On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation
On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation
复制标题
DOI：
10.1016/j.jde.2020.02.003
复制
发表时间：
2020-07
影响因子：
2.4
通讯作者：
D. Dai;Shuai‐Xia Xu;Lun Zhang
中科院分区：
文献类型：
--
作者：
D. Dai;Shuai‐Xia Xu;Lun Zhang
We consider a family of tronquée solutions of the Painlevé II equation q ″(s)= 2 q (s) 3+ s q (s)−(2 α+ 1 2), α>− 1 2, which is characterized by the Stokes multipliers s 1=− e− 2 α π i, s 2= ω, s 3=− e 2 α π i with ω being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if ω= 0. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for α>− 1/2 and ω≥ 0, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters α and ω are chosen to be special values. Some applications of our results in random matrix theory are also discussed.
