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Triki H, Porsezian K, Senthilnathan K, Nithyanandan K. Chirped self-similar solitary waves for the generalized nonlinear Schrödinger equation with distributed two-power-law nonlinearities. Phys Rev E 2019; 100:042208. [PMID: 31770930 DOI: 10.1103/physreve.100.042208] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2019] [Indexed: 11/07/2022]
Abstract
We investigate the propagation characteristics of the chirped self-similar solitary waves in non-Kerr nonlinear media within the framework of the generalized nonlinear Schrödinger equation with distributed dispersion, two-power-law nonlinearities, and gain or loss. This model contains many special types of nonlinear equations that appear in various branches of contemporary physics. We extend the self-similar analysis presented for searching chirped self-similar structures of the cubic model to a more general problem involving two nonlinear terms of arbitrary power. A variety of exact linearly chirped localized solutions with interesting properties are derived in the presence of all physical effects. The solutions comprise bright, kink and antikink, and algebraic solitary wave solutions, illustrating the potentially rich set of self-similar pulses of the model. It is shown that these optical pulses possess a linear chirp that leads to efficient compression or amplification, and thus are particularly useful in the design of optical fiber amplifiers, optical pulse compressors, and solitary wave based communication links. Finally, the stability of the self-similar solutions is discussed numerically under finite initial perturbations.
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Affiliation(s)
- H Triki
- Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, 23000 Annaba, Algeria
| | - K Porsezian
- Department of Physics, Pondicherry University, Puducherry 605014, India
| | - K Senthilnathan
- Department of Physics, School of Advanced Sciences, VIT University, Vellore 632014, India
| | - K Nithyanandan
- Laboratoire Interdisciplinaire de Physique, Université de Grenoble-Alpes, Saint-Martin-d'Heres, 38402, France
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Triki H, Porsezian K, Choudhuri A. Solitons in the nonlinear Schrödinger equation with two power-law nonlinear terms modulated in time and space. Phys Rev E 2017; 95:062208. [PMID: 28709188 DOI: 10.1103/physreve.95.062208] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2016] [Indexed: 11/07/2022]
Abstract
A nonlinear Schrödinger equation that includes two terms with power-law nonlinearity and external potential modulated both on time and on the spatial coordinates is considered. The model appears in various branches of contemporary physics, especially in the case of lower values of the nonlinearity power. A significant generalization of the similarity transformations approach to construct explicit localized solutions for the model with arbitrary power-law nonlinearities is introduced. We obtain the exact analytical bright and kink soliton solutions of the governing equation for different nonlinearities and potentials that are of particular interest in applications to Bose-Einstein condensates and nonlinear optics. Necessary conditions on the physical parameters for propagating envelope formation are presented. The obtained results can be straightforwardly applied to a large variety of nonlinear Schrödinger models and hence would be of value to understand nonlinear phenomena in a diversity of nonlinear media.
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Affiliation(s)
- Houria Triki
- Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, 23000 Annaba, Algeria
| | - K Porsezian
- Department of Physics, School of Physical, Chemical and Applied Sciences, Pondicherry University, Pondicherry 605014, India
| | - Amitava Choudhuri
- Department of Physics, The University of Burdwan, Golapbag 713104, West Bengal, India
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Fujioka J, Espinosa A. Diversity of solitons in a generalized nonlinear Schrödinger equation with self-steepening and higher-order dispersive and nonlinear terms. CHAOS (WOODBURY, N.Y.) 2015; 25:113114. [PMID: 26627574 DOI: 10.1063/1.4936211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and self-steepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or "embedded," fixed or "moving" (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property.
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Affiliation(s)
- J Fujioka
- Instituto de Física, Dpto. de Física Química, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 México D.F., Mexico
| | - A Espinosa
- Instituto de Física, Dpto. de Física Química, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 México D.F., Mexico
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Adhikari SK. Stable spatial and spatiotemporal optical soliton in the core of an optical vortex. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042926. [PMID: 26565323 DOI: 10.1103/physreve.92.042926] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2015] [Indexed: 06/05/2023]
Abstract
We demonstrate a robust, stable, mobile, two-dimensional (2D) spatial and three-dimensional (3D) spatiotemporal optical soliton in the core of an optical vortex, while all nonlinearities are of the cubic (Kerr) type. The 3D soliton can propagate with a constant velocity along the vortex core without any deformation. Stability of the soliton under a small perturbation is established numerically. Two such solitons moving along the vortex core can undergo a quasielastic collision at medium velocities. Possibilities of forming such a 2D spatial soliton in the core of a vortical beam are discussed.
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Affiliation(s)
- S K Adhikari
- Instituto de Física Teórica, Universidade Estadual Paulista, 01.140-070 São Paulo, São Paulo, Brazil
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Fujioka J, Cortés E, Pérez-Pascual R, Rodríguez RF, Espinosa A, Malomed BA. Chaotic solitons in the quadratic-cubic nonlinear Schrödinger equation under nonlinearity management. CHAOS (WOODBURY, N.Y.) 2011; 21:033120. [PMID: 21974655 DOI: 10.1063/1.3629985] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We analyze the response of rational and regular (hyperbolic-secant) soliton solutions of an extended nonlinear Schrödinger equation (NLSE) which includes an additional self-defocusing quadratic term, to periodic modulations of the coefficient in front of this term. Using the variational approximation (VA) with rational and hyperbolic trial functions, we transform this NLSE into Hamiltonian dynamical systems which give rise to chaotic solutions. The presence of chaos in the variational solutions is corroborated by calculating their power spectra and the correlation dimension of the Poincaré maps. This chaotic behavior (predicted by the VA) is not observed in the direct numerical solutions of the NLSE when rational initial conditions are used. The solitary-wave solutions generated by these initial conditions gradually decay under the action of the nonlinearity management. On the contrary, the solutions of the NLSE with exponentially localized initial conditions are robust solitary-waves with oscillations consistent with a chaotic or a complex quasiperiodic behavior.
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Affiliation(s)
- J Fujioka
- Instituto de Física, Departamento de Física Química, Universidad Nacional Autónoma de México, México.
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Alatas H, Iskandar AA, Tjia MO, Valkering TP. Rational solitons in deep nonlinear optical Bragg grating. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:066606. [PMID: 16906996 DOI: 10.1103/physreve.73.066606] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2005] [Indexed: 05/11/2023]
Abstract
We have examined the rational solitons in the Generalized Coupled Mode model for a deep nonlinear Bragg grating. These solitons are the degenerate forms of the ordinary solitons and appear at the transition lines in the parameter plane. A simple formulation is presented for the investigation of the bifurcations induced by detuning the carrier wave frequency. The analysis yields among others the appearance of in-gap dark and antidark rational solitons unknown in the nonlinear shallow grating. The exact expressions for the corresponding rational solitons are also derived in the process, which are characterized by rational algebraic functions. It is further demonstrated that certain effects in the soliton energy variations are to be expected when the frequency is varied across the values where the rational solitons appear.
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Affiliation(s)
- H Alatas
- Department of Physics, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia.
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Mihalache D, Mazilu D, Crasovan LC, Malomed BA, Lederer F. Three-dimensional spinning solitons in the cubic-quintic nonlinear medium. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:7142-7145. [PMID: 11088411 DOI: 10.1103/physreve.61.7142] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/1999] [Indexed: 05/23/2023]
Abstract
We find one-parameter families of three-dimensional spatiotemporal bright vortex solitons (doughnuts, or spinning light bullets), in bulk dispersive cubic-quintic optically nonlinear media. The spinning solitons display a symmetry-breaking azimuthal instability, which leads to breakup of the spinning soliton into a set of fragments, each being a stable nonspinning light bullet. However, in some cases the instability is developing so slowly that the spinning light bullets may be regarded as virtually stable ones, from the standpoint of an experiment with finite-size samples.
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Affiliation(s)
- D Mihalache
- Institute of Solid State Theory and Theoretical Optics, Friedrich-Schiller University Jena, Max-Wien-Platz 1, D-07743, Jena, Germany
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Schürmann HW. Traveling-wave solutions of the cubic-quintic nonlinear Schrödinger equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4312-4320. [PMID: 9965579 DOI: 10.1103/physreve.54.4312] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Micallef RW, Afanasjev VV, Kivshar YS, Love JD. Optical solitons with power-law asymptotics. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:2936-2942. [PMID: 9965412 DOI: 10.1103/physreve.54.2936] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Pelinovsky DE, Afanasjev VV, Kivshar YS. Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:1940-1953. [PMID: 9964457 DOI: 10.1103/physreve.53.1940] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Akhmediev NN, Afanasjev VV, Soto-Crespo JM. Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:1190-1201. [PMID: 9964355 DOI: 10.1103/physreve.53.1190] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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