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Wu Y, Vivas-Cortez M, Rehman HU, Sherif ESM, Rashid A. Bifurcation study, phase portraits and optical solitons of dual-mode resonant nonlinear Schrodinger dynamical equation with Kerr law non-linearity. Heliyon 2024; 10:e34416. [PMID: 39144959 PMCID: PMC11320164 DOI: 10.1016/j.heliyon.2024.e34416] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2024] [Revised: 06/20/2024] [Accepted: 07/09/2024] [Indexed: 08/16/2024] Open
Abstract
This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.
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Affiliation(s)
- Yong Wu
- General Education Department, Anhui Xinhua University, Hefei, China
| | - Miguel Vivas-Cortez
- School of Physical and Mathematical Sciences, Faculty of Exact and Natural Sciences, Pontifical Catholic University of Ecuador, Av. 12 October 1076. Section, Quito 17-01-2184, Ecuador
| | | | - El-Sayed M. Sherif
- Mechanical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Al-Riyadh 11421, Saudi Arabia
| | - Abdul Rashid
- Department of Mathematics, University of Okara, Okara, Pakistan
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Zeng L, Shi J, Belić MR, Mihalache D, Chen J, Li J, Zhu X. Surface gap solitons in the Schrödinger equation with quintic nonlinearity and a lattice potential. OPTICS EXPRESS 2023; 31:35471-35483. [PMID: 38017716 DOI: 10.1364/oe.497973] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2023] [Accepted: 09/02/2023] [Indexed: 11/30/2023]
Abstract
We demonstrate the existence of surface gap solitons, a special type of asymmetric solitons, in the one-dimensional nonlinear Schrödinger equation with quintic nonlinearity and a periodic linear potential. The nonlinearity is suddenly switched in a step-like fashion in the middle of the transverse spatial region, while the periodic linear potential is chosen in the form of a simple sin 2 lattice. The asymmetric nonlinearities in this work can be realized by the Feshbach resonance in Bose-Einstein condensates or by the photorefractive effect in optics. The major peaks in the gap soliton families are asymmetric and they are located at the position of the jump in nonlinearity (at x = 0). In addition, the major peaks of the two-peak and multi-peak solitons at the position x = 0 are higher than those after that position, at x > 0. And such phenomena are more obvious when the value of chemical potential is large, or when the difference of nonlinearity values across the jump is big. Along the way, linear stability analysis of the surface gap solitons is performed and the stability domains are identified. It is found that in this model, the solitons in the first band gap are mostly stable (excepting narrow domains of instability at the edges of the gap), while those in the second band gap are mostly unstable (excepting extremely narrow domains of stability for fundamental solitons). These findings are also corroborated by direct numerical simulations.
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Wang Q, Mihalache D, Belić MR, Zeng L, Lin J. Soliton transformation between different potential wells. OPTICS LETTERS 2023; 48:747-750. [PMID: 36723579 DOI: 10.1364/ol.481216] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2022] [Accepted: 12/31/2022] [Indexed: 06/18/2023]
Abstract
This paper presents a novel, to the best of our knowledge, method for realizing soliton transformation between different potential wells by gradually manipulating their depths in the propagation direction. The only requirements for such a transformation are that the gradient of the manipulated depth is smooth enough and the solitons in different potential wells are both in the regions of stability. The comparison of transformed solitons with the iterative ones obtained by the accelerated imaginary-time evolution method proves that our method is efficient and reliable. An interesting consequence is that in some complex potential wells in which it is difficult to find solitons by iterative numerical methods, stable solitons can be obtained by the transformation method. The controllable soliton transformation provides an excellent opportunity for all-optical switching, optical information processing, and other applications.
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Zeng L, Shi J, Li J, Li J, Wang Q. Dark soliton families in quintic nonlinear lattices. OPTICS EXPRESS 2022; 30:42504-42511. [PMID: 36366703 DOI: 10.1364/oe.472311] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2022] [Accepted: 10/17/2022] [Indexed: 06/16/2023]
Abstract
We prove that the dark solitons can be stable in the purely quintic nonlinear lattices, including the fundamental, tripole and five-pole solitons. These dark soliton families are generated on the periodic nonlinear backgrounds. The propagation constant affects the forms of these solitons, while the number of poles does not lead to the variation of the backgrounds. The dark solitons are stable only when the propagation constant is moderately large.
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Nondegenerate Bright Solitons in Coupled Nonlinear Schrödinger Systems: Recent Developments on Optical Vector Solitons. PHOTONICS 2021. [DOI: 10.3390/photonics8070258] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Nonlinear dynamics of an optical pulse or a beam continue to be one of the active areas of research in the field of optical solitons. Especially, in multi-mode fibers or fiber arrays and photorefractive materials, the vector solitons display rich nonlinear phenomena. Due to their fascinating and intriguing novel properties, the theory of optical vector solitons has been developed considerably both from theoretical and experimental points of view leading to soliton-based promising potential applications. Mathematically, the dynamics of vector solitons can be understood from the framework of the coupled nonlinear Schrödinger (CNLS) family of equations. In the recent past, many types of vector solitons have been identified both in the integrable and non-integrable CNLS framework. In this article, we review some of the recent progress in understanding the dynamics of the so called nondegenerate vector bright solitons in nonlinear optics, where the fundamental soliton can have more than one propagation constant. We address this theme by considering the integrable two coupled nonlinear Schrödinger family of equations, namely the Manakov system, mixed 2-CNLS system (or focusing-defocusing CNLS system), coherently coupled nonlinear Schrödinger (CCNLS) system, generalized coupled nonlinear Schrödinger (GCNLS) system and two-component long-wave short-wave resonance interaction (LSRI) system. In these models, we discuss the existence of nondegenerate vector solitons and their associated novel multi-hump geometrical profile nature by deriving their analytical forms through the Hirota bilinear method. Then we reveal the novel collision properties of the nondegenerate solitons in the Manakov system as an example. The asymptotic analysis shows that the nondegenerate solitons, in general, undergo three types of elastic collisions without any energy redistribution among the modes. Furthermore, we show that the energy sharing collision exhibiting vector solitons arises as a special case of the newly reported nondegenerate vector solitons. Finally, we point out the possible further developments in this subject and potential applications.
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Mohammed Elhadj K, Al Sakkaf L, Al Khawaja U, Boudjemâa A. Singular soliton molecules of the nonlinear Schrödinger equation. Phys Rev E 2020; 101:042221. [PMID: 32422845 DOI: 10.1103/physreve.101.042221] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2019] [Accepted: 04/02/2020] [Indexed: 11/07/2022]
Abstract
We derive an exact solution to the local nonlinear Schrödinger equation (NLSE) using the Darboux transformation method. The solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically decaying seed solution, we obtain a two-soliton solution with diverging peaks, which we denote as singular soliton molecule. We find that this solution has a finite binding energy. We calculate the force and potential of interaction between the two solitons, which turn out to be of molecular-type. The robustness of the bond between the two solitons is also verified. Furthermore, we obtain an exact solution to the nonlocal NLSE using the same method and seed solution. The solution in this case corresponds to an elastic collision of a soliton, a breather soliton on flat background, and a breather soliton on a background with linear ramp. Finally, we consider an NLSE which is nonlocal in time rather than space. Although we did not find a Lax pair to this equation, we derive three exact solutions.
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Affiliation(s)
- Khelifa Mohammed Elhadj
- Department of Physics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University of Chlef, P.O. Box 78, 02000, Chlef, Algeria.,Laboratory of Mechanics and Energy, Hassiba Benbouali University of Chlef, P.O. Box 78, 02000, Chlef, Algeria
| | - L Al Sakkaf
- Department of Physics, United Arab Emirates University, P.O. Box 15551, Al-Ain, United Arab Emirates
| | - U Al Khawaja
- Department of Physics, United Arab Emirates University, P.O. Box 15551, Al-Ain, United Arab Emirates
| | - Abdelâali Boudjemâa
- Department of Physics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University of Chlef, P.O. Box 78, 02000, Chlef, Algeria.,Laboratory of Mechanics and Energy, Hassiba Benbouali University of Chlef, P.O. Box 78, 02000, Chlef, Algeria
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Abstract
This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross–Pitaevskii equation (GPE), which combines the attractive potential ∼ r − 2 and the quartic self-repulsive nonlinearity, induced by the Lee–Huang–Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity ∼ r − 4 / 3 at r → 0 . Modes with embedded angular momentum exist too, but they are unstable. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity ∼ r − 2 / ( 4 − D ) . Such states may be considered the results of screening a “bare” delta-functional attractive potential by the respective nonlinearities.
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