The Perturbed Plane‐Wave Solutions of the Cubic Schrödinger Equation

A detailed analysis is given to the solution of the cubic Schrödinger equation iq t + q xx + 2| q | 2 q = 0 under the boundary conditions as | x |→∞. The inverse‐scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and “pulsates” in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [ q ( x + T,t ) = q ( x,t ) as | x |→∞] is discussed.