Fermi-Pasta-Ulam-Tsingou recurrence of periodic anomalous waves in the complex Ginzburg-Landau and in the Lugiato-Lefever equations
| Autor: | Paolo Maria Santini, F. Coppini |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Physics
Nonlinear Sciences - Exactly Solvable and Integrable Systems Plane (geometry) Order (ring theory) Mathematical Physics Mathematics - Mathematical Physics 01 natural sciences 010305 fluids & plasmas Nonlinear system Special functions 0103 physical sciences Attractor Initial value problem Elementary function Soliton 010306 general physics Nonlinear Sciences::Pattern Formation and Solitons Mathematical physics |
| Popis: | The complex Ginzburg-Landau (CGL) equation, an envelope model relevant in the description of several natural phenomena like binary-fluid convection and second-order phase transitions, and the Lugiato-Lefever (LL) equation, describing the dynamics of optical fields in pumped lossy cavities, can be viewed as nonintegrable generalizations of the nonlinear Schr\"odinger (NLS) equation, including diffusion, linear and nonlinear loss or gain terms, and external forcing. In this paper we treat the nonintegrable terms of both equations as small perturbations of the integrable focusing NLS equation, and we study the Cauchy problem of the CGL and LL equations corresponding to periodic initial perturbations of the unstable NLS background solution, in the simplest case of a single unstable mode. Using the approach developed in a recent paper by the authors with P. G. Grinevich [Phys. Rev. E 101, 032204 (2020)], based on the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic models describing quantitatively how the solution evolves, after a suitable transient, into a Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence of anomalous waves (AWs) described by slowly varying lower dimensional patterns (attractors) in the $(x,t)$ plane, characterized by $\mathrm{\ensuremath{\Delta}}x=L/2$ or $\mathrm{\ensuremath{\Delta}}x=0$ in the case in which loss or gain, respectively, effects prevail, where $\mathrm{\ensuremath{\Delta}}x$ is the $x$-shift of the position of the AW during the recurrence and $L$ is the period. We also obtain, in the CGL case, the analytic condition for which loss and gain exactly balance, stabilizing the ideal FPUT recurrence of periodic NLS AWs; such a stabilization is not possible in the LL case due to the external forcing. These processes are described, to leading order, in terms of elementary functions of the initial data in the CGL case, and in terms of elementary and special functions of the initial data in the LL case. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|