Connection problem of the first Painlevé transcendents with large initial data

Abstract

Abstract In previous work, Bender and Komijani (2015 J. Phys. A: Math. Theor. 48 475202) studied the first Painlevé (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be asymptotically determined using a -symmetric Hamiltonian. In the present work, we consider the initial value problem of the PI equation in a more general setting. We show that the initial conditions ( y ( 0 ) , y ′ ( 0 ) ) = ( a , b ) located on a sequence of curves Γ n , n = 1 , 2 , … , will give rise to separatrix solutions. These curves separate the singular and the oscillating solutions of PI. The limiting form equation b 2 / 4 − a 3 = f n ∼ A n 6 / 5 for the curves Γ n as n → ∞ is derived, where A is a positive constant. The discrete set { f n } could be regarded as the nonlinear eigenvalues. Our analytical asymptotic formula of Γ n matches the numerical results remarkably well, even for small n. The main tool is the method of uniform asymptotics introduced by Bassom et al (1998 Arch. Rational Mech. Anal. 143 241–71) in the studies of the second Painlevé (PII) equation.