Article
Dissipative Localised Structures for the Complex Discrete Ginzburg–Landau Equation
Author/s | Hennig, Dirk
Karachalios, Nikos I. Cuevas-Maraver, Jesús |
Department | Universidad de Sevilla. Departamento de Física Aplicada I |
Publication Date | 2023-04 |
Deposit Date | 2023-05-04 |
Published in |
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Abstract | The discrete complex Ginzburg–Landau equation is a fundamental model for the dynamics of nonlinear lattices incorporating competitive dissipation and energy gain effects. Such mechanisms are of particular importance for ... The discrete complex Ginzburg–Landau equation is a fundamental model for the dynamics of nonlinear lattices incorporating competitive dissipation and energy gain effects. Such mechanisms are of particular importance for the study of survival/destruction of localised structures in many physical situations. In this work, we prove that in the discrete complex Ginzburg–Landau equation dissipative solitonic waveforms persist for significant times by introducing a dynamical transitivity argument. This argument is based on a combination of the notions of “inviscid limits” and of the “continuous dependence of solutions on their initial data”, between the dissipative system and its Hamiltonian counterparts. Thereby, it establishes closeness of the solutions of the Ginzburg–Landau lattice to those of the conservative ideals described by the Discrete Nonlinear Schrödinger and Ablowitz–Ladik lattices. Such a closeness holds when the initial conditions of the systems are chosen to be sufficiently small in the suitable metrics and for small values of the dissipation or gain strengths. Our numerical findings are in excellent agreement with the analytical predictions for the dynamics of the dissipative bright, dark or even Peregrine-type solitonic waveforms. |
Funding agencies | EU (FEDER program2014-2020) through both Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía under the project P18-RT-3480 EU (FEDER program2014-2020) through both Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía under the project US-1380977 MCIN/AEI/10.13039/501100011033 under the project PID2019-110430GB-C21 MCIN/AEI/10.13039/501100011033 under the project PID2020-112620GB-I00 |
Project ID. | P18-RT-3480
US-1380977 PID2019-110430GB-C21 PID2020-112620GB-I00 |
Citation | Hennig, D., Karachalios, N.I. y Cuevas-Maraver, J. (2023). Dissipative Localised Structures for the Complex Discrete Ginzburg–Landau Equation. Journal of Nonlinear Science, 33 (51). https://doi.org/10.1007/s00332-023-09904-2. |
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