On the Lax pairs of the sixth Painlevé equation
نویسنده
چکیده
The dependence of the sixth equation of Painlevé on its four parameters (2α,−2β, 2γ, 1− 2δ) = (θ ∞ , θ 0 , θ 2 1 , θ 2 x) is holomorphic, therefore one expects all its Lax pairs to display such a dependence. This is indeed the case of the second order scalar “Lax” pair of Fuchs, but the second order matrix Lax pair of Jimbo and Miwa presents a meromorphic dependence on θ∞ (and a holomorphic dependence on the three other θj). We analyze the reason for this feature and make suggestions to suppress it.
منابع مشابه
Trivially related lax pairs of the Sawada-Kotera equation
We show that a recently introduced Lax pair of the Sawada-Kotera equation is nota new one but is trivially related to the known old Lax pair. Using the so-called trivialcompositions of the old Lax pairs with a differentially constrained arbitrary operators,we give some examples of trivial Lax pairs of KdV and Sawada-Kotera equations.
متن کاملA Lax pair for a lattice modified KdV equation, reductions to q-Painlevé equations and associated Lax pairs
We present a new, nonautonomous Lax pair for a lattice nonautomous modified Korteweg–deVries equation and show that it can be consistently extended multidimensionally, a property commonly referred to as being consistent around a cube. This nonautonomous equation is reduced to a series of q-discrete Painlevé equations, and Lax pairs for the reduced equations are found. A 2× 2 Lax pair is given f...
متن کاملOn the Linearization of the Painlevé III-VI Equations and Reductions of the Three-Wave Resonant System
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, fourth, and fifth Painlevé equations, together with the previously known Fuchs–Garnier pair for the sixth Painlevé equation. These Fuchs–Garnier pairs have an important feature: they are linear with respect to ...
متن کاملOn the Linearization of the Painlevé Equations
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 Painlevé equations. These Fuchs–Garnier pairs have an important feature: they are linear with respect...
متن کامل3 × 3 Lax pairs for the fourth , fifth and sixth Painlevé equations
We obtain 3 × 3 matrix Lax pairs for systems of ODEs that are solvable in terms of the fourth, fifth and sixth Painlevé equations by considering similarity reductions of the scattering Lax pair for the (2+1)-dimensional three-wave resonant interaction system. These results allow us to construct new 3× 3 Lax representations for the fourth and fifth Painlevé equations, together with the previousl...
متن کامل