Hirota bilinear approach to GUE, NLS, and Painlev\'e IV

Tracy and Widom showed that the level spacing function of the Gaussian unitary ensemble is related to a particular solution of the fourth Painlev'e equation. We reconsider this problem from the viewpoint of Hirota’s bilinear method in soliton theory and present another proof. We also consider the asymptotic behavior of the level spacing function as $s\to \mathrm{\infty }$, and its relation to the “Clarkson-McLeod solution” to the Painlev'e IV equation.