Affiliation:
1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
2. Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma,
Roma, Italy
3. Dipartimento di Fisica, Universita di Roma "La Sapienza", Roma, Italy
Abstract
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.
Bibliography: 86 titles.
Funder
Russian Science Foundation
Italian Ministry of Education, University and Research
Publisher
Steklov Mathematical Institute
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献