Cauchy problem for a nonlinear Schrödinger equation with a large initial gradient in the weakly dispersive limit

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation with a large gradient of the initial function and a small dispersion parameter. The renormalization method is used to construct an asymptotic solution in the explicit form of integral convolution. An asymptotic analogue of the renormalization group property is established under scaling transformations determined by the dispersion parameter. In the case of a negative focusing coefficient, a clarifying expression is obtained for the asymptotic solution in terms of known elliptic special functions.