Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 068, 10 pages      arXiv:1807.04442      https://doi.org/10.3842/SIGMA.2018.068
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

Numerical Approach to Painlevé Transcendents on Unbounded Domains

Christian Klein and Nikola Stoilov
Institut de Mathématiques de Bourgogne, UMR 5584, Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon Cedex, France

Received April 18, 2018, in final form July 02, 2018; Published online July 12, 2018

Abstract
A multidomain spectral approach for Painlevé transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and approximates the solution on straight lines lying entirely within said sector without the need of evaluating truncations of the series at any finite point. The accuracy of the method is illustrated for the example of the tritronquée solution to the Painlevé I equation.

Key words: Painlevé equations; spectral methods.

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References

  1. Bornemann F., On the numerical evaluation of distributions in random matrix theory: a review, Markov Process. Related Fields 16 (2010), 803-866, arXiv:0904.1581.
  2. Bornemann F., On the numerical evaluation of Fredholm determinants, Math. Comp. 79 (2010), 871-915, arXiv:0804.2543.
  3. Boutroux P., Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre, Ann. Sci. École Norm. Sup. (3) 30 (1913), 255-375.
  4. Boutroux P., Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre (suite), Ann. Sci. École Norm. Sup. (3) 31 (1914), 99-159.
  5. Clarkson P.A., Painlevé equations - nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 331-411.
  6. Costin O., Huang M., Tanveer S., Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of $P_{\rm I}$, Duke Math. J. 163 (2014), 665-704, arXiv:1209.1009.
  7. Crespo S., Klein C., Stoilov N., Vallée C., Multidomain spectral method for the hypergeometric function, in preparation.
  8. Driscoll T.A., Bornemann F., Trefethen L.N., The chebop system for automatic solution of differential equations, BIT 48 (2008), 701-723.
  9. Driscoll T.A., Weideman J.A.C., Optimal domain splitting for interpolation by Chebyshev polynomials, SIAM J. Numer. Anal. 52 (2014), 1913-1927.
  10. Dubrovin B., Grava T., Klein C., On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation, J. Nonlinear Sci. 19 (2009), 57-94.
  11. Fasondini M., Fornberg B., Weideman J.A.C., Methods for the computation of the multivalued Painlevé transcendents on their Riemann surfaces, J. Comput. Phys. 344 (2017), 36-50.
  12. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents: the Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
  13. Fornberg B., Weideman J.A.C., A numerical methodology for the Painlevé equations, J. Comput. Phys. 230 (2011), 5957-5973.
  14. Frauendiener J., Klein C., Computational approach to hyperelliptic Riemann surfaces, Lett. Math. Phys. 105 (2015), 379-400, arXiv:1408.2201.
  15. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000.
  16. Grava T., Kapaev A., Klein C., On the tritronquée solutions of ${\rm P}_{\rm I}^2$, Constr. Approx. 41 (2015), 425-466, arXiv:1306.6161.
  17. Grava T., Klein C., Numerical study of a multiscale expansion of Korteweg-de Vries and Camassa-Holm equation, in Integrable Systems and Random Matrices, Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 81-98, math-ph/0702038.
  18. Grava T., Klein C., A numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions, Phys. D 241 (2012), 2246-2264, arXiv:1202.0962.
  19. Hastings S.P., McLeod J.B., A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 73 (1980), 31-51.
  20. Ince E.L., Ordinary differential equations, Dover Publications, New York, 1956.
  21. Joshi N., Kitaev A.V., On Boutroux's tritronquée solutions of the first Painlevé equation, Stud. Appl. Math. 107 (2001), 253-291.
  22. Kapaev A.A., Monodromy deformation approach to the scaling limit of the Painlevé first equation, in The Kowalevski Property (Leeds, 2000), CRM Proc. Lecture Notes, Vol. 32, Amer. Math. Soc., Providence, RI, 2002, 157-179, nlin.SI/0105002.
  23. Kapaev A.A., Monodromy approach to the scaling limits in isomonodromy systems, Theoret. and Math. Phys. 137 (2003), 1691-1702, nlin.SI/0211022.
  24. Kapaev A.A., Quasi-linear stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen. 37 (2004), 11149-11167, nlin.SI/0404026.
  25. Lanczos C., Trigonometric interpolation of empirical and analytic functions, J. Math. and Phys. 17 (1938), 123-199.
  26. Novokshenov V.Yu., Padé approximations for Painlevé I and II transcendents, Theoret. and Math. Phys. 159 (2009), 853-862.
  27. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, available at https://dlmf.nist.gov/.
  28. Olver S., Numerical solution of Riemann-Hilbert problems: Painlevé II, Found. Comput. Math. 11 (2011), 153-179.
  29. Trefethen L.N., Spectral methods in MATLAB, Software, Environments, and Tools, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
  30. Trefethen L.N., Approximation theory and approximation practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.
  31. Weideman J.A.C., Reddy S.C., A MATLAB differentiation matrix suite, ACM Trans. Math. Software 26 (2000), 465-519.

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