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

The implicit solution of the geometric optics equations (i.e. the modulation equations arising from the WKB Ansatz) of the defocusing nonlinear Schrödinger (NLS) equation is known to be expressible in terms of the classical hodograph transform. In this note, the solution procedure for the 2 × 2 system of quasilinear modulation equations is implemented, analogous to the implicit solution of the inviscid Burgers' equation, for smooth monotone initial data consistent with the modulation Ansatz. The implicit system is solved exactly using a classical method of Riemann. The relevant Riemann-Green functions can be found explicitly, hence allowing the exact location and time of shock formation to be calculated. The entire evolution of the exact solution can be observed through the shock formation.