Whitham modulation theory of defocusing nonlinear Schrödinger equation and the classification and evolutions of solutions with initial discontinuity

Abstract

Since the Whitham modulation theory was first proposed in 1965, it has been widely concerned because of its superiority in studying dispersive fluid dynamics and dealing with discontinuous initial value problems. In this paper, the Whitham modulation theory of the defocusing nonlinear Schrödinger equation is developed, the classification and evolution of the solutions of discontinuous initial value problem are studied. Moreover, the region of dispersive shock wave, the region rarefaction wave, the region of unmodulated wave and the plateau region are distinguished. Particularly, the correctness of the results is verified by direct numerical simulation. Specifically, the solutions of 0-phase and 1-phase and their corresponding Whitham equations are derived by the finite gap integration method. Also the Whitham equation of genus <i>N</i> corresponding to the <i>N</i>-phase periodic wave solution is derived. The basic structures of rarefaction wave and dispersive shock wave are given, in which the boundaries of the regions are calculated in detail. The Riemann invariants and density distributions of dispersive fluids in each case are discussed. When the initial value is fixed as a special one, the vacuum point is considered and analyzed in detail. In addition, the oscillating front and the soliton front in the dispersive shock wave are considered. In fact, the Whitham modulation theory has many wonderful applications in real physics and engineering. The dam problem is investigated as a special Riemann problem, the piston problem of dispersive fluid is analyzed, and the novel undular bores are found.

Generation and Control of Shock Waves in Exciton-Polariton Condensates

“…[26,27] Shock waves have been found in different fields of physics, such as plasmas, [28] optics, [29] water waves, [30] and polariton BECs. [13,31] Under weak dispersion approximation conditions, through the discussion of the Riemann problem [32][33][34][35][36] of the defocusing nonlinear Schrödinger equation (NLSE), the wave structure of the solution is composed of two kinds of rarefaction waves and two kinds of shock waves, and the parameter ranges of various shock waves are shown. [26,[37][38][39] However, there is no report about the effects of saturated nonlinearity and normal dispersion on shock waves.…”