The finite-gap method and the periodic Cauchy problem for $(2+1)$-dimensional anomalous waves for the focusing Davey–Stewartson $2$ equation

The focusing nonlinear Schrödinger equation is the simplest universal model describing the modulation instability of $(1+1)$-dimensional quasi monochromatic waves in weakly nonlinear media, and modulation instability is considered to be the main physical mechanism for the appearance of anomalous (rogue) waves in nature. By analogy with the recently developed analytic theory of periodic anomalous waves of the focusing nonlinear Schrödinger equation, in this paper we extend these results to a $(2+1)$-dimensional context, concentrating on the focusing Davey–Stewartson $2$ equation, an integrable $(2+1)$-dimensional generalization of the focusing nonlinear Schrödinger equation. More precisely, we use the finite gap theory to solve, to the leading order, the doubly periodic Cauchy problem for the focusing Davey–Stewartson $2$ equation, for small initial perturbations of the unstable background solution, which we call the doubly periodic Cauchy problem for anomalous waves. As in the case of the nonlinear Schrödinger equation, we show that, to the leading order, the solution of this Cauchy problem is expressed in terms of elementary functions of the initial data.
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