On the exact solution of the geometric optics approximation of the defocusing nonlinear Schrodinger equation

The implicit solution of the geometric optics equations (i.e, the modulatio n equations arising from the WKB Ansatz) of the defocusing nonlinear Schrod inger (NLS) equation is known to be expressible in terms of the classical h odograph transform. In this note, the solution procedure for the 2 x 2 syst em of quasilinear modulation equations is implemented, analogous to the imp licit solution of the inviscid Burgers' equation, for smooth monotone initi al data consistent with the modulation Ansatz. The implicit system is solve d exactly using a classical method of Riemann. The relevant Riemann-Green f unctions can be found explicitly, hence allowing the exact location and tim e of shock formation to be calculated. The entire evolution of the exact so lution can be observed through the shock formation. (C) 1999 Elsevier Scien ce B.V. All rights reserved.