Journal articles: 'Solitons'

1

Aycock, Lauren M., Hilary M. Hurst, Dmitry K. Efimkin, Dina Genkina, Hsin-I. Lu, Victor M. Galitski, and I. B. Spielman. "Brownian motion of solitons in a Bose–Einstein condensate." Proceedings of the National Academy of Sciences 114, no. 10 (February 14, 2017): 2503–8. http://dx.doi.org/10.1073/pnas.1615004114.

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We observed and controlled the Brownian motion of solitons. We launched solitonic excitations in highly elongatedRb87Bose–Einstein condensates (BECs) and showed that a dilute background of impurity atoms in a different internal state dramatically affects the soliton. With no impurities and in one dimension (1D), these solitons would have an infinite lifetime, a consequence of integrability. In our experiment, the added impurities scatter off the much larger soliton, contributing to its Brownian motion and decreasing its lifetime. We describe the soliton’s diffusive behavior using a quasi-1D scattering theory of impurity atoms interacting with a soliton, giving diffusion coefficients consistent with experiment.

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Segovia, Francis Armando, and Emilse Cabrera. "SOLUCIÓN DE LA ECUACIÓN NO LINEAL DE SCHRODINGER (1+1) EN UN MEDIO KERR." Redes de Ingeniería 6, no. 2 (December 26, 2015): 26. http://dx.doi.org/10.14483/udistrital.jour.redes.2015.2.a03.

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Se presenta un marco teórico y se muestra una simulación numérica de la propagación de solitones. Con especial atención a los solitones ópticos espaciales, se calcula analíticamente el perfil de solitón correspondiente a la ecuación Schrodinger no-lineal para un medio Kerr. Los resultados muestran que los solitones ópticos son pulsos estables cuya forma y espectro son preservados en grandes distancias.Solution of the nonlinear Schrodinger equation (1+1) in a Kerr mediumABSTRACTThis document presents a theoretical framework and shows a numerical simulation for the propagation of solitons. With special attention to the spatial optical solitons, we calculates analytically the profile of solitón corresponding to the non-linear Schrodinger equation for a Kerr medium. The results show that the optical solitons are stable pulses whose shape and spectrum are preserved at great distances.Keywords: nonlinear optics, nonlinear Schrodinger equation, solitons.

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3

Zhao, Xue-Hui, Bo Tian, Yong-Jiang Guo, and Hui-Min Li. "Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves." Modern Physics Letters B 32, no. 08 (March 12, 2018): 1750268. http://dx.doi.org/10.1142/s0217984917502682.

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Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

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GONZÁLEZ, JORGE A., and JOSE R. CARBÓ. "STATIONARITY-BREAKING BIFURCATIONS OF SOLITONS UNDER NONLINEAR DAMPING." Modern Physics Letters B 08, no. 12 (May 20, 1994): 739–48. http://dx.doi.org/10.1142/s0217984994000741.

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The existence and dynamics of solitons in general systems with nonlinear damping are investigated. The mechanism of a new bifurcation after which the soliton can no longer be in a stationary state is discussed. Some particular cases are studied in detail and exact solutions are presented. The possibility and importance of self-sustained solitons, solitonic limit cycles, and chaotic solitons in these systems are analyzed.

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Peng, Yangyang, Guangyu Xu, Keyun Zhang, Meisong Liao, Yongzheng Fang, and Yan Zhou. "Modulating anti-dark vector solitons." Laser Physics 33, no. 9 (July 12, 2023): 095101. http://dx.doi.org/10.1088/1555-6611/ace251.

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Abstract Theoretical analysis of the modulation of anti-dark vector solitons is conducted in this work. The simulation depends on a single-mode optical fiber out-cavity modulation system model that works at 1 μm. The anti-dark vector soliton’s initial state is assumed to be polarization-/group-velocity-locked, with same/different central wavelengths in orthogonally polarized directions. After soliton parameter modulation, modulated anti-dark vector solitons at the output port will demonstrate different properties in orthogonal directions. For example, two symmetrically located frequency peaks always exist for output orthogonal modes when the input state is polarization-locked. And a dual-wavelength anti-dark vector soliton with temporal pulse oscillation can be generated by changing the projection angle with the help of a polarization beam splitter, when the input vector soliton’s group-velocity is locked. These modulation results are instructive for the study of out-cavity modulating optical fiber vector soltions with different pulsed properties.

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6

Xiao, Zi-Jian, Bo Tian, and Yan Sun. "Soliton interactions and Bäcklund transformation for a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili equation in fluid dynamics." Modern Physics Letters B 32, no. 02 (January 20, 2018): 1750170. http://dx.doi.org/10.1142/s0217984917501706.

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In this paper, we investigate a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili (mKP) equation in fluid dynamics. With the binary Bell-polynomial and an auxiliary function, bilinear forms for the equation are constructed. Based on the bilinear forms, multi-soliton solutions and Bell-polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton interactions are presented. Based on the graphic analysis, Parametric conditions for the existence of the shock waves, elevation solitons and depression solitons are given, and it is shown that under the condition of keeping the wave vectors invariable, the change of [Formula: see text] and [Formula: see text] can lead to the change of the solitonic velocities, but the shape of each soliton remains unchanged, where [Formula: see text] and [Formula: see text] are the variable coefficients in the equation. Oblique elastic interactions can exist between the (i) two shock waves, (ii) two elevation solitons, and (iii) elevation and depression solitons. However, oblique interactions between (i) shock waves and elevation solitons, (ii) shock waves and depression solitons are inelastic.

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7

Zhang, Ling-Ling, and Xiao-Min Wang. "Bright–dark soliton dynamics and interaction for the variable coefficient three-coupled nonlinear Schrödinger equations." Modern Physics Letters B 34, no. 05 (December 20, 2019): 2050064. http://dx.doi.org/10.1142/s0217984920500645.

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Under investigation in this paper is the variable coefficient three-coupled nonlinear Schrödinger (CNLS) equations, which govern the dynamics of solitonic excitations along three-spine [Formula: see text]-helical protein with inhomogeneous effect. Via the Hirota method and symbolic computation, the exact two-bright-one-dark (TBD) and one-bright-two-dark (BTD) soliton solutions are constructed analytically. The propagation properties are discussed for TBD and BTD solitons when the variable coefficient is a hyperbolic secant function. Figures are plotted to reveal the following interactions of TBD and BTD two solitons: (1) Evolution without interactions of double-parabola-shaped solitons, of double-[Formula: see text]-shaped solitons and of parabola-[Formula: see text]-shaped solitons; (2) Evolution with periodic interaction of double-parabola-shaped solitons and of parabola-[Formula: see text]-shaped solitons; (3) Collision of double-[Formula: see text]-shaped solitons and of parabola-[Formula: see text]-shaped solitons.

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8

PENG, GANG-DING, and ADRIAN ANKIEWICZ. "FUNDAMENTAL AND SECOND-ORDER SOLITION TRANSMISSION IN NONLINEAR DIRECTIONAL FIBER COUPLERS." Journal of Nonlinear Optical Physics & Materials 01, no. 01 (January 1992): 135–50. http://dx.doi.org/10.1142/s021819919200008x.

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Transmission characteristics of first-order and second-order solitons propagating through a nonlinear optical fiber coupler are investigated by analysing the coupled nonlinear Schrödinger equations (NLSEs). We show that it is most advantageous to use fundamental solitions to make an ideal optical switch which can be used in multiplexing and/or demultiplexing soliton signals from different sources, and that such a switch can have a high switching efficiency and intact soliton output. Also, we have analyzed the relation between critical power of a soliton switch and that of a cw switch, and have given the soliton “critical energy” in an explicit form in terms of the physical parameters. Further, we give evidence to show that soliton bound-states and different solitons can be generated through soliton conversion in a nonlinear coupler.

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9

Ivanov, S. K., and A. M. Kamchatnov. "Motion of dark solitons in a non-uniform flow of Bose–Einstein condensate." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 11 (November 2022): 113142. http://dx.doi.org/10.1063/5.0123514.

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We study motion of dark solitons in a non-uniform one-dimensional flow of a Bose–Einstein condensate. Our approach is based on Hamiltonian mechanics applied to the particle-like behavior of dark solitons in a slightly non-uniform and slowly changing surrounding. In one-dimensional geometry, the condensate’s wave function undergoes the jump-like behavior across the soliton, and this leads to generation of the counterflow in the background condensate. For a correct description of soliton’s dynamics, the contributions of this counterflow to the momentum and energy of the soliton are taken into account. The resulting Hamilton equations are reduced to the Newton-like equation for the soliton’s path, and this Newton equation is solved in several typical situations. The analytical results are confirmed by numerical calculations.

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10

Singh, Abhishek, and Shyam Kishor. "SOME TYPES OF η-RICCI SOLITONS ON LORENTZIAN PARA-SASAKIAN MANIFOLDS." Facta Universitatis, Series: Mathematics and Informatics 33, no. 2 (September 7, 2018): 217. http://dx.doi.org/10.22190/fumi1802217s.

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In this paper we study some types of η-Ricci solitons on Lorentzianpara-Sasakian manifolds and we give an example of η-Ricci solitons on 3-dimensional Lorentzian para-Sasakian manifold. We obtain the conditions of η-Ricci soliton on ϕ-conformally flat, ϕ-conharmonically flat and ϕ-projectivelyflat Lorentzian para-Sasakian manifolds, the existence of η-Ricci solitons implies that (M,g) is η-Einstein manifold. In these cases there is no Ricci solitonon M with the potential vector field

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11

Jia, Xiao-Yue, Bo Tian, Zhong Du, Yan Sun, and Lei Liu. "Lump and rogue waves for the variable-coefficient Kadomtsev–Petviashvili equation in a fluid." Modern Physics Letters B 32, no. 10 (April 10, 2018): 1850086. http://dx.doi.org/10.1142/s0217984918500860.

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Under investigation in this paper is the variable-coefficient Kadomtsev–Petviashvili equation, which describes the long waves with small amplitude and slow dependence on the transverse coordinate in a single-layer shallow fluid. Employing the bilinear form and symbolic computation, we obtain the lump, mixed lump-stripe soliton and mixed rogue wave-stripe soliton solutions. Discussions indicate that the variable coefficients are related to both the lump soliton’s velocity and amplitude. Mixed lump-stripe soliton solutions display two different properties, fusion and fission. Mixed rogue wave-stripe soliton solutions show that a rogue wave arises from one of the stripe solitons and disappears into the other. When the time approaches 0, rogue wave’s energy reaches the maximum. Interactions between a lump soliton and one-stripe soliton, and between a rogue wave and a pair of stripe solitons, are shown graphically.

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12

LIANG, Z. X., and Z. D. ZHANG. "EXACT SOLITONS IN THE GROSS–PITAEVSKII EQUATION WITH TIME-MODULATED NONLINEARITY." Modern Physics Letters B 21, no. 07 (March 20, 2007): 383–90. http://dx.doi.org/10.1142/s0217984907012864.

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Exact solitonic solutions of the Gross–Pitaevskii equation with time-modulated nonlinearity of a(t) = a0 / (t + t0) are obtained. With help of these solutions, we analyze the properties of Feshbach-managed solitons in Bose–Einstein condensates in details. Our results show that the parameters of atomic matter waves can be manipulated by proper variation of the scattering length. In particular, an exact two-soliton solution is given, from which, it is shown that the separation between the neighboring solitons can be effectively maintained by allowing the solitons to have unequal initial amplitudes.

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13

Bo, Wen-Bo, Ru-Ru Wang, Wei Liu, and Yue-Yue Wang. "Symmetry breaking of solitons in the PT-symmetric nonlinear Schrödinger equation with the cubic–quintic competing saturable nonlinearity." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 9 (September 2022): 093104. http://dx.doi.org/10.1063/5.0091738.

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The symmetry breaking of solitons in the nonlinear Schrödinger equation with cubic–quintic competing nonlinearity and parity-time symmetric potential is studied. At first, a new asymmetric branch separates from the fundamental symmetric soliton at the first power critical point, and then, the asymmetric branch passes through the branch of the fundamental symmetric soliton and finally merges into the branch of the fundamental symmetric soliton at the second power critical point, while the power of the soliton increases. This leads to the symmetry breaking and double-loop bifurcation of fundamental symmetric solitons. From the power-propagation constant curves of solitons, symmetric fundamental and tripole solitons, asymmetric solitons can also exist. The stability of symmetric fundamental solitons, asymmetric solitons, and symmetric tripole solitons is discussed by the linear stability analysis and direct simulation. Results indicate that symmetric fundamental solitons and symmetric tripole solitons tend to be stable with the increase in the soliton power. Asymmetric solitons are unstable in both high and low power regions. Moreover, with the increase in saturable nonlinearity, the stability region of fundamental symmetric solitons and symmetric tripole solitons becomes wider.

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14

Castro, Camilo J., and Deterlino Urzagasti. "Seesaw drift of bright solitons of the nonlinear Schrödinger equation with a periodic potential." Journal of Nonlinear Optical Physics & Materials 25, no. 03 (September 2016): 1650038. http://dx.doi.org/10.1142/s0218863516500387.

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Soliton solutions are investigated employing the nonlinear Schrödinger equation (NLSE) with an additional term corresponding to an external periodic field. In particular, we use this equation to describe the behavior of solitons in fiber optics in the case of anomalous dispersion. Employing the framework of variational analysis and analytical approximations, single peaked soliton solutions are derived, which exhibit variations of the solitonic parameters due to the effect of the periodic potential and a harmonic oscillator motion of the soliton center, when the frequency of the external field is small, whereas high values of the frequency of the external field produce static solitons. Finally, a variational-numerical analysis was developed and compared with a purely numerical model.

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15

Wang, Chunxia, Xiaojun Yin, and Liguo Chen. "Soliton molecules, bifurcation solitons and interaction solutions of a generalized (2 + 1)-dimensional korteweg-de vries system for the shallow-water waves." Physica Scripta 99, no. 10 (September 23, 2024): 105272. http://dx.doi.org/10.1088/1402-4896/ad79a1.

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Abstract The central purpose of this paper is exploring the soliton molecules, bifurcation solitons and interaction solutions of the Korteweg–de Vries system based on the Hirota bilinear method. The studied system acts as an extension of the classic KdV system for the shallow-water waves, and is very useful to contribute in nonlinear wave phenomena. Firstly, the soliton molecules are obtained by adding resonance parameters in N-soliton. Then the interaction solutions between soliton/breather and soliton molecules are studied, as well as the interaction between two soliton molecules by using N-soliton. Moreover, a class of novel bifurcation solitons are derived, including Y-type bifurcation solitons, X-type bifurcation solitons and multiple-bifurcation solitons. In the end, the dynamic properties of soliton molecules, bifurcation solitons as well as the interaction solutions are presented graphically. The developed solutions of this research are all new and can enable us apprehend the nonlinear dynamic behaviors of the generalized (2+1)-dimensional Korteweg–de Vries system better.

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Dai, Chao-Qing, Hai-Ping Zhu, and Chun-Long Zheng. "Tunnelling Effects of Solitons in Optical Fibers with Higher-Order Effects." Zeitschrift für Naturforschung A 67, no. 6-7 (July 1, 2012): 338–46. http://dx.doi.org/10.5560/zna.2012-0033.

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We construct four types of analytical soliton solutions for the higher-order nonlinear Schrödinger equation with distributed coefficients. These solutions include bright solitons, dark solitons, combined solitons, and M-shaped solitons. Moreover, the explicit functions which describe the evolution of the width, peak, and phase are discussed exactly.We finally discuss the nonlinear soliton tunnelling effect for four types of femtosecond solitons

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Ma, Hongcai, Qiaoxin Cheng, and Aiping Deng. "N-soliton solutions and localized wave interaction solutions of a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyamaf equation." Modern Physics Letters B 35, no. 10 (March 31, 2021): 2150277. http://dx.doi.org/10.1142/s0217984921502778.

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[Formula: see text]-soliton solutions are derived for a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by using bilinear transformation. Some local waves such as period soliton, line soliton, lump soliton and their interaction are constructed by selecting specific parameters on the multi-soliton solutions. By selecting special constraints on the two soliton solutions, period and lump soliton solution can be obtained; three solitons can reduce to the interaction solution between period soliton and line soliton or lump soliton and line soliton under special parameters; the interaction solution among period soliton and two line solitons, or the interaction solution for two period solitons or two lump solitons via taking specific constraints from four soliton solutions. Finally, some images of the results are drawn, and their dynamic behavior is analyzed.

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Seadawy, Aly R., and Mujahid Iqbal. "Optical soliton solutions for nonlinear complex Ginzburg–Landau dynamical equation with laws of nonlinearity Kerr law media." International Journal of Modern Physics B 34, no. 19 (July 27, 2020): 2050179. http://dx.doi.org/10.1142/s0217979220501799.

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In this research article, our aim is to construct new optical soliton solutions for nonlinear complex Ginzburg–Landau equation with the help of modified mathematical technique. In this work, we studied both laws of nonlinearity (Kerr and power laws). The obtained solutions represent dark and bright solitons, singular and combined bright-dark solitons, traveling wave, and periodic solitary wave. The determined solutions provide help in the development of optical fibers, soliton dynamics, and nonlinear optics. The constructed solitonic solutions prove that the applicable technique is more reliable, efficient, fruitful and powerful to investigate higher order complex nonlinear partial differential equations (PDEs) involved in mathematical physics, quantum plasma, geophysics, mechanics, fiber optics, field of engineering, and many other kinds of applied sciences.

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Polyakov, Sergey, and Anatoly Sukhorukov. "Gap solitons." Izvestiya VUZ. Applied Nonlinear Dynamics 6, no. 4 (1998): 45–56. http://dx.doi.org/10.18500/0869-6632-1998-6-4-45-56.

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In this paper we present the studies of the problem of slow and immobile solitons which are excited inside the stop bands of inhomogeneous media with either quadratic or cubic nonlinearity. We found an integral criterion of slow soliton formation, determined the main properties of gар solitons. The dynamics of parametric gap soliton formation and propagation in quadratic media is illustrated by means of computer simulations and compared with such solitons due to cubic nonlinearity.

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Yao, Y., C. J. Luo, X. X. Wang, and H. Zhang. "Research on solitons’ interactions in one-dimensional indium chains on Si(111) surfaces." Journal of Physics: Conference Series 2639, no. 1 (November 1, 2023): 012051. http://dx.doi.org/10.1088/1742-6596/2639/1/012051.

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Abstract Solitons have garnered significant attention across various fields, yet a contentious debate persists regarding the precise structure of solitons on indium chains. Currently, multiple forms of solitons in one-dimensional atomic chains have been reported. STM provides an effective means to study the precise atomic structure of solitons, particularly their dynamics and interactions. However, limited research has been conducted on soliton interactions and soliton-chain interactions, despite their profound impact on relative soliton motions and the overall physical properties of the system. In this work, we characterized the structures of the soliton dimer and trimer, observed the displacements induced by the soliton entity and statisticized the dynamic behaviors of soliton dimers over time evolution or temperature. To reveal the soliton mechanism, we further utilized STM to investigate the CDWs between two solitons when two monomers were encountered. Additionally, we achieved the manipulation of the monomer on the indium chain by the STM tip. Our work serves as an important approach to elucidate interactions in correlated electronic systems and advance the development of potential topological soliton computers.

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21

Zhao, Chen, Yi-Tian Gao, Zhong-Zhou Lan, Jin-Wei Yang, and Chuan-Qi Su. "Bilinear forms and dark-soliton solutions for a fifth-order variable-coefficient nonlinear Schrödinger equation in an optical fiber." Modern Physics Letters B 30, no. 24 (September 10, 2016): 1650312. http://dx.doi.org/10.1142/s0217984916503127.

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In this paper, a fifth-order variable-coefficient nonlinear Schrödinger equation is investigated, which describes the propagation of the attosecond pulses in an optical fiber. Via the Hirota’s method and auxiliary functions, bilinear forms and dark one-, two- and three-soliton solutions are obtained. Propagation and interaction of the solitons are discussed graphically: We observe that the solitonic velocities are only related to [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the coefficients of the second-, third-, fourth- and fifth-order terms, respectively, with [Formula: see text] being the scaled distance, while the solitonic amplitudes are related to [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] as well as the wave number. When [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are the constants, or the linear, quadratic and trigonometric functions of [Formula: see text], we obtain the linear, parabolic, cubic and periodic dark solitons, respectively. Interactions between (among) the two (three) solitons are depicted, which can be regarded to be elastic because the solitonic amplitudes remain unchanged except for some phase shifts after each interaction in an optical fiber.

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Sun, Yan, Bo Tian, Hui-Ling Zhen, Xiao-Yu Wu, and Xi-Yang Xie. "Soliton solutions for a (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a plasma." Modern Physics Letters B 30, no. 20 (July 30, 2016): 1650213. http://dx.doi.org/10.1142/s0217984916502134.

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Under investigation in this paper is a (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation, which describes the nonlinear behaviors of ion-acoustic waves in a magnetized plasma where the cooler ions are treated as a fluid with adiabatic pressure and the hot isothermal electrons are described by a Boltzmann distribution. With the Hirota method and symbolic computation, we obtain the one-, two- and three-soliton solutions for such an equation. We graphically study the solitons related with the coefficient of the cubic nonlinearity [Formula: see text]. Amplitude of the one soliton increases with increasing [Formula: see text], but the width of one soliton keeps unchanged as [Formula: see text] increases. The two solitons and three solitons are parallel, and the amplitudes of the solitons increase with increasing [Formula: see text], but the widths of the solitons are unchanged. It is shown that the interactions between the two solitons and among the three solitons are elastic.

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Konyukhov, Andrey I. "Transformation of Eigenvalues of the Zakharov–Shabat Problem under the Effect of Soliton Collision." Izvestiya of Saratov University. New series. Series: Physics 20, no. 4 (2020): 248–57. http://dx.doi.org/10.18500/1817-3020-2020-20-4-248-257.

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Background and Objectives: The Zakharov–Shabat spectral problem allows to find soliton solutions of the nonlinear Schrodinger equation. Solving the Zakharov–Shabat problem gives both a discrete set of eigenvalues λj and a continuous one. Each discrete eigenvalue corresponds to an individual soliton with the real part Re(λj) providing the soliton velocity and the imaginary part Im(λj) determining the soliton amplitude. Solitons can be used in optical communication lines to compensate both non-linearity and dispersion. However, a direct use of solitons in return-to-zero signal encoding is inhibited. The interaction between solitions leads to the loss of transmitted data. The problem of soliton interaction can be solved using eigenvalues. The latter do not change when the solitons obey the nonlinear Schrodinger equation. Eigenvalue communication was realized recently using electronic signal processing. To increase the transmission speed the all-optical method for controlling eigenvalues should be developed. The presented research is useful to develop optical methods for the transformation of the eigenvalues. The purpose of the current paper is twofold. First, we intend to clarify the issue of whether the dispersion perturbation can not only split a bound soliton state but join solitons into a short oscillating period breather. The second goal of the paper is to describe the complicated dynamics and mutual interaction of complex eigenvalues of the Zakharov–Shabat spectral problem. Materials and Methods: Pulse propagation in single-mode optical fibers with a variable core diameter can be described using the nonlinear Schrödinger equation (NLSE) which coefficients depends on the evolution coordinate. The NLSE with the variable dispersion coefficient was considered. The dispersion coefficient was described using a hyperbolic tangent function. The NLSE and the Zakharov– Shabat spectral problem were solved using the split-step method and the layer-peeling method, respectively. Results: The results of numerical analysis of the modification of soliton pulses under the effect of variable dispersion coefficient are presented. The main attention is paid to the process of transformation of eigenvalues of the Zakharov–Shabat problem. Collision of two in-phase solitons, which are characterized by two complex eigenvalues is considered. When the coefficients of the nonlinear Schrodinger equation change, the collision of the solitons becomes inelastic. The inelastic collision is characterized by the change of the eigenvalues. It is shown that the variation of the coefficients of the NLSE allows to control both real and imaginary parts of the eigenvalues. Two scenarios for the change of the eigenvalues were identified. The first scenario is characterized by preserving the zero real part of the eigenvalues. The second one is characterized by the equality of their imaginary parts. The transformation of eigenvalues is most effective at the distance where the field spectrum possesses a two-lobe shape. Variation of the NLSE coefficient can introduce splitting or joining of colliding soliton pulses. Conclusion: The presented results show that the eigenvalues can be changed only with a small variation of the NLSE coefficients. On the one hand, a change in the eigenvalues under the effect of inelastic soliton collision is an undesirable effect since the inelastic collision of solitons will lead to unaccounted modulation in soliton optical communication links. On the other hand, the dependence of the eigenvalues on the parameters of the colliding solitons allows to modulate the eigenvalues using all-fiber optical devices. Currently, the modulation of the eigenvalues is organized using electronic devices. Therefore, the transmission of information is limited to nanosecond pulses. For picosecond pulse communication, the development of all-optical modulation methods is required. The presented results will be useful in the development of methods for controlling optical solitons and soliton states of the Bose–Einstein condensate.

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Hossain, Md Nur, M. Mamun Miah, Moataz Alosaimi, Faisal Alsharif, and Mohammad Kanan. "Exploring Novel Soliton Solutions to the Time-Fractional Coupled Drinfel’d–Sokolov–Wilson Equation in Industrial Engineering Using Two Efficient Techniques." Fractal and Fractional 8, no. 6 (June 13, 2024): 352. http://dx.doi.org/10.3390/fractalfract8060352.

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The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the Ω′Ω, 1Ω method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model’s behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation’s underlying dynamics and offer potential applications across various scientific fields.

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Ma, Lei-Nuo, Si Li, Tian-Mu Wang, Xi-Yang Xie, and Zhong Du. "Multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations via Riemann-Hilbert approach." Physica Scripta 98, no. 7 (June 22, 2023): 075222. http://dx.doi.org/10.1088/1402-4896/acde12.

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Abstract In this paper, we study multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fibers. We analyze the spectrum of the Lax pair to establish the Riemann-Hilbert problem. Using such Riemann-Hilbert problem, we calculate various multi-soliton solutions without reflection, including breather-like and mixed solitons. We illustrate the propagation and interaction dynamics of the solitons through appropriate parameter selection and asymptotic analysis. We find that the interaction between solitons is elastic, the amplitudes of solitons are only determined by the initial velocity and interaction, and the soliton with lower energy always yields a position shift when elastic interaction occurs. In addition, we observe that the existence time of the wave changes with energy and that multiple elastic interactions between solitons can be obtained when we choose appropriate variable coefficients. Then, we investigate the influences of group velocity dispersions and fourth-order dispersions on the interactions of solitons through parameter modulation mode and asymptotic analysis. Furthermore, we present several new types of nonlinear phenomena graphically, including elastic interactions between parabolic solitons and hump-type solitons, elastic interactions between cubic solitons and hump-type solitons, and periodic-changing propagations.

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Huang, Qian-Min, and Yi-Tian Gao. "Bilinear form, bilinear Bäcklund transformation and dynamic features of the soliton solutions for a variable-coefficient (3+1)-dimensional generalized shallow water wave equation." Modern Physics Letters B 31, no. 22 (August 10, 2017): 1750126. http://dx.doi.org/10.1142/s0217984917501263.

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Under investigation in this letter is a variable-coefficient (3[Formula: see text]+[Formula: see text]1)-dimensional generalized shallow water wave equation. Bilinear form and Bäcklund transformation are obtained. One-, two- and three-soliton solutions are derived via the Hirota bilinear method. Interaction and propagation of the solitons are discussed graphically. Stability of the solitons is studied numerically. Soliton amplitude is determined by the spectral parameters. Soliton velocity is not only related to the spectral parameters, but also to the variable coefficients. Phase shifts are the only difference between the two-soliton solutions and the superposition of the two relevant one-soliton solutions. Numerical investigation on the stability of the solitons indicates that the solitons could resist the disturbance of small perturbations and propagate steadily.

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Mao, Dong, Zhiwen He, Qun Gao, Chao Zeng, Ling Yun, Yueqing Du, Hua Lu, Zhipei Sun, and Jianlin Zhao. "Birefringence-Managed Normal-Dispersion Fiber Laser Delivering Energy-Tunable Chirp-Free Solitons." Ultrafast Science 2022 (July 30, 2022): 1–12. http://dx.doi.org/10.34133/2022/9760631.

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Chirp-free solitons have been mainly achieved with anomalous-dispersion fiber lasers by the balance of dispersive and nonlinear effects, and the single-pulse energy is constrained within a relatively small range. Here, we report a class of chirp-free pulse in normal-dispersion erbium-doped fiber lasers, termed birefringence-managed soliton, in which the birefringence-related phase-matching effect dominates the soliton evolution. Controllable harmonic mode locking from 5 order to 85 order is obtained at the same pump level of ~10 mW with soliton energy fully tunable beyond ten times, which indicates a new birefringence-related soliton energy law, which fundamentally differs from the conventional soliton energy theorem. The unique transformation behavior between birefringence-managed solitons and dissipative solitons is directly visualized via the single-shot spectroscopy. The results demonstrate a novel approach of engineering fiber birefringence to create energy-tunable chirp-free solitons in normal-dispersion regime and open new research directions in fields of optical solitons, ultrafast lasers, and their applications.

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Hammad, A., T. D. Swinburne, H. Hasan, S. Del Rosso, L. Iannucci, and A. P. Sutton. "Theory of the deformation of aligned polyethylene." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2180 (August 2015): 20150171. http://dx.doi.org/10.1098/rspa.2015.0171.

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Solitons are proposed as the agents of plastic and viscoelastic deformation in aligned polyethylene. Interactions between straight, parallel molecules are mapped rigorously onto the Frenkel–Kontorova model. It is shown that these molecular interactions distribute an applied load between molecules, with a characteristic transfer length equal to the soliton width. Load transfer leads to the introduction of tensile and compressive solitons at the chain ends to mark the onset of plasticity at a well-defined yield stress, which is much less than the theoretical pull-out stress. Interaction energies between solitons and an equation of motion for solitons are derived. The equation of motion is based on Langevin dynamics and the fluctuation–dissipation theorem and it leads to the rigorous definition of an effective mass for solitons. It forms the basis of a soliton dynamics in direct analogy to dislocation dynamics. Close parallels are drawn between solitons in aligned polymers and dislocations in crystals, including the configurational force on a soliton. The origins of the strain rate and temperature dependencies of the viscoelastic behaviour are discussed in terms of the formation energy of solitons. A failure mechanism is proposed involving soliton condensation under a tensile load.

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He, Feng-Tao, Xiao-Lin Wang, and Zuo-Liang Duan. "The Effects of Five-Order Nonlinear on the Dynamics of Dark Solitons in Optical Fiber." Scientific World Journal 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/130734.

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We study the influence of five-order nonlinear on the dynamic of dark soliton. Starting from the cubic-quintic nonlinear Schrodinger equation with the quadratic phase chirp term, by using a similarity transformation technique, we give the exact solution of dark soliton and calculate the precise expressions of dark soliton's width, amplitude, wave central position, and wave velocity which can describe the dynamic behavior of soliton's evolution. From two different kinds of quadratic phase chirps, we mainly analyze the effect on dark soliton’s dynamics which different fiver-order nonlinear term generates. The results show the following two points with quintic nonlinearities coefficient increasing: (1) if the coefficients of the quadratic phase chirp term relate to the propagation distance, the solitary wave displays a periodic change and the soliton’s width increases, while its amplitude and wave velocity reduce. (2) If the coefficients of the quadratic phase chirp term do not depend on propagation distance, the wave function only emerges in a fixed area. The soliton’s width increases, while its amplitude and the wave velocity reduce.

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Ahmed, Iftikhar, Aly R. Seadawy, and Dianchen Lu. "Mixed lump-solitons, periodic lump and breather soliton solutions for (2 + 1)-dimensional extended Kadomtsev–Petviashvili dynamical equation." International Journal of Modern Physics B 33, no. 05 (February 20, 2019): 1950019. http://dx.doi.org/10.1142/s021797921950019x.

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In this study, based on the Hirota bilinear method, mixed lump-solitons, periodic lump and breather soliton solutions are derived for (2 + 1)-dimensional extended KP equation with the aid of symbolic computation. Furthermore, dynamics of these solutions are explained with 3d plots and 2d contour plots by taking special choices of the involved parameters. Through the mixed lump-soliton solutions, we observe two fusion phenomena, first from interaction of lump and single soliton and other from interaction of lump with two solitons. In both cases, lump moves gradually towards soliton and transfers energy until it completely merges with the solitons. We also observe new characteristics of periodic lump solutions and kinky breather solitons.

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Deshmukh, Sharief, and Hana Alsodais. "A Note on Ricci Solitons." Symmetry 12, no. 2 (February 17, 2020): 289. http://dx.doi.org/10.3390/sym12020289.

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In this paper, we characterize trivial Ricci solitons. We observe the important role of the energy function f of a Ricci soliton (half the squared length of the potential vector field) in the charectrization of trivial Ricci solitons. We find three characterizations of connected trivial Ricci solitons by imposing different restrictions on the energy function. We also use Hessian of the potential function to characterize compact trivial Ricci solitons. Finally, we show that a solution of a Poisson equation is the energy function f of a compact Ricci soliton if and only if the Ricci soliton is trivial.

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Bhrawy, A. H., A. A. Alshaery, E. M. Hilal, Wayne N. Manrakhan, Michelle Savescu, and Anjan Biswas. "Dispersive optical solitons with Schrödinger–Hirota equation." Journal of Nonlinear Optical Physics & Materials 23, no. 01 (March 2014): 1450014. http://dx.doi.org/10.1142/s0218863514500143.

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The dynamics of dispersive optical solitons, modeled by Schrödinger–Hirota equation, are studied in this paper. Bright, dark and singular optical soliton solutions to this model are obtained in presence of perturbation terms that are considered with full nonlinearity. Soliton perturbation theory is also applied to retrieve adiabatic parameter dynamics of bright solitons. Optical soliton cooling is also studied. Finally, exact bright, dark and singular solitons are addressed for birefringent fibers with perturbation terms included.

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Ying, Jin-ping, and Sen-yue Lou. "Abundant Coherent Structures of the (2+1)-dimensional Broer-Kaup-Kupershmidt Equation." Zeitschrift für Naturforschung A 56, no. 9-10 (October 1, 2001): 619–25. http://dx.doi.org/10.1515/zna-2001-0903.

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Abstract By using of the Bäcklund transformation, which is related to the standard truncated Painleve analysis, some types of significant exact soliton solutions of the (2+1)-dimensional Broer-Kaup-Kupershmidt equation are obtained. A special type of soliton solutions may be described by means of the variable coefficient heat conduction equation. Due to the entrance of infinitely many arbitrary functions in the general expressions of the soliton solution the solitons of the (2+1)- dimensional Broer-Kaup equation possess very abundant structures. By fixing the arbitrary functions appropriately, we may obtain some types of multiple straight line solitons, multiple curved line solitons, dromions, ring solitons and etc.

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Wang, Hui, and Tian-Tian Zhang. "Stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms." International Journal of Numerical Methods for Heat & Fluid Flow 29, no. 3 (March 4, 2019): 878–89. http://dx.doi.org/10.1108/hff-08-2018-0448.

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Purpose The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms, which can be used to describe the propagation properties of optical soliton solutions. Design/methodology/approach The authors apply the ansatz method and the Hamiltonian system technique to find its bright, dark and Gaussian wave solitons and analyze its modulation instability analysis and stability analysis solution. Findings The results imply that the generalized nonlinear Schrödinger equation has bright, dark and Gaussian wave solitons. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior. Some constraint conditions are provided which can guarantee the existence of solitons. The authors analyze its modulation instability analysis and stability analysis solution. Originality/value These results may help us to further study the local structure and the interaction of solutions in generalized nonlinear Schrödinger -type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of the generalized nonlinear Schrödinger--type equations.

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Jasim AL-Taie, Mohammed Salim, and Wisam Roiss Matrood. "Optimize Nonlinear Effects on Fundamental and High-order Soliton in Photonic Crystal Fiber." Malaysian Journal of Fundamental and Applied Sciences 20, no. 2 (April 24, 2024): 320–27. http://dx.doi.org/10.11113/mjfas.v20n2.3299.

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Nonlinear effects in optical fibers are mainly caused by two sources: inelastic scattering behaviour or the intensity sensitivity of the medium's refractive index. The propagation process in photonic crystal fibers is more complex than the propagation process of first-order solitons, second-order solitons, and third-order solitons. This article discusses the effects of propagation on first-, second- and third-order solitons. A popular approach to supercontinuum generation through soliton fission is the higher-order soliton technique for spectral generation.

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ZHEN, HUI-LING, BO TIAN, PAN WANG, RONG-XIANG LIU, and HUI ZHONG. "SOLITON INTERACTION OF THE ZAKHAROV–KUZNETSOV EQUATIONS IN PLASMA DYNAMICS." International Journal of Modern Physics B 27, no. 09 (April 10, 2013): 1350029. http://dx.doi.org/10.1142/s021797921350029x.

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In this paper we investigate the constant- and variable-coefficient Zakharov–Kuznetsov (ZK) equations respectively for the electrostatic solitons and two-dimensional ion-acoustic waves obliquely propagating in the inhomogeneous magnetized two-ion-temperature dusty plasmas. By virtue of the symbolic computation and Hirota method, new bilinear forms and N-soliton solutions are both derived. Asymptotic analysis on two-soliton solutions indicates that the soliton interaction is elastic. Propagation characteristics and interaction behavior of the solitons are discussed via graphical analysis. Effects of the dispersive and disturbed coefficients are analyzed. For the constant-coefficient ZK equation, amplitude of the one soliton becomes larger when the absolute value of dispersive coefficient B increases, while interaction between the two solitons varies with the product of B and disturbed coefficient C: when BC>0, two solitons are always parallel, or they interact with each other that way. For the variable-coefficient ZK equation, periodical soliton arises when the disturbed coefficient γ(t) is a periodical function, and periods of the solitons are inversely correlated to the period of γ(t).

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Xie, Xi-Yang, and Gao-Qing Meng. "Dark-soliton collisions for a coupled AB system in the geophysical fluids or nonlinear optics." Modern Physics Letters B 32, no. 04 (February 9, 2018): 1850039. http://dx.doi.org/10.1142/s0217984918500392.

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Under investigation in this paper is a coupled AB system, which describes the marginally unstable baroclinic wave packets in the geophysical fluids or ultra-short pulses in nonlinear optics. As the dark solitons are more resistant against various perturbations than the bright ones, we aim to investigate the dark solitons in the geophysical fluids or nonlinear optics. Dark one- and two-soliton solutions for such a system are derived based on the bilinear forms and propagations of the one solitons and collisions between the two solitons are graphically illustrated and analyzed. Further, influences of the coefficients [Formula: see text] and [Formula: see text] on the solitons are discussed, where [Formula: see text] is a parameter measuring the state of the basic flow and [Formula: see text] is the group velocity. The dark-one solitons with invariant shapes and amplitudes are viewed, and elastic collisions between the dark-two solitons are observed. Also, elastic collision between the bright and dark solitons is viewed, and the dark soliton is found to possess two peaks. [Formula: see text] is found to affect the widths of the dark-one solitons and the travelling directions of the dark-two solitons. It is shown that [Formula: see text] cannot influence shapes of [Formula: see text] and [Formula: see text], but affect the plane of the one soliton for [Formula: see text], and the two-peak dark soliton for [Formula: see text] changes to the single-peak one with the value of [Formula: see text] decreasing, where [Formula: see text] and [Formula: see text] are the packets of short waves and [Formula: see text] is the mean flow.

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Asjad, Muhammad Imran, Naeem Ullah, Hamood Ur Rehman, and Tuan Nguyen Gia. "Novel soliton solutions to the Atangana–Baleanu fractional system of equations for the ISALWs." Open Physics 19, no. 1 (January 1, 2021): 770–79. http://dx.doi.org/10.1515/phys-2021-0085.

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Abstract This work deals the construction of novel soliton solutions to the Atangana–Baleanu (AB) fractional system of equations for the ion sound and Langmuir waves by using Sardar-subequation method (SSM). The outcomes are in the form of bright, singular, dark and combo soliton solutions. These solutions have wide applications in the arena of optoelectronics and wave propagation. The bright solitons will be a vast advantage in controlling the soliton disorder, dark solitons are also beneficial for soliton communication when a background wave exists and singular solitons only elaborate the shape of solitons and show a total spectrum of soliton solutions created from the model. These results would be very helpful to study and understand the physical phenomena in nonlinear optics. The performance of the SSM shows that this is powerful, talented, suitable and direct technique to discover the exact solutions for a number of nonlinear fractional models.

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Hong, Woo-Pyo. "Dynamics of Pulsating, Erupting, and Creeping Solitons in the Cubic- Quintic Complex Ginzburg-Landau Equation under the Modulated Field." Zeitschrift für Naturforschung A 61, no. 10-11 (November 1, 2006): 525–35. http://dx.doi.org/10.1515/zna-2006-10-1103.

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It is shown that the dynamics of the pulsating, erupting, and creeping (PEC) solitons in the one-dimensional cubic-quintic complex Ginzburg-Landau equation can be drastically modified in the presence of a modulated field. We first perform the linear instability analysis of continuous-wave (CW) and obtain the gain by the modulational instability (MI). It is found that the CW states applied by the weakly modulated field always transform into fronts for the parameters of the PEC solitons. We then show that, when the modulated field is applied to the pulse-like initial profile, multiple solitons are formed for the parameters of the pulsating and erupting solitons. Furthermore, as the strength of the gain term increases, the multiple pulsating or erupting solitons transform into fixedshape stable solitons. This may be important for a practical use such as to generate multiple stable femtosecond pulses. For the case of creeping soliton parameters, the presence of a modulated field does not generate multiple solitons, however, the initial profile transforms into an irregularly pulsating soliton or evolves into a fixed-shape soliton as the strength of the gain term is increased. - PACS numbers: 42.65.Tg, 03.40.Kf, 05.70.Ln, 47.20.Ky

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Siddiqi, Mohd Danish, and Fatemah Mofarreh. "Hyperbolic Ricci soliton and gradient hyperbolic Ricci soliton on relativistic prefect fluid spacetime." AIMS Mathematics 9, no. 8 (2024): 21628–40. http://dx.doi.org/10.3934/math.20241051.

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<p>In this research note, we investigated the characteristics of perfect fluid spacetime when coupled with the hyperbolic Ricci soliton. We additionally interacted with the perfect fluid spacetime, with a $ \varphi(\mathcal{Q}) $-vector field and a bi-conformal vector field that admits the hyperbolic Ricci solitons. Furthermore, we analyze the gradient hyperbolic Ricci soliton in perfect fluid spacetime, employing a scalar concircular field, and discuss about the gradient hyperbolic Ricci soliton's rate of change. In the end, we determined the energy conditions for perfect fluid spacetime in terms of gradient hyperbolic Ricci soliton with a scalar concircular field.</p>

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Hussain, Ibrar, Tahirullah, and Suhail Khan. "Four-dimensional Lorentzian plane symmetric static Ricci solitons." International Journal of Modern Physics D 28, no. 16 (October 14, 2019): 2040010. http://dx.doi.org/10.1142/s0218271820400106.

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Our focus is to investigate the Ricci solitons of the plane symmetric and static four-dimensional Lorentzian metrics. It is found that these metrics admit shrinking and concircular potential Ricci soliton vector fields with either 6- or 10-dimensional Lie algebra. Further, it is observed that the 4-dimensional Lorentzian static Ricci soliton manifolds are Einsteinian and hence the Ricci solitons are the trivial Ricci solitons.

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RAGIADAKOS, C. N. "GEOMETRODYNAMIC SOLITONS." International Journal of Modern Physics A 14, no. 16 (June 30, 1999): 2607–30. http://dx.doi.org/10.1142/s0217751x99001305.

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A formally renormalizable extended conformal gauge field action is proposed to take the place of the Rainich conditions in geometrodynamics. The moduli parameters of the Lorentzian complex structure of space–time are the dynamical variables of the present action. It admits two kinds of solitons: the pure geometric ("leptonic") solitons with vanishing gauge field and the "hadronic" ones with gauge field contributions. The gauge field modes are perturbatively confined, because the present gauge field action asymptotically generates a linear potential. The pure geometric solitons are topologically separated into three classes. One static massive soliton is found in the first class and one massless stationary soliton in the degenerate sector of every class. The corresponding (complex) conjugate Hermitian structures are the antisolitons. In the static soliton sector the electromagnetic field is explicitly defined via the Lorentzian complex structure tensor. The mass and the charge variables of the static soliton take unique values. This soliton has spin and a fermionic gyromagnetic ratio. The model has no other simple pure geometric static soliton. The energy is properly defined as a function of the moduli parameters of the complex structure. This permits the definition of the corresponding excitation modes. One must be the photonic vector mode, which appears in the static soliton sector, and the other must be massive. There can be one scalar and three vector modes. Based on this soliton (particle) spectrum, an effective Lagrangian is derived with a spontaneously broken SU (2)× U (1) symmetry, implied by the unitarity condition. A general description of asymptotically flat Lorentzian complex structures, using ordinary (not local) twistors, is also found.

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DAI, CHAO-QING, and JIE-FANG ZHANG. "TRAVELLING WAVE SOLUTIONS TO THE COUPLED DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS." International Journal of Modern Physics B 19, no. 13 (May 20, 2005): 2129–43. http://dx.doi.org/10.1142/s0217979205029778.

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In this paper, we have utilized the extended Jacobian elliptic function approach to construct seven families of new Jacobian elliptic function solutions for the coupled discrete nonlinear Schrödinger equations. When the modulus m → 1 or 0, some of these obtained solutions degenerate to the soliton solutions (the moving bright-bright and dark-dark solitons), the solitonic solutions and the trigonometric function solutions. This integrable model possesses the moving solitons because there is no PN barrier to block their motion in the lattice. We also find that some solutions in differential-difference equations (DDEs) are essentially identical to the continuous cases, while some solutions such as sec-type and tan-type in differential-difference models present different properties.

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Kachulin, Dmitry, and Andrey Gelash. "On the phase dependence of the soliton collisions in the Dyachenko–Zakharov envelope equation." Nonlinear Processes in Geophysics 25, no. 3 (August 15, 2018): 553–63. http://dx.doi.org/10.5194/npg-25-553-2018.

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Abstract. We study soliton collisions in the Dyachenko–Zakharov equation for the envelope of gravity waves in deep water. The numerical simulations of the soliton interactions revealed several fundamentally different effects when compared to analytical two-soliton solutions of the nonlinear Schrodinger equation. The relative phase of the solitons is shown to be the key parameter determining the dynamics of the interaction. We find that the maximum of the wave field can significantly exceed the sum of the soliton amplitudes. The solitons lose up to a few percent of their energy during the collisions due to radiation of incoherent waves and in addition exchange energy with each other. The level of the energy loss increases with certain synchronization of soliton phases. Each of the solitons can gain or lose the energy after collision, resulting in increase or decrease in the amplitude. The magnitude of the space shifts that solitons acquire after collisions depends on the relative phase and can be either positive or negative.

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Chen, G. W., K. L. Jia, S. K. Ji, J. Zhu, and H. Y. Li. "Soliton rains with isolated solitons induced by acoustic waves in a nonlinear multimodal interference-based fiber laser." Laser Physics 33, no. 3 (February 2, 2023): 035101. http://dx.doi.org/10.1088/1555-6611/acb355.

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Abstract We experimentally and theoretically demonstrate soliton rains with isolated solitons and soliton bunches in a passively mode-locked fiber laser, using a mode-locker based on a nonlinear multimodal interference technique. The isolated solitons are captured at the moments of integer multiples of 21 ns and 33 ns, which correspond to the acoustic modes in TR2m and R0m branches. The theory of nonlinear fiber optics is used to characterize the mechanism of capturing isolated solitons. The modal delay induced by the multimode fiber gives a revealing insight into the loss of capacity to capture and transition from soliton rains to bunches. Our study introduces a new way to precisely control the behaviors of multi-solitons and provides a potential tool for understanding complex nonlinear dynamics.

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Ahmed, Tanvir, and Javid Atai. "Moving Bragg Solitons in a Dual-Core System Composed of a Linear Bragg Grating with Dispersive Reflectivity and a Uniform Nonlinear Core." Photonics 11, no. 4 (March 30, 2024): 324. http://dx.doi.org/10.3390/photonics11040324.

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The existence and stability of moving Bragg grating solitons are systematically investigated in a dual-core system, where one core is uniform and has Kerr nonlinearity, and the other is linear with Bragg grating and dispersive reflectivity. It is found that moving soliton solutions exist throughout the upper and lower bandgaps, whereas no soliton solutions exist in the central bandgap. Similar to the quiescent solitons in the system, it is found that when dispersive reflectivity is nonzero, for certain values of parameters, sidelobes appear in the solitons’ profiles. The stability of the moving solitons is characterized using systematic numerical stability analysis. Additionally, the impact and interplay of dispersive reflectivity, soliton velocity, and group velocity on the stability border are analyzed.

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Xu, Haitao, Zhelang Pan, Zhihuan Luo, Yan Liu, Suiyan Tan, Zhijie Mai, and Jun Xu. "Zigzag Solitons and Spontaneous Symmetry Breaking in Discrete Rabi Lattices with Long-Range Hopping." Symmetry 10, no. 7 (July 12, 2018): 277. http://dx.doi.org/10.3390/sym10070277.

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A new type of discrete soliton, which we call zigzag solitons, is founded in two-component discrete Rabi lattices with long-range hopping. The spontaneous symmetry breaking (SSB) of zigzag solitons is also studied. Through numerical simulation, we found that by enhancing the intensity of the long-range linearly-coupled effect or increasing the total input power, the SSB process from the symmetric soliton to the asymmetric soliton will switch from the supercritical to subcritical type. These results can help us better understand both the discrete solitons and the Rabi coupled effect.

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Bin Turki, Nasser, Sameh Shenawy, H. K. EL-Sayied, N. Syied, and C. A. Mantica. "ρ-Einstein Solitons on Warped Product Manifolds and Applications." Journal of Mathematics 2022 (October 18, 2022): 1–10. http://dx.doi.org/10.1155/2022/1028339.

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The purpose of this research is to investigate how a ρ-Einstein soliton structure on a warped product manifold affects its base and fiber factor manifolds. Firstly, the pertinent properties of ρ-Einstein solitons are provided. Secondly, numerous necessary and sufficient conditions of a ρ-Einstein soliton warped product manifold to make its factor ρ-Einstein soliton are examined. On a ρ-Einstein gradient soliton warped product manifold, necessary and sufficient conditions for making its factor ρ-Einstein gradient soliton are presented. ρ-Einstein solitons on warped product manifolds admitting a conformal vector field are also considered. Finally, the structure of ρ-Einstein solitons on some warped product space-times is investigated.

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Mandal, Gurudas, Kaushik Roy, Anindita Paul, Asit Saha, and Prasanta Chatterjee. "Overtaking Collision and Phase Shifts of Dust Acoustic Multi-Solitons in a Four Component Dusty Plasma with Nonthermal Electrons." Zeitschrift für Naturforschung A 70, no. 9 (September 1, 2015): 703–11. http://dx.doi.org/10.1515/zna-2015-0106.

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AbstractThe nonlinear propagation and interaction of dust acoustic multi-solitons in a four component dusty plasma consisting of negatively and positively charged cold dust fluids, non-thermal electrons, and ions were investigated. By employing reductive perturbation technique (RPT), we obtained Korteweged–de Vries (KdV) equation for our system. With the help of Hirota’s bilinear method, we derived two-soliton and three-soliton solutions of the KdV equation. Phase shifts of two solitons and three solitons after collision are discussed. It was observed that the parameters α, β, β1, μe, μi, and σ play a significant role in the formation of two-soliton and three-soliton solutions. The effect of the parameter β1 on the profiles of two soliton and three soliton is shown in detail.

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Sun, Ya, Bo Tian, Yu-Feng Wang, Yun-Po Wang, and Zhi-Ruo Huang. "Bright solitons and their interactions of the (3 + 1)-dimensional coupled nonlinear Schrödinger system for an optical fiber." Modern Physics Letters B 29, no. 35n36 (December 30, 2015): 1550245. http://dx.doi.org/10.1142/s0217984915502450.

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Under investigation in this paper is the [Formula: see text]-dimensional coupled nonlinear Schrödinger system for an optical fiber with birefringence. With the Hirota method, bilinear forms of the system are derived via an auxiliary function, and the bright one- and two-soliton solutions are constructed. Based on those soliton solutions, soliton propagation and interaction are investigated analytically and graphically. Non-singular cases of the bright one-soliton solutions are presented, from which the single-peak and two-peak solitons can arise, respectively. Through the analysis on the bright two-soliton solutions, the elastic and inelastic interactions are investigated. Three kinds of the elastic interactions are presented, between the two one-peak solitons, a one-peak soliton and a two-peak soliton, and the two two-peak solitons.

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Journal articles on the topic 'Solitons'

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