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HEUN EQUATION AND PAINLEVÉ EQUATION

by Kouichi Takemura , 2005
"... Abstract. We relate two parameter solutions of the sixth Painlevé equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on differentials of elliptic modular functions, and obtain the ..."
Abstract - Cited by 2 (1 self) - Add to MetaCart
Abstract. We relate two parameter solutions of the sixth Painlevé equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on differentials of elliptic modular functions, and obtain

Folding transformations of the Painlevé equations

by Teruhisa Tsuda, Kazuo Okamoto, Hidetaka Sakai, Teruhisa Tsuda, Kazuo Okamoto, Hidetaka Sakai , 2005
"... New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3, or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We ma ..."
Abstract - Cited by 34 (3 self) - Add to MetaCart
New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3, or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We

Dynamics of the sixth Painleve equation

by Michi-Aki Inaba , Katsunori Iwasaki , Masa-Hiko Saito - Proceedings of Conference Internationale Theories Asymptotiques et Equations de Painleve, Seminaires et Congres, Soc
"... Abstract. -The sixth Painlevé equation is hiding extremely rich geometric structures behind its outward appearance. This article tries to give as a total picture as possible of its dynamical natures, based on the Riemann-Hilbert approach recently developed by the authors, using various techniques f ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
Abstract. -The sixth Painlevé equation is hiding extremely rich geometric structures behind its outward appearance. This article tries to give as a total picture as possible of its dynamical natures, based on the Riemann-Hilbert approach recently developed by the authors, using various techniques

On the Linearization of the Painlevé Equations

by N. Joshi, A A. V. Kitaev, P. A. Treharne , 2007
"... We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system, to its Lax pair. Thus we obtain new 3×3 matrix Fuchs–Garnier pairs for the third and fifth Painlevé equations, together with the previously known Fuchs– Garnier pair for the fourth and sixth Pain ..."
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We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system, to its Lax pair. Thus we obtain new 3×3 matrix Fuchs–Garnier pairs for the third and fifth Painlevé equations, together with the previously known Fuchs– Garnier pair for the fourth and sixth

NONLINEAR QUASICLASSICS AND THE PAINLEVÉ EQUATIONS

by Vadim L. Vereschagin , 2008
"... A century-old history of calculation of asymptotics for solutions to Painlevé equations (usually denoted Pj, j=1,2,...,6) as their variable x tends to infinity was started by pioneer works by Painlevé, Gambier and Boutroux [1]. In 1980-1981 papers by Jimbo, Miwa and Flashka, Newell [2] initiated the ..."
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A century-old history of calculation of asymptotics for solutions to Painlevé equations (usually denoted Pj, j=1,2,...,6) as their variable x tends to infinity was started by pioneer works by Painlevé, Gambier and Boutroux [1]. In 1980-1981 papers by Jimbo, Miwa and Flashka, Newell [2] initiated the

QUIVERS AND DIFFERENCE PAINLEVÉ EQUATIONS

by Philip Boalch , 2007
"... We will describe natural Lax pairs for the difference Painlevé equations with affine Weyl symmetry groups of types E6, E7 and E8, showing that they do indeed arise as symmetries of certain Fuchsian systems of differential equations. ..."
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We will describe natural Lax pairs for the difference Painlevé equations with affine Weyl symmetry groups of types E6, E7 and E8, showing that they do indeed arise as symmetries of certain Fuchsian systems of differential equations.

Hypergeometric solutions to the q-Painlevé equations

by K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada - Internat. Math. Res. Notices
"... Hypergeometric solutions to seven q-Painlevé equations in Sakai’s classification are constructed. Geometry of plane curves is used to reduce the q-Painlevé equations to the three-term recurrence relations for q-hypergeometric functions. 1 ..."
Abstract - Cited by 38 (5 self) - Add to MetaCart
Hypergeometric solutions to seven q-Painlevé equations in Sakai’s classification are constructed. Geometry of plane curves is used to reduce the q-Painlevé equations to the three-term recurrence relations for q-hypergeometric functions. 1

SYMMETRIES IN THE THIRD PAINLEVÉ EQUATION

by Yusuke Sasano , 704
"... Abstract. The third Painlevé equation PIII is known to have symmetry of the affine Weyl group of type C (1) 2 with respect to the Bäcklund transformations. We introduce a new representation of this system. 0. Statement of main results It is well-known by K. Okamoto that the third Painlevé equation h ..."
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Abstract. The third Painlevé equation PIII is known to have symmetry of the affine Weyl group of type C (1) 2 with respect to the Bäcklund transformations. We introduce a new representation of this system. 0. Statement of main results It is well-known by K. Okamoto that the third Painlevé equation

Lax forms of the q-Painlevé equations

by Mikio Murata , 810
"... All q-Painlevé equations which are obtained from the q-analog of the sixth Painlevé equation are expressed in a Lax formalism. They are characterized by the data of the associated linear q-difference equations. The degeneration pattern from the q-Painlevé equation of type A2 is also presented. ..."
Abstract - Cited by 4 (0 self) - Add to MetaCart
All q-Painlevé equations which are obtained from the q-analog of the sixth Painlevé equation are expressed in a Lax formalism. They are characterized by the data of the associated linear q-difference equations. The degeneration pattern from the q-Painlevé equation of type A2 is also presented.

New expressions for discrete Painlevé equations

by Mikio Murata - Funkcial. Ekvac
"... It is known that discrete Painlevé equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlevé equations in which the symmetries become clearly visible. We know how to obtain discrete Painlevé equations from certain rational surfaces in con ..."
Abstract - Cited by 6 (0 self) - Add to MetaCart
It is known that discrete Painlevé equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlevé equations in which the symmetries become clearly visible. We know how to obtain discrete Painlevé equations from certain rational surfaces
Results 1 - 10 of 831