_E^{(n)}\) is defined. \(y_E^{(n)}\) is a solution of a nonlinear equation with Neumann boundary conditions on an interval \([a_n, b_n\)], accompanied with a constant \(C_n\). When \(n \to \infty\), \(y_E^{(n)}\) converges to the solution \(y(z)\) and \(a_n, b_n\) and \(C_n\) also converges to \(a, b\) and \(C\), respectively, if the nonlinear term \(|y(z)|^3\) is small. A numerical example is also shown.
It is not clear why this method works well although the second Painlevé equation is closely related to the Airy function.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
33E17 Painlevé-type functions
34B15 Nonlinear boundary value problems for ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations

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References:

[1] Ablowitz, M. J.; Segur., H., “Exact Linearization of a Painlevé Transcendent.”, Phys. Rev. Letts, 38, 1103-1106 (1977) · doi:10.1103/PhysRevLett.38.1103
[2] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1964), New York: Dover, New York · Zbl 0171.38503
[3] Amster, P.; Kwong, M. K.; Rogers., C., “On a Neumann Boundary Value Problem for the Painleve II Equation in Two-Ion Electro-Diffusion.”, Nonlin. Anal. Th. Meth. App, 74, 2897-2907 (2011) · Zbl 1218.34018
[4] Bass., L., “Electrical structures of interfaces in steady electrolysis.”, Trans. Faraday Soc, 60, 1656-1663 (1964)
[5] Bass, L.; Bracken., A. J., “Emergent Behaviour in Electrodiffusion: Planck’s Other Quanta.”, Rep. Math. Phys, 73, 65-75 (2014) · Zbl 1314.82037
[6] Bass, L.; Nimmo, J. J. C.; Rogers, C.; Schief., W. K., “Electrical Structures of Interfaces: A Painlevé II Model.”, Proc. Roy. Soc. (London) A, 466, 2117-2136 (2010) · Zbl 1192.35173
[7] Bassom, A. P.; Clarkson, P. A.; Law, C. K.; McLeod., J. B., “Application of Uniform Asymptotics to the Second Painlevé Transcendent.”, Arch. Rat. Mech. Anal, 143, 241-271 (1998) · Zbl 0912.34007
[8] Bracken, A. J.; Bass., L., “Differential Equations of Electrodiffusion: Constant Field Solutions, Uniqueness, and New Formulas of Goldman-Hodgkin-Katz Type.”, SIAM J. App. Math, 76, 2286-2305 (2016) · Zbl 1404.82055
[9] Bracken, A. J.; Bass., L., “Series Solution of Painlevé II in Electrodiffusion: Conjectured Convergence.”, J. Phys. A: Math. Theor, 51, 035202 (2018) · Zbl 1383.34017
[10] Bracken, A. J.; Bass, L.; Rogers., C., “Bäcklund Flux-Quantization in a Model of Electrodiffusion Based on Painlevé II.”, J. Phys. A: Math. Theor, 45, 105204 (2012) · Zbl 1242.82037
[11] Clarkson, P. A.; Márcellan, F.; van Assche, W., Lecture Notes in Mathematics, 1883, Painlevé Equations: Nonlinear Special Functions.” In Orthogonal Polynomials and Special Functions: Computation and Application, 331-411 (2006), Berlin: Springer-Verlag, Berlin · Zbl 1095.33001
[12] Clarkson., P. A., “On Airy Solutions of the Second Painlevé Equation.”, Stud. App. Math, 137, 93-109 (2016) · Zbl 1346.35179
[13] Forrester, P. J.; Witte, N. S., Painlevé II in Random Matrix Theory and Related Fields, Constructive Approx., 41, 589-613 (2015) · Zbl 1318.81048
[14] Ince, E. L., Ordinary Differential Equations (1956), New York: Dover, New York · JFM 53.0399.07
[15] Joshi, N.; Kruskal., M. D., “A Direct Proof That Solutions of the 6 Painlevé Equations Have No Movable Singularities Except Poles.”, Stud. Appl. Math, 93, 187-207 (1994) · Zbl 0823.34004
[16] Kajiwara, K.; Noumi, M.; Yamada., Y., “Geometric Aspects of Painlevé Equations.”, J. Phys. A: Math. Theor, 50, 073001 (2017) · Zbl 1441.34095
[17] Lukashevich, N. A., “The Second Painlevé Equation.”, Diff. Equ, 7, 853-854 (1971) · Zbl 0271.34011
[18] MacGillivray., A. D., “Nernst-Planck Equations and Electroneutrality and Donnan Equilibrium Assumptions.”, J. Chem. Phys, 48, 2903-2907 (1968)
[19] MATLAB (2016), Natick, MA: MathWorks, Natick, MA · Zbl 1358.65086
[20] Montroll, E. W.; Shuler., K. E., “ Dynamics of Technological Evolution: Random Walk Model for the Research Enterprise.”, Proc. Natl. Acad. Sci. USA, 76, 6030-6034 (1979)
[21] Painlevé., P., “Sur les Équations Différentielles du Second Ordre et d’Ordre Supérieur dont l’Intégrale Générale est Uniforme.”, Acta Math, 25, 1-85 (1902) · JFM 32.0340.01
[22] Park, J.-H.; Jerome., J. W., “Qualitative Properties of Steady-State Poisson-Nernst-Planck Systems: Mathematical Study.”, SIAM J. Appl. Math, 57, 609-630 (1997) · Zbl 0874.34017
[23] Rogers, C.; Bassom, A. P.; Schief., W. K., “On a Painleve II Model in Steady Electrolysis: Application of a Bäcklund Transformation.”, J. Math. Anal. App, 240, 367-381 (1999) · Zbl 0944.34076
[24] Rogers, C.; Clarkson, P. A., Ermakov-Painlevé II Reduction in Cold Plasma Physics. Application of a Bäcklund Transformation, J. Nonlin. Math. Phys., 25, 247-261 (2018) · Zbl 1411.37061
[25] Sakai., H., “Rational Surfaces Associated with Affine Root Systems and Geometry of the Painlevé Equations.”, Commun. Math. Phys, 220, 165-229 (2001) · Zbl 1010.34083
[26] Thompson., H. B., “Existence for 2-Point Boundary-Value-Problems in 2-Ion lectrodiffusion.”, J. Math. Anal. App, 184, 82-94 (1994) · Zbl 0802.34025
[27] Vorob’ev, A. P., “On Rational Solutions of the Second Painlevé Equation.”, Diff. Equ, 1, 79-81 (1965)
[28] Yablonskii, A. I., “On Rational Solutions of the Second Painlevé Equation.”, Vesti. Akad. Nauk BSSR. Ser. Fiz. Tkh. Nauk, 3, 30-35 (1959)
[29] Zaltzman, B.; Rubinstein, I., “Electro-Osmotic Slip and Electroconvective Instability.”, J. Fluid Mech, 579, 173-226 (2007) · Zbl 1175.76168
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