Total integrals of Ablowitz‐Segur solutions for the inhomogeneous Painlevé II equation

Abstract

In this paper, we establish a formula determining the value of the Cauchy principal value integrals of the real and purely imaginary Ablowitz-Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the analysis of the corresponding Riemann-Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. Obtained integral formulas are consistent with already known analogous results for the Ablowitz-Segur solutions of the homogeneous Painlevé II equation. K E Y W O R D S asymptotic expansion, geometric flow, Painlevé II equation, Riemann-Hilbert-problem 1 − 2 + 3 + 1 2 3 = −2 sin( ).(2)Roughly speaking, any choice of ( 1 , 2 , 3 ) ∈ ℂ 3 satisfying the condition (2) gives us a solution Φ( , ) of the corresponding RH problem, which is a 2 × 2 matrix-valued function sectionally holomorphic in and meromorphic with respect to the variable (see Refs. 1-5). If we assume that ( , ) ∶= (4 3 ∕3 + ) is a phase function and 3 is the third Pauli matrix (see definition (15)), then the function 504

Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant

“…Our next result is the evaluation of the total integrals of the Clarkson-McLeod solutions q(x; α, κ). Similar results for the integrals of the Painlevé II transcendents have been derived in [2,3,10,22,23].…”