Phys. Rev. E (3), 2020, том 101, выпуск 3, 32204, 32204 стр.
Effect of a small loss or gain in the periodic nonlinear Schrodinger anomalous wave dynamics
F. Coppiniab, P. G. Grinevichcd, P. M. Santiniba
a Dipartimento di Fisica, University of Rome "La Sapienza"
b Istituto Nazionale di Fisica Nucleare, Sezione di Roma
c Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
d L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences
The focusing nonlinear Schrodinger (NLS) equation is the simplest universal model describing the modulation instability of quasimonochromatic waves in weakly nonlinear media, the main physical mechanism for the appearance of anomalous (rogue) waves (AWs) in nature. In this paper, concentrating on the simplest case of a single unstable mode, we study the special Cauchy problem for the NLS equation perturbed by a linear loss or gain term, corresponding to periodic initial perturbations of the unstable background solution of the NLS. Using the finite gap method and the theory of perturbations of soliton partial differential equations, we construct the proper analytic model describing quantitatively how the solution evolves after a suitable transient into slowly varying lower dimensional patterns (attractors) on the (x, t) plane, characterized by Delta X = L/2 in the case of loss and by Delta X = 0 in the case of gain, where Delta X is the x shift of the position of the AW during the recurrence, and L is the period. This process is described, to leading order, in terms of elementary functions of the initial data. Since dissipation can hardly be avoided in all natural phenomena involving AWs, and since a small dissipation induces O(1) effects on the periodic AW dynamics, generating the slowly varying loss or gain attractors analytically described in this paper, we expect that these attractors together with their generalizations corresponding to more unstable modes will play a basic role in the theory of periodic AWs in nature.
|Российский научный фонд
|Sapienza Università di Roma
|The work of P.G.G. was supported by the Russian Science Foundation Grant No. 18-11-00316. P.M.S. was partially supported by the University La Sapienza funds, year 2017.
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