Riemann surfaces close to degenerate ones in the theory of rogue waves (Q6633149)

The aim of this paper id to give an overview of the results obtained in the field of Riemann Surfaces close to degenerate ones in the theory of rogue waves also known as anomalous waves or freak waves. At present the development of the theory of rogue waves is one of priority directions in mathematical physics. At the moment there is no unanimous consensus about the generation mechanism of these waves and it cannot be ruled out that the main contribution in different systems is made by different mechanisms. However, the main candidate is considered to be modulation instability in nonlinear media. One of the research directions in the theory of rogue waves involves the use of integrable models, including the nonlinear Schrödinger (NLS) equation: \N\[\Niu_t+u_{xx}\mp2|u|^2u=0,\N\]\Nwhere \(u=u(x,t)\in\mathbb{C}\) is a complex-valued function of two real variables. This equation has two real forms : first when the minus sign is chosen, we obtain the defocusing equation; in optics it corresponds to a medium whose refractive index decreases when exposed to an electromagnetic wave; second when the plus sign is chosen, we obtain the focusing equation; in optics it corresponds to a medium whose refractive index increases when exposed to an electromagnetic wave. The choice of the latter is motivated by the fact that in some classical studies the NLS equation was derived as a model for describing modulation instability in nonlinear optics and in the theory of deep water waves, respectively. The integrability of the NLS equation was established by V. E. Zakharov and A. B. Shabat in 1972. \N\NThe most powerful method for constructing spatially periodic solutions of soliton equations is the finite-gap technique. Recall that a finite-gap (algebro-geometric) approach to constructing solutions of soliton equations was proposed by \textit{S. P. Novikov} [Funct. Anal. Appl. 8, 236--246 (1974; Zbl 0299.35017); translation from Funkts. Anal. Prilozh. 8, No. 3, 54--66 (1974)] devoted to the spatially periodic problem for the Korteweg-de Vries (KdV) equation. A significant contribution to the development of this method has also been made by other authors. However, as Novikov pointed out, despite the apparent simplicity of the theta-function formulas, one usually needs to make them more explicit in order to apply them. Fortunately in the problem of generation of rogue waves due to modulation instability, the Cauchy data at the initial time have a special form and are a small perturbation of the unstable background. In this case the spectral curves in the finite-gap approach turn out to be small perturbations of rational curves, and one can obtain very simple asymptotic formulas that surprisingly well agree with the results of numerical integration for a small number of unstable modes. This work consists of the following basic parts: 1) Introduction. 2) Spectral transformation for the nonlinear Schrödinger equation. 3) Cauchy problem for rogue waves. 4) Linear instability of the Akhmediev breather. 5) The effect of small loss on the recurrece of Akhmediev breathers. 6) Focusing Davey-Stewartson II equation as a generation model of rogue waves in spatially two-dimensional systems.