Riemann Surfaces Close to Degenerate Ones in the Theory of Rogue Waves

At present the development of the theory of rogue waves (also known as anomalous waves or freak 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. The choice of the latter is motivated by the fact that in the classical studies of V. I. Bespalov–V. I. Talanov and V. E. Zakharov in the mid-1960s, 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. The most powerful method for constructing spatially periodic (quasiperiodic) solutions of soliton equations is the finite-gap technique. The development of this approach was initiated by S. P. Novikov in 1974. 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, as noticed by P. M. Santini and the present author, 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. The present paper provides an overview of the results obtained in this direction.