Weak Nonlinear Asymptotic Solutions for the Fourth Order Analogue of the Second Painlevé Equation

The fourth-order analogue of the second Painlevé equation is considered. The monodromy manifold for a Lax pair associated with the $P_2^2$ equation is constructed. The direct monodromy problem for the Lax pair is solved. Asymptotic solutions expressed via trigonometric functions in the Boutroux variables along the rays $\phi = \frac{2}{5}\pi(2n+1)$ on the complex plane have been found by the isomonodromy deformations technique.