Radiation solution of the nonlinear Schrödinger equation

The sine-Gordon, sinh-Gordon, modified Korteweg-deVries (KdV), and nonlinear (cubic) Schrödinger equations are four of the most important (from a physical point of view) nonlinear evolution equations that fit into the Ablowitz–Kaup–Newell–Segur (AKNS) inverse scattering framework. Historically, the soliton solutions of these equations have been exhaustively studied in the literature, the radiation solutions being almost entirely neglected. Using an expansion approach (expanding the reflection coefficient in powers of the area A of the input potential), we have previously studied the complete spatial and temporal evolution of the radiation solutions of the first three equations cited above. In this paper we demonstrate, using an illustrative example, that the expansion approach also works for the nonlinear Schrödinger equation. The qualitative features of the radiation solution thus obtained are easily understood in the physical context of the self-defocusing of an intense optical beam. Making use of numerical simulation, we show that the features persist as A is increased to very large values. The solution is also found to be analytically consistent with the asymptotic (t → ∞) form quoted in the literature for a general input profile.