Published: April 23, 2019

Whitham theory and dispersive shock waves (DSWs) for the radial nonlinear Schroedinger (rNLS) equation. 

Dispersive shock waves of the defocusing rNLS equation in two spatial dimensions are studied. This equation arises naturally in Bose-Einstein condensates, water waves and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable and nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and Riemann type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, comparison with the corresponding DSW solutions of the one-dimensional NLS equation exhibits significant qualitative and quantitative differences. The rNLS Whitham theory and numerics indicate some novel phenomena. Their detailed study is a matter of future work. The rNLS Whitham system also provides the starting point for studying the formation of DSWs for the full (2+1)-dimensional NLS equation.