Integrable Equations with a Forcing of a Distribution Type

Abstract

We consider the nonlinear Schrodinger equation in the variable q(x,t) with the forcing iu(t)δ(x)+iu1(t)δ′(x). We assume that q(x,0),u(t),u1(t) are given and that these functions as well as their first two derivatives belong to L1∩L2(ℝ+). We show that the solution of this problem can be reduced to solving two Riemann—Hilbert (RH) problems in the complex k‐plane with jumps on Im(k2) = 0. Each RH problem is equivalent to a linear integral equation that has a unique global solution. These linear integral equations are uniquely defined in terms of certain functions (scattering data) b(k), c(k), d(k), and f(k). The functions b(k) and d(k) can be effectively computed in terms of q(x,0). However, although the analytic properties of c(k) and f(k) are completely determined, the relationship between c(k), f(k), q(x,0), u(t), and u1(t) is highly nonlinear. In spite of this difficulty, we can give an effective description of the asymptotic behavior of q(x,t) for large t. In particular, we show that as t → ∞, solitons are generated moving away from the origin. It is important to emphasize that the analysis of this problem, in addition to techniques of exact integrability, requires the essential use of general PDE techniques.