Analytical solution for the modified nonlinear Schrödinger equation describing optical shock formation

We present an exact analytical solution by the use of an ansatz method for the modified nonlinear Schr\"odinger equation iU_ζ+1/2σU_ττ+N²|U|²U+isN²(|U|²U)_τ=0, describing the propagation of light pulses in optical fibers. The inclusion of the term isN²(|U|²U)_τ in the usual nonlinear Schr\"odinger equation arises from an intensity-dependent group velocity and produces a temporal pulse distortion leading to the development of an optical shock. Previous work [Xu Bingzhen and Wang Wenzheng, Phys. Rev. E 51, 1493 (1995)] using the traveling-wave method does not exhibit this important physical picture.