We get the leading term of the Gurevich-Pitaevskii special solution of the KdV equation in the oscillation zone without using averaging methods.

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We show that the known Flaschka-Newell L-A pair for the second Painlevé equation gives solutions to linear evolutionary equations similar to the quantum Schrödinger equations. Using the Fourier transform for distributions, we derive this pair from the classical Garnier pair.

Abstract. The effect of the small dispersion on the self-focusing of solutions of the equations of nonlinear geometric optics in one-dimensional case is investigated. In the main order this influence is described by means of the universal special solution of the nonlinear Schrцdinger equation, which is isomonodromic. Analytic and asymptotic properties of this solution are described.

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