Integral Equations and Connection Formulae for the Painlevé Equations

Level-spacing distributions and the Airy kernel

“…Hastings and McLeod [13] (see also [5,6]) have shown that q(s; 1), the unique solution to equation (1.11) which is asymptotic to Ai(s) as s → +∞, is asymptotically equal to −s/2 as s → −∞. In fact there is a complete asymptotic expansion in decreasing powers of −s, beginnning with this term [19].…”

Bäcklund Transformations and Solution Hierarchies for the Fourth Painlevé Equation

“…The observation by Ablowitz and Segur [3J led to the formulation of the so-called Painleve conjecture [7,8] (see also [9,10]). Subsequently, solutions of several of the Painleve equations have been expressed The Fourth Painleve Equation ;3 in terms of the solution of linear integral equations from which it is possible to derive many properties of the Painleve equations including global existence and uniqueness of solutions and connection formulae relating the asymptotic behavior of the solutions as z --+ +00 to the asymptotic behavior as z --+ -00 (cf., [3,5,6,9,[11][12][13][14][15][16][17]).…”

Hirota bilinear approach to GUE, NLS, and Painlevé IV

“…Real solutions to the Painlevé IV equation in the case β = 0 and real α with the boundary condition (9) are called "Clarkson-McLeod solution" [18][19][20]. Moreover, in the case β = 0 and real α, any solutions of the Painlevé IV Eq.…”

“…We also consider the asymptotic behavior of F 2,N (s) as s → ±∞, and its relation to the "ClarksonMcLeod solution" [18][19][20] to the Painlevé IV equation. This paper is dedicated to the memory of Professor Ryogo Hirota, who introduced his bilinear method that is a powerful tool for obtaining a wide class of exact solutions of nonlinear integrable equations, such as the NLS equation and the Painlevé IV equation.…”