Rational solutions of Painlevé-II equation as Gram determinant

2022/03/05 by Ling, Liming, Lu, Bing-Ying, Zhang, Xiaoen
#33E17 #34M50 #37J65 #37K10 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2203.02647

Abstract

Under the Flaschka-Newell Lax pair, the Darboux transformation for the Painlevé-II equation is constructed by the limiting technique. With the aid of the Darboux transformation, the rational solutions are represented by the Gram determinant, and then we give the large y asymptotics of the determinant and the rational solutions. Finally, the solution of the corresponding Riemann-Hilbert problem is obtained from the Darboux matrices.

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