Small-amplitude dispersion oscillations on the background of the nonlinear geometric optic approximation

Analogues of the Pearcey integral describe the small dispersion influence on the beginning of spontaneous-vanishing processes for the nonlinear geometric optic approximation amplitude, which is a solution of equations of the focusing nonlinear Schrödinger equation type. The asymptotic behavior as $x^2+t^2\to\infty$ of these analogues is considered. For $x^2+t^2\to\infty$, the special functions under consideration have a domain of small-amplitude high-frequency oscillations, which occur on the background of the nonzero-amplitude nonlinear geometric optic approximation.