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The integrable focusing Davey-Stewarson (DS) equations, multidimensionalgeneralizations of the focusing cubic nonlinear Schr\"odinger (NLS) equation,provide ideal mathematical models for describing analytically the dynamics of2+1 dimensional anomalous (rogue) waves (AWs). In this paper i) we constructthe $N$-breather AW solution of Akhmediev type of the DS1 and DS2 equations,describing the nonlinear interaction of $N$ unstable modes over the constantbackground solution. ii) For the simplest multidimensional solution of DS2 weconstruct its limiting subcases, and we identify the constraint on itsarbitrary parameters giving rise to blow up at finite time. iii) We use matchedasymptotic expansions to describe the relevance of the constructed AW solutionsin the doubly periodic Cauchy problem for small initial perturbations of thebackground, in the case of one and two unstable modes. We also show, in thecase of two unstable modes, that no blow up takes place generically, althoughthe AW amplitude can be arbitrarily large. All the results are expressed interms of elementary functions.

The periodic N breather anomalous wave solution of the Davey-Stewartson equations; first appearance, recurrence, and blow up properties