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Gonpe Tafo JB, Nana L, Kofane TC. Effects of nonlinear gradient terms on the defect turbulence regime in weakly dissipative systems. Phys Rev E 2017; 96:022205. [PMID: 28950606 DOI: 10.1103/physreve.96.022205] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2017] [Indexed: 06/07/2023]
Abstract
We investigate the behavior of traveling waves in a defect turbulence regime with the periodic boundary conditions by using the lowest-order complex Ginzburg-Landau equation (CGLE), and we show the effect of the nonlinear gradient terms in the system. It is found that the nonlinear gradient terms which appear at the same order as the quintic term can change the behavior of the wave patterns. The presence of the nonlinear gradient terms can cause major changes in the behavior of the solution. They can be considered like the stabilizing terms. The system which was initially unstable or chaotic can become stable by including the nonlinear gradient terms.
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Affiliation(s)
- J B Gonpe Tafo
- Laboratoire de Physique Fondamentale, Groupe Phénomènes Non Linéaires et Systèmes Complexes, UFD de Mathématique, Informatique Appliquée et Physique Fondamentale, Université de Douala, Boîte Postale 24157, Douala, Cameroon
- Centre d'Excellence Africain en Technologies de l'Information et de la Communication, Boîte Postale 812, Yaoundé, Cameroon
| | - L Nana
- Laboratoire de Physique Fondamentale, Groupe Phénomènes Non Linéaires et Systèmes Complexes, UFD de Mathématique, Informatique Appliquée et Physique Fondamentale, Université de Douala, Boîte Postale 24157, Douala, Cameroon
| | - T C Kofane
- Centre d'Excellence Africain en Technologies de l'Information et de la Communication, Boîte Postale 812, Yaoundé, Cameroon
- Laboratoire de Mécanique, Département de Physique, Faculté des Sciences, Université de Yaoundé I, Boîte Postale 812, Yaoundé, Cameroon
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Ding E, Tang AYS, Chow KW, Malomed BA. Pinned modes in two-dimensional lossy lattices with local gain and nonlinearity. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:rsta.2014.0018. [PMID: 25246677 DOI: 10.1098/rsta.2014.0018] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We introduce a system with one or two amplified nonlinear sites ('hot spots', HSs) embedded into a two-dimensional linear lossy lattice. The system describes an array of evanescently coupled optical or plasmonic waveguides, with gain applied to selected HS cores. The subject of the analysis is discrete solitons pinned to the HSs. The shape of the localized modes is found in quasi-analytical and numerical forms, using a truncated lattice for the analytical consideration. Stability eigenvalues are computed numerically, and the results are supplemented by direct numerical simulations. In the case of self-focusing nonlinearity, the modes pinned to a single HS are stable and unstable when the nonlinearity includes the cubic loss and gain, respectively. If the nonlinearity is self-defocusing, the unsaturated cubic gain acting at the HS supports stable modes in a small parametric area, whereas weak cubic loss gives rise to a bistability of the discrete solitons. Symmetric and antisymmetric modes pinned to a symmetric set of two HSs are also considered.
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Affiliation(s)
- Edwin Ding
- Department of Mathematics and Physics, Azusa Pacific University, Box 7000, Azusa, CA 91702-7000, USA
| | - A Y S Tang
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong
| | - K W Chow
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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Malomed BA, Ding E, Chow KW, Lai SK. Pinned modes in lossy lattices with local gain and nonlinearity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:036608. [PMID: 23031046 DOI: 10.1103/physreve.86.036608] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2012] [Revised: 07/24/2012] [Indexed: 06/01/2023]
Abstract
We introduce a discrete linear lossy system with an embedded "hot spot" (HS), i.e., a site carrying linear gain and complex cubic nonlinearity. The system can be used to model an array of optical or plasmonic waveguides, where selective excitation of particular cores is possible. Localized modes pinned to the HS are constructed in an implicit analytical form, and their stability is investigated numerically. Stability regions for the modes are obtained in the parameter space of the linear gain and cubic gain or loss. An essential result is that the interaction of the unsaturated cubic gain and self-defocusing nonlinearity can produce stable modes, although they may be destabilized by finite-amplitude perturbations. On the other hand, the interplay of the cubic loss and self-defocusing gives rise to a bistability.
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Affiliation(s)
- Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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Ndzana FII, Mohamadou A, Kofané TC. Discrete Lange-Newell criterion for dissipative systems. Phys Rev E 2009; 79:056611. [PMID: 19518586 DOI: 10.1103/physreve.79.056611] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2008] [Indexed: 11/07/2022]
Abstract
We report on the derivation of the discrete complex Ginzburg-Landau equation with first- and second-neighbor couplings using a nonlinear electrical network. Furthermore, we discuss theoretically and numerically modulational instability of plane carrier waves launched through the line. It is pointed out that the underlying analysis not only spells out the discrete Lange-Newell criterion by the means of the linear stability analysis at which the modulational instability occurs for the generation of a train of ultrashort pulses, but also characterizes the long-time dynamical behavior of the system when the instability grows.
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Affiliation(s)
- Fabien I I Ndzana
- Department of Physics, Faculty of Science, Laboratory of Mechanics, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.
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Ndzana F, Mohamadou A, Kofané TC. Modulational instability in a purely nonlinear coupled complex Ginzburg-Landau equations through a nonlinear discrete transmission line. CHAOS (WOODBURY, N.Y.) 2008; 18:043121. [PMID: 19123631 DOI: 10.1063/1.2988260] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We study wave propagation in a nonlinear transmission line with dissipative elements. We show analytically that the telegraphers' equations of the electrical transmission line can be modeled by a pair of continuous coupled complex Ginzburg-Landau equations, coupled by purely nonlinear terms. Based on this system, we investigated both analytically and numerically the modulational instability (MI). We produce characteristics of the MI in the form of typical dependence of the instability growth rate on the wavenumbers and system parameters. Generic outcomes of the nonlinear development of the MI are investigated by dint of direct simulations of the underlying equations. We find that the initial modulated plane wave disintegrates into waves train. An apparently turbulent state takes place in the system during the propagation.
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Affiliation(s)
- Fabien Ndzana
- Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.
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Ndzana FI, Mohamadou A, Kofané TC, English LQ. Modulated waves and pattern formation in coupled discrete nonlinear LC transmission lines. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:016606. [PMID: 18764073 DOI: 10.1103/physreve.78.016606] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2008] [Revised: 05/01/2008] [Indexed: 05/26/2023]
Abstract
The conditions for the propagation of modulated waves on a system of two coupled discrete nonlinear LC transmission lines with negative nonlinear resistance are examined, each line of the network containing a finite number of cells. Our theoretical analysis shows that the telegrapher equations of the electrical transmission line can be reduced to a set of two coupled discrete complex Ginzburg-Landau equations. Using the standard linear stability analysis, we derive the expression for the growth rate of instability as a function of the wave numbers and system parameters, then obtain regions of modulational instability. Using numerical simulations, we examine the long-time dynamics of modulated waves in the line. This leads to the generation of nonlinear modulated waves which have the shape of a soliton for the fast and low modes. The effects of dissipative elements on the propagation are also shown. The analytical results are corroborated by numerical simulations.
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Affiliation(s)
- Fabien Ii Ndzana
- Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.
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Wamba E, Mohamadou A, Kofané TC. Modulational instability of a trapped Bose-Einstein condensate with two- and three-body interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:046216. [PMID: 18517719 DOI: 10.1103/physreve.77.046216] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2007] [Revised: 01/08/2008] [Indexed: 05/26/2023]
Abstract
We investigate analytically and numerically the modulational instability of a Bose-Einstein condensate with both two- and three-body interatomic interactions and trapped in an external parabolic potential. Analytical investigations performed lead us to establish an explicit time-dependent criterion for the modulational instability of the condensate. The effects of the potential as well as of the quintic nonlinear interaction are studied. Direct numerical simulations of the Gross-Pitaevskii equation with two- and three-body interactions describing the dynamics of the condensate agree with the analytical predictions.
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Affiliation(s)
- Etienne Wamba
- Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P. O. Box 812, Yaounde, Cameroon.
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Mohamadou A, Ayissi BE, Kofané TC. Instability criteria and pattern formation in the complex Ginzburg-Landau equation with higher-order terms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:046604. [PMID: 17155189 DOI: 10.1103/physreve.74.046604] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2006] [Revised: 06/13/2006] [Indexed: 05/12/2023]
Abstract
We study the modulational instability and spatial pattern formation in extended media, taking the one-dimensional complex Ginzburg-Landau equation with higher-order terms as a perturbation of the nonlinear Schrödinger equation as a model. By stability analysis for the original partial differential equation, we derive its stability condition as well as the threshold for amplitude perturbations and we show how nonlinear higher-order terms qualitatively change the behavior of the system. The analytical results are found to be in agreement with numerical findings. Modulational instability mediates pattern formation through the lattice. The main feature of the traveling plane waves is its disintegration in pulse train during the propagation through the system.
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Affiliation(s)
- Alidou Mohamadou
- Laboratory of Mechanic, Department of Physics, Faculty of Science, University of Yaoundé I, Yaoundé, Cameroon
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Mohamadou A, Kofané TC. Modulational instability and pattern formation in discrete dissipative systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:046607. [PMID: 16711942 DOI: 10.1103/physreve.73.046607] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2005] [Indexed: 05/09/2023]
Abstract
We report in this paper the study of modulated wave trains in the one-dimensional (1D) discrete Ginzburg-Landau model. The full linear stability analysis of the nonlinear plane wave solutions is performed by considering both the wave vector (q) of the basic states and the wave vector (Q) of the perturbations as free parameters. In particular, it is shown that a threshold exists for the amplitude and above this threshold, the induced modulational instability leads to the formation of ordered and disordered patterns. The theoretical findings have been numerically tested through direct simulations and have been found to be in agreement with the theoretical prediction. We show numerically that modulational instability is also an indicator of the presence of discrete solitons as were early predicted to exist in Ginzburg-Landau lattices.
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Affiliation(s)
- Alidou Mohamadou
- Laboratory of Mechanic, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon
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