Painlevé equations

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Painlevé equations are nonlinear ordinary differential equations of the second order in complex , the solutions of which have moving singularities that are at most poles . They were introduced around 1900 and in the years thereafter by Paul Painlevé in search of new special functions that are defined by such differential equations and play a major role in the theory of precisely integratable systems in mathematical physics. The solutions to the six types of Painlevé equations are called Painlevé transcendentals.

definition

Painlevé equations have the form (subscripts indicate partial derivatives):

with an in and rational and in x locally analytic function . The solutions can have isolated singularities ( poles and essential singularities) which, since the equation is non-linear, can also be dependent on the function sought or its initial values ​​(movable singularities). Painlevé demanded that the moving singularities of the solutions of the differential equation should only be ordinary poles and not essential singularities or branching points ( Painlevé property ). This can also be formulated in such a way that the solutions of the equation remain unambiguous in the vicinity of the moving singularities.

Painlevé's motivation was to find new special functions. Many of the previously known special functions had proven to be the solution of the hypergeometric differential equation , an ordinary linear differential equation of the 2nd order, the coefficients of which in the complex have at most three poles. Another productive group of special functions were the elliptic functions, such as the Weierstrass function, which satisfies a nonlinear differential equation of the first order in the complex. In the case of differential equations of the first order, Lazarus Fuchs and Henri Poincaré had already shown in 1884 that only the Riccati differential equation fulfilled the Painlevé condition, and Sofia Kowalewskaja had shown in the case of the equations of the heavy gyroscope that the solutions with Painlevé condition were exactly integrable .

Inspired by this, Painlevé systematically examined all 2nd order equations with the above properties and sorted out all cases in which they could be traced back to linear differential equations or the solutions were already known (elliptic functions, etc.). In the end, through his efforts and that of Bertrand Gambier , who found three of the equations that Painlevé had overlooked, a list of six Painlevé's equations came together. These are independent of each other for “generic” values ​​of the parameters, which was controversial in Painlevé's time (there was a dispute with Roger Liouville who doubted this), but from the late 1980s onwards by Japanese mathematicians such as Keiji Nishioka and Hiroshi Umemura has been proven.

The six Painlevé equations

The equations are called Painlevé I to VI (Roman numerals, here 1 to 6) (Greek letters such as denote complex constants).

  • Type 1:
  • Type 2:
  • Type 3:
  • Type 4::
  • Type 5:
  • Type 6:

Types 1 to 3 are from Painlevé, type 4.5 from Gambier. Type 6 was also added by Gambier, but originally found by Richard Fuchs , the son of Lazarus Fuchs, as a monodromic equation. To this end, he considered a Fuchs differential equation , a linear differential equation of the second order on the Riemann sphere with four regular singular places (fixed essential singularities, o. B. d. A. in the points and a movable singularity in the point ), and that Behavior of the solutions on paths around the singularities (monodromy). This can be described by a substitution in the space of the fundamental solutions (with the monodrome matrix) and Fuchs looked for an equation whose monodrome did not depend on the movable singularity (isomonodrome deformation) and thus came up with the Painleve equation of type 6. The additional equations, the Fuchs stated, can today be regarded as lax couples . Soon after Fuchs, the investigation of the isomonodromy of the Painlevé equations was expanded by Ludwig Schlesinger and René Garnier , and Garnier also found connections with integrable systems that are described by Abelian functions. From the 1970s, the theory of isomonodromic deformations of the Painleve equation was further developed by Michio Jimbō , Tetsuji Miwa , Mikio Satō , K. Ueno as well as Hermann Flaschka and Alan C. Newell (isomonodromic deformation method) and others. It is connected with Riemann-Hilbert problems. Fuchs also found a connection with the incomplete elliptic integral and the Picard-Fuchs differential equation, which was taken up by Painlevé and in 1998 by Yuri Manin . The equation of type 6 contains the others as special cases for certain values ​​of the parameters.

The Painlevé equations can also be formulated as canonical equations within the framework of Hamiltonian mechanics (J. Malmquist 1922/23, K. Okamoto 1979/80). Furthermore, they have hidden symmetries that can be expressed by Bäcklund transformations of the dependent and independent variables of the equations. Kazuo Okamoto, who in 1979 developed a geometric theory of the space of the initial values ​​of the Painleve equations (Okamoto space), linked these symmetries with certain Lie algebras and the associated affine Weyl algebras. Geometric theory was used by H. Sakai in 2001 to classify all discrete and continuous Painleve equations.

Jean Chazy tried to extend the program to third order differential equations. However, the equations in question have solutions that are related to the module functions and their singularities do not satisfy the Painlevé condition (as with the module functions, movable edges appear as singularities). Special cases have elliptic functions and integrals as a solution and are connected to the solutions of the sixth Painlevé equation.

Applications

In physics, they were used in statistical mechanics ( Ising model and various other spin systems, Bosegas, theory of random matrices, etc.), in two-dimensional quantum gravity and string theory. For example, the correlation function for the two-dimensional Ising model can be expressed by Painleve transcendent of type 3.

Painlevé himself applied it in the theory of algebraic surfaces. This was taken up by K. Okamoto in 1979 in his geometric theory of the Painlevé equations. They also have applications in the differential geometry of surfaces.

In general, it also plays a major role in the theory of precisely integrable systems. As already mentioned, Sofia Kowalewskaja found a connection between the absence of movable singularities that are not of the pole type and the exact integrability of the top. This was taken up in 1977 by Mark J. Ablowitz and Harvey Segur in the case of non-linear partial differential equations (PDE) for solitons , which can be exactly solved with the inverse scattering transformation (IST) . and Ablowitz, Ramani and Segur hypothesized that every ordinary differential equation that arises from the reduction of such PDE integrable with the IST possesses the Painlevé property (with a possible variable transformation being necessary). Ablowitz and Segur and others demonstrated this with many examples such as the Korteweg-de-Vries equation , which leads to the Painleve equation of type 2, the Sinus-Gordon equation , which leads to type 3, and the Boussinesq equation , which leads to type 1. The assumption has been confirmed many times and also used to find new soliton equations and serves as the basis for tests for exact integrability (Painlevé tests), but has not yet been proven. A variant that can be applied directly to PDE was given by J. Weiss, M. Tabor and G. Carneval in 1983.

literature

Original works:

  • Painlevé: Memoire sur les equations différentielles dont l'intégrale générale est uniforme, Bull. Soc. Math. France, Vol. 28, 1900, pp. 201-261. Digitized
  • Painlevé: Sur les equations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math., Volume 21, 1902, pp. 1-85.
  • Gambier: Sur les equations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critique fixés, Acta. Math., Vol. 33, 1910, 1-55
  • R. Fuchs: Sur quelques equations différentielles linéaires du second ordre, CR Acad. Sci. Paris, Volume 141, 1905, pp. 555-558
  • R. Fuchs: About linear homogeneous differential equations of the second order with three essentially singular places located in the finite, Math. Annalen, Volume 63, 1907, pp. 301–321, SUB Göttingen

Textbooks, introductions:

  • MA Ablowitz, PA Clarkson: Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press 1991
  • Robert Conte (Ed.): The Painlevé property, one century later, CRM Series in Mathematical Physics, Springer 1999
  • Masatoshi Noumi: Painlevé equations through symmetry, American Mathematical Society 2004

Web links

Individual evidence

  1. The singularities that are not poles are also referred to as critical points and the Painlevé condition then states that there must be no moving critical points. With differential equations in three and more dimensions, even more unpleasant singularities can occur and the edges become singular (Jean Chazy, see below)
  2. For special values ​​of the parameters, however, it may be possible to derive some of the equations from one of the others
  3. Masatoshi Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs 223, Providence, RI: American Mathematical Society 2004
  4. Schlesinger, About a class of differential systems of any order with fixed critical points, J. für Reine und Angewandte Mathematik, Volume 141, 1912, pp. 96-145, SUB Göttingen
  5. Garnier, Sur des équations différentielles du troisième ordre dont l'intégrale est uniform et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses point critiques fixés, Ann. Sci. de l'ENS, Volume 29, 1912, pp. 1-16
  6. Garnier, Etudes de l'intégrale générale de l'équation VI de M. Painlevé dans le voisinage de ses singularité transcendentes, Ann. Sci. Ecole Norm. Sup. (3), Vol. 34, 1917, pp. 239-353
  7. Garnier, Sur une classe de systèmes differentiels abéliens deduits de la théorie des equations linéaires, Rend. Circ. Mat. Palermo, Volume 43, 1918-19, pp. 155-191
  8. H. Flaschka, AC Newell, Monodromy and spectrum preserving deformations, Part 1, Comm. Math. Phys., Vol. 76, 1980, pp. 65-116
  9. ^ PA Clarkson et al. a., One hundred years of PVI, the Fuchs-Painlevé equation, J. Phys. A, Volume 39, 2006, foreword
  10. Alexander Its, Victor Novokshenov, The isomonodromy deformation method in the theory of Painlevé equations, Springer, Lecture Notes in Mathematics 1191, 1986
  11. Athanassios Fokas, Alexander Its, Andrei Kapaev, Victor Novokshenov, Painlevé Transcendents - The Riemann-Hilbert approach, AMS, Math. Surveys and Monographs 128, 2006
  12. ^ Painlevé, Sur les equations différentielles du second ordre à point critiques fixes, CR Acad. Sci. (Paris), Vol. 143, 1906, pp. 1111-1117
  13. ^ Manin, Sixth Painlevé equation, universal elliptic curve, and mirror of P2, AMS Transl. (2), Volume 186, 1998, pp. 131-151
  14. Fokas, Ablowitz, On a unified approach to the transformations and elementary solutions of the Painlevé equations, J. Math. Phys., Volume 23, 1982, pp. 2033-2042
  15. H. Sakai, Rational surfaces associated to affine root systems and geometry of the Painlevé equations, Comm. Math. Phys., Vol. 220, 2001, pp. 165-229
  16. Chazy, Sur les equations différentielles dont l'intégrale générale possede un coupure essential mobile, CR Acad. Sci. (Paris), Volume 150, 1910, pp. 456–458, Sur les equations différentielles de troisième ordre et d'ordre supérieur dont l'ntégrale générale a ses points critiques fixés, Acta Math., Volume 33, 1911, p. 317 -385
  17. Barry McCoy , Spin Systems, Statistical Mechanics and Painlevé Functions, in D. Levi, P. Winternitz (eds.), Painlevé Transcendents: Their Asymptotics and Physical Applications, NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, 1992, pp. 377-391
  18. Craig Tracy , Harold Widom , Fredholm determinants, differential equations and matrix models, Comm. Math. Phys., Vol. 163, 1994, pp. 33-72. Arxiv
  19. Tracy, Widom, Level spacing distributions and the Airy Kernel, Comm. Math. Phys., Vol. 159, 1994, pp. 151-174. Arxiv . The Tracy-Widom distribution, the probability distribution of the normalized greatest eigenvalue of a Hermitian random matrix, can be expressed by a Painlevé transcendent of type 2.
  20. ^ TT Wu, BM McCoy, C..A. Tracy, E. Barouch, Spin-spin correlation functions for the two dimensional Ising model: exact theory in the scaling region, Physical Review B, Volume 13, 1976, 316-374. See Barry McCoy, Ising model, exact results, Scholarpedia
  21. Painlevé, Lemons de Stockholm, oeuvre, Volume 1, Paris 1972
  22. Okamoto, Sur les feuilletages associés aux equations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math., Vol. 5, 1979, pp. 1-79
  23. Alexander Bobenko, Ulrich Eitner, Painlevé equations in the differential geometry of surfaces, Springer, Lecture Notes in Mathematics 1753, 2000
  24. Ablowitz, Segur, Exact linearization of a Painlevé transcendent; Phys. Rev. Lett., Vol. 38, 1977, pp. 1103-1106
  25. M. Ablowitz, A. Ramani, H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type, 2 parts, J. Math. Phys., Volume 21, 1980, pp. 715-721, 1006 -1015
  26. ^ Ablowitz, Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, 1991, p. 359
  27. Weiss, Tabor, Carneval, The Painlevé property for partial differential equations, J. Math. Phys., Volume 24, 1983, pp. 522-526