Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves (Q2821963)

The paper addresses a solution to the one-dimensional nonlinear Schrödinger equation (NLSE) with the self-attractive sign of the nonlinearity and the initial condition in the form of a box, using the semi-classical approximation (i.e., the dispersive term in the NLSE is treated as a small one, in comparison with the nonlinearity). It is necessary to mention that, for the first time, the self-attractive NLSE with the box-shaped input was considered in the classical work by \textit{S. V. Manakov} [Zh. Eksp. Teor. Fiz. 65, 1392--1398 (1973); transl. in Sov. Phys. JETP 38, 693--696 (1974)]. Similar to the case of the NLSE with the self-repulsive sign of the nonlinearity, the decay of the box-shaped input generates dispersive-shock-wave patterns, propagating into the empty space (they are referred to as ``dam-break waves'' in this paper). In the region of the original box-shaped input, counter-propagating dispersive shock waves pass through each other. The nonlinear interaction between them gives rise to a pattern in the form of a breather lattice, which may be interpreted as a rogue-wave pattern, in the form of an array of Akhmediev of, eventually, Peregrine solitons. The approximate investigation of the analytical solution developed in the present paper uses two technical tools: the Witham equations (unlike the case of the self-repulsive NLSE, in the case of the self-attraction the Witham equations form an elliptic, rather than hyperbolic, system), and the steepest-descent approximation for the consideration of the Riemann-Hilbert problem.