Structure and dynamic evolution of Riemann problem solutions for the Kundu equation with step-like initial data

The Kundu equation can be used to describe the propagation of surface waves in various scenarios, such as coastal regions and tidal phenomena, the propagation of shock waves and turbulence in fluids, and the behavior of light pulses and optical phenomena in media with nonlinear properties. In this paper, we investigate the periodic solutions of the Kundu equation, and classification and propagation of nonlinear wave patterns for its Riemann problem due to the Whitham modulation theory. The non-monotonic dependence of the Riemann invariant on physical variables indeed leads to the emergence of various wave structures containing the rarefaction waves, cnoidal dispersive shock waves (DSWs), contact DSWs, and combined shock waves. We give the Whitham modulation equations of the Kundu equation with step-like initial values by using the Flaschka-Forest-McLaughlin approach. Moreover, we present a full classification of possible wave structures and discuss the distributions of Riemann invariants, intensity variables, and chirp variables for each classification to offer a detailed insight into the dynamical behaviors of such systems. Moreover, the effectiveness of the Whitham theory in the Kundu equation is verified through direct numerical simulations. These results are useful to understand the different wave structures of the Kundu equation, and to design the related physical experiments.