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Tracy and Widom showed that the level spacing function of the Gaussianunitary ensemble is related to a particular solution of the fourth Painlev\'eequation. We reconsider this problem from the viewpoint of Hirota's bilinearmethod in soliton theory and present another proof. We also consider theasymptotic behavior of the level spacing function as $s\to\infty$, and itsrelation to the "Clarkson-McLeod solution" to the Painlev\'e IV equation.

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