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Journal articles on the topic 'Three-dimensional solitons'

Author: Grafiati

Published: 28 May 2022

Last updated: 18 July 2025

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1

Liu, Siyao. "Algebraic Schouten Solitons of Three-Dimensional Lorentzian Lie Groups." Symmetry 15, no. 4 (2023): 866. http://dx.doi.org/10.3390/sym15040866.

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In 2016, Wears defined and studied algebraic T-solitons. In this paper, we define algebraic Schouten solitons as a special T-soliton and classify the algebraic Schouten solitons associated with Levi-Civita connections, canonical connections, and Kobayashi–Nomizu connections on three-dimensional Lorentzian Lie groups that have some product structure.

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2

De, Uday Chand, and Chiranjib Dey. "On Three Dimensional Cosymplectic Manifolds Admitting Almost Ricci Solitons." Tamkang Journal of Mathematics 51, no. 4 (2020): 303–12. http://dx.doi.org/10.5556/j.tkjm.51.2020.3077.

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In the present paper we study three dimensional cosymplectic manifolds admitting almost Ricci solitons. Among others we prove that in a three dimensional compact orientable cosymplectic manifold M^3 withoutboundary an almost Ricci soliton reduces to Ricci soliton under certain restriction on the potential function lambda. As a consequence we obtain several corollaries. Moreover we study gradient almost Ricci solitons.

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3

Zhmudskii, A. A., and B. A. Ivanov. "Magnon bubbles: a new type of three-dimensional magnetic soliton." Soviet Journal of Low Temperature Physics 12, no. 6 (1986): 367–68. https://doi.org/10.1063/10.0031524.

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4

Meng, Yong, Hafiz Wajahat Ahmed Riaz, and Ji Lin. "New types of nondegenerate solitons for a (2+1)-dimensional coupled system*." Communications in Theoretical Physics 77, no. 9 (2025): 095001. https://doi.org/10.1088/1572-9494/adc240.

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Abstract In this paper, we investigate the (2+1)-dimensional three-component long-wave-short-wave resonance interaction system, which describes complex systems and nonlinear wave phenomena in physics. By employing the Hirota bilinear method, we derive the general nondegenerate N-soliton solution of the system, where each short-wave component contains N arbitrary functions of the independent variable y. The presence of these arbitrary functions in the analytical solution enables the construction of a wide range of nondegenerate soliton types. Finally, we illustrate the structural features of several novel nondegenerate solitons, including M-shaped, multiple double-hump, and sawtooth double-striped solitons, as well as interactions between nondegenerate solitons, such as dromion-like solitons and solitoffs, with the aid of figures.

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5

Mondal, Ashis. "Three-dimensional para-Kenmotsu manifolds admitting eta-Ricci solitons." Gulf Journal of Mathematics 11, no. 2 (2021): 44–52. http://dx.doi.org/10.56947/gjom.v11i2.584.

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In the present paper we study η-Ricci solitons on three-dimensional para-Kenmotsu manifolds with the curvature condition R.Q=0. Also we study conformal flat, projectively flat and concircularly flat η-Ricci soliton on a three-dimensional para-Kenmotsu manifold. Finally, we construct an example of a three-dimensional para-Kenmotsu manifold which admits η-Ricci solitons.

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6

FERRETTI, G., S. G. RAJEEV, and Z. YANG. "BARYONS AS SOLITONS IN THREE-DIMENSIONAL QUANTUM CHROMODYNAMICS." International Journal of Modern Physics A 07, no. 32 (1992): 8001–19. http://dx.doi.org/10.1142/s0217751x92003628.

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We show that baryons of three-dimensional quantum chromodynamics can be understood as solitons of its effective Lagrangian. In the parity-preserving phase we study, these baryons are fermions for odd Nc and bosons for even Nc, never anyons. We quantize the collective variables of the solitons and thereby calculate the flavor quantum numbers. magnetic moments and mass splittings of the baryon. The flavor quantum numbers are in agreement with naive quark model for the low-lying states. The magnetic moments and mass splittings are smaller in the soliton model by a factor of logFπ/Ncmπ. We also show that there is a dibaryon solution that is an analog of the deuteron. These solitons can describe defects in a quantum antiferromagnet.

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7

Turan, Mine, Chand De, and Ahmet Yildiz. "Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sasakian manifolds." Filomat 26, no. 2 (2012): 363–70. http://dx.doi.org/10.2298/fil1202363t.

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The object of the present paper is to study 3-dimensional trans-Sasakian manifolds admitting Ricci solitons and gradient Ricci solitons. We prove that if (1,V, ?) is a Ricci soliton where V is collinear with the characteristic vector field ?, then V is a constant multiple of ? and the manifold is of constant scalar curvature provided ?, ? =constant. Next we prove that in a 3-dimensional trans-Sasakian manifold with constant scalar curvature if 1 is a gradient Ricci soliton, then the manifold is either a ?-Kenmotsu manifold or an Einstein manifold. As a consequence of this result we obtain several corollaries.

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8

Lu, Dianchen, Aly R. Seadawy, and Iftikhar Ahmed. "Applications of mixed lump-solitons solutions and multi-peaks solitons for newly extended (2+1)-dimensional Boussinesq wave equation." Modern Physics Letters B 33, no. 29 (2019): 1950363. http://dx.doi.org/10.1142/s0217984919503639.

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In this work, based on the Hirota bilinear method, mixed lump-solitons solutions and multi-peaks solitons are derived for a new extended (2[Formula: see text]+[Formula: see text]1)-dimensional Boussinesq equation by using ansatz function technique with symbolic computation. Through the mixed lump-solitons, we obtained two types of interaction phenomena, first from lump-single soliton solution and other from lump-two soliton solutions and their dynamics is given by three-dimensional plots and two-dimensional contour plots by taking appropriate values of given parameters. Furthermore, we obtained new patterns of multi-peaks solitons.

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9

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 (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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10

Aburdzhaniya, G. D., V. P. Lakhin, and A. B. Mikhailovskii. "Nonlinear regular structures of drift magneto-acoustic waves." Journal of Plasma Physics 38, no. 3 (1987): 373–86. http://dx.doi.org/10.1017/s0022377800012666.

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Nonlinear regular structures in a magnetized plasma connected with drift magneto-acoustic waves (DMA) are investigated theoretically. Three-dimensional nonlinear equations of weakly dispersive DMA waves are obtained. These equations contain both the scalar nonlinearity and the vector one, and generalize the two-dimensional Kadomtsev–Petviashvili (KP) equation. The existence is shown of regular stationary structures due to the scalar nonlinearity: one-dimensional solitons, two-dimensional rational solitons, chains of solitons and so-called ‘crosses’. The stability of one-dimensional DMA solitons is investigated. It is shown that soliton stability depends on the sign of the wave dispersion as in the case of systems described by the KP-type equation.

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11

Li, Zhao, and Chen Peng. "Dynamics and Embedded Solitons of Stochastic Quadratic and Cubic Nonlinear Susceptibilities with Multiplicative White Noise in the Itô Sense." Mathematics 11, no. 14 (2023): 3185. http://dx.doi.org/10.3390/math11143185.

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The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities in the Itô sense, which can further help researchers understand the propagation of soliton nonlinear systems. Firstly, a two-dimensional dynamics system and its perturbation system are obtained by using a traveling wave transformation. Secondly, the phase portraits of the two-dimensional dynamics system are plotted. Furthermore, the chaotic behavior, two-dimensional phase portraits, three-dimensional phase portraits and sensitivity of the perturbation system are analyzed via Maple software. Finally, the embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities are obtained. Moreover, three-dimensional and two-dimensional solitons of stochastic quadratic and cubic nonlinear susceptibilities are plotted.

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12

Veretenov, N. A., S. V. Fedorov, and N. N. Rosanov. "Topological three-dimensional dissipative optical solitons." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2124 (2018): 20170367. http://dx.doi.org/10.1098/rsta.2017.0367.

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This article presents a review of recent investigations of topological three-dimensional (3D) dissipative optical solitons in homogeneous laser media with fast nonlinearity of amplification and absorption. The solitons are found numerically, with their formation, by embedding two-dimensional laser solitons or their complexes in 3D space after their rotation around a vortex straight line with their simultaneous twist. After a transient, the ‘hula-hoop’ solitons can form with a number of closed and unclosed infinite vortex lines, i.e. the solitons are tangles in topological notation. They are attractors and are characterized by extreme stability. The solitons presented here can be realized in lasers with fast saturable absorption and are promising for information applications. The tangle solitons of the type described present an example of self-organization that can be found not only in optics but also in various distributed dissipative systems of different types. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)’.

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13

Borisov, A. B., and F. N. Rybakov. "Three-dimensional magnetic solitons." Physics of Metals and Metallography 112, no. 7 (2011): 745–66. http://dx.doi.org/10.1134/s0031918x11070040.

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14

Zhu, Jia-Rong, and Bo Ren. "Multilinear Variable Separation Approach in (4+1)-Dimensional Boiti–Leon–Manna–Pempinelli Equation." Symmetry 16, no. 11 (2024): 1529. http://dx.doi.org/10.3390/sym16111529.

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In this paper, we use the multilinear variable separation approach involving two arbitrary variable separation functions to construct a new variable separation solution of the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Through considering different parameters, three types of local excitations including dromions, lumps, and ring solitons are constructed. Dromion molecules, lump molecules, ring soliton molecules, and their interactions are analyzed through the velocity resonance mechanism. In addition, the results reveal the elastic and inelastic interactions between solitons. We discuss some dynamical properties of these solitons and soliton molecules obtained analytically. Three-dimensional diagrams and contour plots of the solution are given to help understand the physical mechanism of the solutions.

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15

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 (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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16

Zhao, Xue-Hui, Bo Tian, De-Yin Liu, Xiao-Yu Wu, Jun Chai, and Yong-Jiang Guo. "Dark solitons, Lax pair and infinitely-many conservation laws for a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in the inhomogeneous Heisenberg ferromagnetic spin chain." Modern Physics Letters B 31, no. 03 (2017): 1750013. http://dx.doi.org/10.1142/s0217984917500130.

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Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain. Lax pair and infinitely-many conservation laws are derived, indicating the existence of the multi-soliton solutions for such an equation. Via the Hirota method with an auxiliary function, bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interactions for the dark solitons are illustrated graphically: Velocity of the solitons is linearly related to the coefficients of the second- and fourth-order dispersion terms, while amplitude of the solitons does not depend on them. Interactions between the two solitons are shown to be elastic, while those among the three solitons are pairwise elastic.

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17

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 (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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18

Frycz, P., R. Rankin, and J. C. Samson. "Stability of electron inertia Alfvén solitons." Journal of Plasma Physics 48, no. 2 (1992): 335–43. http://dx.doi.org/10.1017/s0022377800016585.

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Das, Kamp and Sluijter have proposed equations describing three-dimensional electron inertia Alfvén waves for which the characteristic length scales in directions parallel and perpendicular to the ambient magnetic field are of the same order. Planar, obliquely propagating soliton solutions of these equations are known to be linearly unstable. Numerical simulations reveal the nonlinear phase of the evolution of these solitons: a transition from flat to cylindrical solitons is observed, followed by breaking-up into three-dimensional localized cavities. The final stage corresponds to wave breaking; no final stable structure is achieved within the model.

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19

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 (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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20

Matuszewski, M., E. Infeld, G. Rowlands, and M. Trippenbach. "Stability analysis of three-dimensional breather solitons in a Bose–Einstein condensate." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2063 (2005): 3561–74. http://dx.doi.org/10.1098/rspa.2005.1531.

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We investigated the stability properties of breather soliton trains in a three-dimensional Bose–Einstein condensate (BEC) with Feshbach-resonance management of the scattering length. This is done so as to generate both attractive and repulsive interaction. The condensate is confined only by a one-dimensional optical lattice and we consider strong, moderate and weak confinement. By strong confinement we mean a situation in which a quasi two-dimensional soliton is created. Moderate confinement admits a fully three-dimensional soliton. Weak confinement allows individual solitons to interact. Stability properties are investigated by several theoretical methods such as a variational analysis, treatment of motion in effective potential wells, and collapse dynamics. Armed with all the information forthcoming from these methods, we then undertake a numerical calculation. Our theoretical predictions are fully confirmed, perhaps to a higher degree than expected. We compare regions of stability in parameter space obtained from a fully three-dimensional analysis with those from a quasi two-dimensional treatment, when the dynamics in one direction are frozen. We find that in the three-dimensional case the stability region splits into two parts. However, as we tighten the confinement, one of the islands of stability moves toward higher frequencies and the lower frequency region becomes more and more like that for the quasi two-dimensional case. We demonstrate these solutions in direct numerical simulations and, importantly, suggest a way of creating robust three-dimensional solitons in experiments in a BEC in a one-dimensional lattice.

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21

Bersons, I., R. Veilande, and A. Pirktinsh. "Three-dimensional collinearly propagating solitons." Physica Scripta 89, no. 4 (2014): 045102. http://dx.doi.org/10.1088/0031-8949/89/04/045102.

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22

Driben, Rodislav, Yaroslav V. Kartashov, Boris A. Malomed, Torsten Meier, and Lluis Torner. "Three-dimensional hybrid vortex solitons." New Journal of Physics 16, no. 6 (2014): 063035. http://dx.doi.org/10.1088/1367-2630/16/6/063035.

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23

Zhao, Chen, Yi-Tian Gao, Zhong-Zhou Lan, and Jin-Wei Yang. "Bäcklund Transformation and Soliton Solutions for a (3+1)-Dimensional Variable-Coefficient Breaking Soliton Equation." Zeitschrift für Naturforschung A 71, no. 9 (2016): 797–805. http://dx.doi.org/10.1515/zna-2016-0127.

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AbstractIn this article, a (3+1)-dimensional variable-coefficient breaking soliton equation is investigated. Based on the Bell polynomials and symbolic computation, the bilinear forms and Bäcklund transformation for the equation are derived. One-, two-, and three-soliton solutions are obtained via the Hirota method.N-soliton solutions are also constructed. Propagation characteristics and interaction behaviors of the solitons are discussed graphically: (i) solitonic direction and position depend on the sign of the wave numbers; (ii) shapes of the multisoliton interactions in the scaled space and time coordinates are affected by the variable coefficients; (iii) multisoliton interactions are elastic for that the velocity and amplitude of each soliton remain unchanged after each interaction except for a phase shift.

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24

Rizvi, Syed T. R., Sana Ghafoor, Aly R. Seadawy, Ahmed H. Arnous, Hakim AL Garalleh, and Nehad Ali Shah. "Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model." AIMS Mathematics 9, no. 8 (2024): 21144–76. http://dx.doi.org/10.3934/math.20241027.

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In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

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25

Biswas, Gour, Xiaomin Chen, and Uday De. "Riemann solitons on almost co-Kähler manifolds." Filomat 36, no. 4 (2022): 1403–13. http://dx.doi.org/10.2298/fil2204403b.

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The aim of the present paper is to characterize almost co-K?hler manifolds whose metrics are the Riemann solitons. At first we provide a necessary and sufficient condition for the metric of a 3-dimensional manifold to be Riemann soliton. Next it is proved that if the metric of an almost co-K?hler manifold is a Riemann soliton with the soliton vector field ?, then the manifold is flat. It is also shown that if the metric of a (?, ?)-almost co-K?hler manifold with ? < 0 is a Riemann soliton, then the soliton is expanding and ?, ?, ? satisfies a relation. We also prove that there does not exist gradient almost Riemann solitons on (?, ?)-almost co-K?hler manifolds with ? < 0. Finally, the existence of a Riemann soliton on a three dimensional almost co-K?hler manifold is ensured by a proper example.

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26

Vylegzhanin, D. V., P. N. Klepikov, E. D. Rodionov, and O. P. Khromova. "On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton." Izvestiya of Altai State University, no. 4(120) (September 10, 2021): 86–90. http://dx.doi.org/10.14258/izvasu(2021)4-13.

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Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

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Lan, Zhong-Zhou, Yi-Tian Gao, Jin-Wei Yang, Chuan-Qi Su, and Qi-Min Wang. "Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional B-type Kadomtsev–Petviashvili equation in the fluid/plasma mechanics." Modern Physics Letters B 30, no. 25 (2016): 1650265. http://dx.doi.org/10.1142/s0217984916502651.

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Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili equation for the shallow water wave in a fluid or electrostatic wave potential in a plasma. Bilinear form, Bäcklund transformation and Lax pair are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota’s method. Propagation and interaction of the solitons are illustrated graphically: (i) Through the asymptotic analysis, elastic and inelastic interactions between the two solitons are discussed analytically and graphically, respectively. The elastic interaction, amplitudes, velocities and shapes of the two solitons remain unchanged except for a phase shift. However, in the area of the inelastic interaction, amplitudes of the two solitons have a linear superposition. (ii) Elastic interactions among the three solitons indicate that the properties of the elastic interactions among the three solitons are similar to those between the two solitons. Moreover, oblique and overtaking interactions between the two solitons are displayed. Oblique interactions among the three solitons and interactions among the two parallel solitons and a single one are presented as well. (iii) Inelastic–elastic interactions imply that the interaction between the inelastic region and another one is elastic.

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Zhong, Rong-Xuan, Nan Huang, Huang-Wu Li, et al. "Matter-wave solitons supported by quadrupole–quadrupole interactions and anisotropic discrete lattices." International Journal of Modern Physics B 32, no. 09 (2018): 1850107. http://dx.doi.org/10.1142/s0217979218501072.

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We numerically and analytically investigate the formations and features of two-dimensional discrete Bose–Einstein condensate solitons, which are constructed by quadrupole–quadrupole interactional particles trapped in the tunable anisotropic discrete optical lattices. The square optical lattices in the model can be formed by two pairs of interfering plane waves with different intensities. Two hopping rates of the particles in the orthogonal directions are different, which gives rise to a linear anisotropic system. We find that if all of the pairs of dipole and anti-dipole are perpendicular to the lattice panel and the line connecting the dipole and anti-dipole which compose the quadrupole is parallel to horizontal direction, both the linear anisotropy and the nonlocal nonlinear one can strongly influence the formations of the solitons. There exist three patterns of stable solitons, namely horizontal elongation quasi-one-dimensional discrete solitons, disk-shape isotropic pattern solitons and vertical elongation quasi-continuous solitons. We systematically demonstrate the relationships of chemical potential, size and shape of the soliton with its total norm and vertical hopping rate and analytically reveal the linear dispersion relation for quasi-one-dimensional discrete solitons.

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29

Karpov, Petr, and Serguei Brazovskii. "Pattern Formation and Aggregation in Ensembles of Solitons in Quasi One-Dimensional Electronic Systems." Symmetry 14, no. 5 (2022): 972. http://dx.doi.org/10.3390/sym14050972.

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Broken symmetries of quasi one-dimensional electronic systems give rise to microscopic solitons taking roles of carriers of the charge or spin. The double degeneracy gives rise to solitons as kinks of the scalar order parameter A; the continuous degeneracy for the complex order parameter Aexp(iθ) gives rise to phase vortices, amplitudes solitons, and their combinations. These degrees of freedom can be controlled or accessed independently via either the spin polarization or the charge doping. The long-range ordering in dimensions above one imposes super-long-range confinement forces upon the solitons, leading to a sequence of phase transitions in their ensembles. The higher-temperature T transition enforces the confinement of solitons into topologically bound complexes: pairs of kinks or the amplitude solitons dressed by exotic half-integer vortices. At a second lower T transition, the solitons aggregate into rods of bi-kinks or into walls of amplitude solitons terminated by rings of half-integer vortices. With lowering T, the walls multiply, passing sequentially across the sample. Here, we summarize results of a numerical modeling for different symmetries, for charged and neutral soliton, in two and three dimensions. The efficient Monte Carlo algorithm, preserving the number of solitons, was employed which substantially facilitates the calculations, allowing to extend them to the three-dimensional case and to include the long-range Coulomb interactions.

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30

Alzahrani, Saleh Mousa, and Talal Alzahrani. "Multiple solitons with bifurcations, lump waves, M-shaped and interaction solitons of three component generalized (3+1)-dimensional Breaking soliton system." AIMS Mathematics 8, no. 8 (2023): 17803–26. http://dx.doi.org/10.3934/math.2023908.

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<abstract><p>The generalized (3+1)-dimensional Breaking soliton system (gBSS) has numerous applications across various scientific fields. This manuscript presents a study on important exact solutions of the gBSS, with a focus on novel solutions. Using the Hirota bilinear technique, we derive the general solution of the proposed system and obtain the novel solutions by considering different types of auxiliary functions. Our analysis includes the study of multi-solitons, multiple bifurcation solitons, lump wave solutions, M-shaped solitons, and their interactions. We also observe several hybrid solitons, including tuning fork-shaped, X-Y shaped, and double Y shaped. Our results are presented through graphical representations.</p></abstract>

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31

Wang, Yong. "Left-Invariant Riemann Solitons of Three-Dimensional Lorentzian Lie Groups." Symmetry 13, no. 2 (2021): 218. http://dx.doi.org/10.3390/sym13020218.

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Riemann solitons are generalized fixed points of the Riemann flow. In this note, we study left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups. We completely classify left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups.

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32

Chen, Meidan, and Biao Li. "Hybrid soliton solutions in the (2+1)-dimensional nonlinear Schrödinger equation." Modern Physics Letters B 31, no. 32 (2017): 1750298. http://dx.doi.org/10.1142/s0217984917502980.

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Rational solutions and hybrid solutions from N-solitons are obtained by using the bilinear method and a long wave limit method. Line rogue waves and lumps in the (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Schrödinger (NLS) equation are derived from two-solitons. Then from three-solitons, hybrid solutions between kink soliton with breathers, periodic line waves and lumps are derived. Interestingly, after the collision, the breathers are kept invariant, but the amplitudes of the periodic line waves and lumps change greatly. For the four-solitons, the solutions describe as breathers with breathers, line rogue waves or lumps. After the collision, breathers and lumps are kept invariant, but the line rogue wave has a great change.

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33

Eldidamony, Hanaa A., Ahmed H. Arnous, Mohammad Mirzazadeh, Mir Sajjad Hashemi, and Mustafa Bayram. "Comparative approaches to solving the (2 + 1)-dimensional generalized coupled nonlinear Schrödinger equations with four-wave mixing." Nonlinear Analysis: Modelling and Control 30 (January 2, 2025): 1–25. https://doi.org/10.15388/namc.2025.30.38324.

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This paper extensively studies the propagation of optical solitons within the framework of (2 + 1)-dimensional generalized coupled nonlinear Schrödinger equations. The investigation employs three worldly integration techniques: the enhanced direct algebraic method, the enhanced Kudryashov method, and the new projective Riccati equation method. Through the application of these methods, a broad spectrum of soliton solutions has been uncovered, including bright, dark, singular, and straddled solitons. Additionally, this study reveals solutions characterized by Jacobi andWeierstrass elliptic functions, enriching the understanding of the dynamics underpinning optical solitons in complex systems. The diversity of the soliton solutions obtained demonstrates the versatility and efficacy of the employed integration techniques and contributes significantly to the theoretical and practical knowledge of nonlinear optical systems.

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34

Siddiqi, Mohd Danish, and Sudhakar K. Chaubey. "Almost $\eta$-conformal Ricci solitons in $(LCS)_{3}$-manifolds." Sarajevo Journal of Mathematics 16, no. 2 (2022): 245–59. http://dx.doi.org/10.5644/sjm.16.02.10.

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The aim of the present paper is to study the properties of three dimensional Lorentzian concircular structure manifolds ($(LCS)_{3}$-manifolds) endowed with almost $\eta$-conformal Ricci solitons. Also, we discuss the $\eta$-conformal gradient shrinking Ricci solitons on $(LCS)_{3}$-manifolds. Finally, the examples of almost $\eta$-conformal Ricci soliton on an $(LCS)_{3}$-manifold are provided in the region where $(LCS)_{3}$-manifold is expanding.

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35

Azami, Shahroud, Mehdi Jafari, Abdul Haseeb, and Abdullah Ali H. Ahmadini. "Cross Curvature Solitons of Lorentzian Three-Dimensional Lie Groups." Axioms 13, no. 4 (2024): 211. http://dx.doi.org/10.3390/axioms13040211.

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36

Li, Li-Li, Bo Tian, Hai-Qiang Zhang, Xing Lü, and Wen-Jun Liu. "Soliton Solutions, Pairwise Collisions and Partially Coherent Interactions for A Generalized (1+1)-Dimensional Coupled Nonlinear Schrödinger System via Symbolic Computation." Zeitschrift für Naturforschung A 63, no. 10-11 (2008): 679–87. http://dx.doi.org/10.1515/zna-2008-10-1111.

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With the aid of symbolic computation, we investigate a generalized (1+1)-dimensional coupled nonlinear Schrödinger system with mixed nonlinear interactions, which has potential applications in nonlinear optics and elastic solids. The exact analytical one-, two-, and three-soliton solutions are firstly obtained by employing the bilinear method under two constraints. Some main propagation and interaction properties of the solitons are discussed simultaneously.Moreover, some figures are plotted to graphically analyze the pairwise collisions and partially coherent interactions of three solitons.

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37

Cao, Huai-Dong, Shu-Cheng Chang, and Chih-Wei Chen. "On Three-Dimensional CR Yamabe Solitons." Journal of Geometric Analysis 28, no. 1 (2017): 335–59. http://dx.doi.org/10.1007/s12220-017-9822-3.

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38

Brozos-Vázquez, M., G. Calvaruso, E. García-Río, and S. Gavino-Fernández. "Three-dimensional Lorentzian homogeneous Ricci solitons." Israel Journal of Mathematics 188, no. 1 (2011): 385–403. http://dx.doi.org/10.1007/s11856-011-0124-3.

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39

Calviño-Louzao, E., J. Seoane-Bascoy, M. E. Vázquez-Abal, and R. Vázquez-Lorenzo. "Three-dimensional homogeneous Lorentzian Yamabe solitons." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 82, no. 2 (2012): 193–203. http://dx.doi.org/10.1007/s12188-012-0072-9.

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40

Seadawy, Aly R., Naila Nasreen, and Dianchen Lu. "Complex model ultra-short pulses in optical fibers via generalized third-order nonlinear Schrödinger dynamical equation." International Journal of Modern Physics B 34, no. 17 (2020): 2050143. http://dx.doi.org/10.1142/s021797922050143x.

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In this paper, several types of solitons such as dark soliton, bright soliton, periodic soliton, kink soliton and solitary waves in three-dimensional and two-dimensional contour plot have been derived for the generalized third-order nonlinear Schrödinger dynamical equations (NLSEs). The generalized third-order NLSE is a significant model ultra-short pulses in optical fibers. The computational work and outcomes achieved show the influence and efficiency of current method. Furthermore, we can solve many other higher-order NLSEs with the help of simple and effective technique.

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41

Kim, Daehwan, and Juncheol Pyo. "Translating solitons foliated by spheres." International Journal of Mathematics 28, no. 01 (2017): 1750006. http://dx.doi.org/10.1142/s0129167x17500069.

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In this paper, we consider translating solitons in [Formula: see text] which is foliated by spheres. In three-dimensional Euclidean space, we show that such a translating soliton is a surface of revolution and the axis of revolution is parallel to the translating direction of the translating soliton. We also show that the same result holds for a higher dimension case with a hypersurface foliated by spheres in parallel hyperplanes that are perpendicular to the translating direction.

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42

Chen, Wenxia, Weixu Ni, Lixin Tian, and Xiyan Yang. "Abundant families of Jacobi elliptic function solutions and peculiar dynamical behavior for a generalized derivative nonlinear Schrödinger equation employing extended F-expansion method." Physica Scripta 99, no. 12 (2024): 125257. http://dx.doi.org/10.1088/1402-4896/ad9090.

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Abstract This study investigates a generalized derivative nonlinear Schrödinger (GDNLS) equation , demonstrating how ultrashort pulses propagate in a single-mode optical fiber. The extended F-expansion method, which is a modification of Kudryashov’s auxiliary equation approach, is applied in this investigation to generate Jacobi elliptic solutions for the GDNLS. Three distinct solution instances are examined, and a variety of explicit solutions, including breathers, solitary waves, bright/dark solitons, bright-dark interaction solitons, a soliton-like solution, and a rogue-like solution, are obtained. To demonstrate the complex dynamical behavior of GDNLS equation, several representative solutions are chosen and their moduli are shown in three-dimensional, two-dimensional, and contour plots using Maple software.

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43

Abdul Kayum, Md, Aly R. Seadawy, Ali M. Akbar, and Taghreed G. Sugati. "Stable solutions to the nonlinear RLC transmission line equation and the Sinh–Poisson equation arising in mathematical physics." Open Physics 18, no. 1 (2020): 710–25. http://dx.doi.org/10.1515/phys-2020-0183.

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AbstractThe Sinh–Poisson equation and the RLC transmission line equation are important nonlinear model equations in the field of engineering and power transmission. The modified simple equation (MSE) procedure is a realistic, competent and efficient mathematical scheme to ascertain the analytic soliton solutions to nonlinear evolution equations (NLEEs). In the present article, the MSE approach is put forward and exploited to establish wave solutions to the previously referred NLEEs and accomplish analytical broad-ranging solutions associated with parameters. Whenever parameters are assigned definite values, diverse types of solitons originated from the general wave solutions. The solitons are explained by sketching three-dimensional and two-dimensional graphs, and their physical significance is clearly stated. The profiles of the attained solutions assimilate compacton, bell-shaped soliton, peakon, kink, singular periodic, periodic soliton and singular kink-type soliton. The outcomes assert that the MSE scheme is an advance, convincing and rigorous scheme to bring out soliton solutions. The solutions obtained may significantly contribute to the areas of science and engineering.

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44

Chen, Junchao, and Biao Li. "Three-Dimensional Bright–Dark Soliton, Bright Soliton Pairs, and Rogue Wave of Coupled Nonlinear Schr¨odinger Equation with Time–Space Modulation." Zeitschrift für Naturforschung A 67, no. 8-9 (2012): 483–90. http://dx.doi.org/10.5560/zna.2012-0045.

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We systematically provide a similarity transformation reducing the (3+1)-dimensional inhomogeneous coupled nonlinear Schrodinger (CNLS) equation with variable coefficients and parabolic potential to the (1+1)-dimensional coupled nonlinear Schrodinger equation with constant coefficients. Based on the similarity transformation, we discuss the dynamics of the propagation of the three-dimensional bright-dark soliton, the interaction between two bright solitons, and the feature of the three-dimensional rogue wave with different parameters. The obtained results may raise the possibility of relative experiments and potential applications.

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45

Jaradat, Imad, and Marwan Alquran. "Construction of Solitary Two-Wave Solutions for a New Two-Mode Version of the Zakharov-Kuznetsov Equation." Mathematics 8, no. 7 (2020): 1127. http://dx.doi.org/10.3390/math8071127.

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A new two-mode version of the generalized Zakharov-Kuznetsov equation is derived using Korsunsky’s method. This dynamical model describes the propagation of two-wave solitons moving simultaneously in the same direction with mutual interaction that depends on an embedded phase-velocity parameter. Three different methods are used to obtain exact bell-shaped soliton solutions and singular soliton solutions to the proposed model. Two-dimensional and three-dimensional plots are also provided to illustrate the interaction dynamics of the obtained two-wave exact solutions upon increasing the phase-velocity parameter.

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46

Peyghan, Esmaeil, Leila Nourmohammadifar, Akram Ali, and Ion Mihai. "(Almost) Ricci Solitons in Lorentzian–Sasakian Hom-Lie Groups." Axioms 13, no. 10 (2024): 693. http://dx.doi.org/10.3390/axioms13100693.

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We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional sl(2,R) and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is a contact 1-form, conditions under which the Ricci curvature tensor is v-parallel are given. Ricci solitons for Lorentzian–Sasakian Hom-Lie algebras are also studied. It is shown that a Ricci soliton vector field ζ is conformal whenever the Lorentzian–Sasakian Hom-Lie algebra is Ricci semisymmetric. To illustrate the use of the theory, a two-parameter family of three-dimensional Lorentzian–Sasakian Hom-Lie algebras which are not Lie algebras is given and their Ricci solitons are computed.

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47

Sarkar, Avijit, Suparna Halder, and Uday De. "Riemann and Ricci bourguignon solitons on three-dimensional quasi-Sasakian manifolds." Filomat 36, no. 19 (2022): 6573–84. http://dx.doi.org/10.2298/fil2219573s.

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48

Zhu, Wei, Hai-Fang Song, Wan-Li Wang, and Bo Ren. "Soliton Molecules, Multi-Lumps and Hybrid Solutions in Generalized (2 + 1)-Dimensional Date–Jimbo–Kashiwara–Miwa Equation in Fluid Mechanics." Symmetry 17, no. 4 (2025): 538. https://doi.org/10.3390/sym17040538.

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The generalized (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (gDJKM) equation, which can be used to describe some phenomena in fluid mechanics, is investigated based on the multi-soliton solution. Soliton molecules of the gDJKM equation are given by the velocity resonance mechanism. A soliton molecule containing three solitons is portrayed at different times. The invariance of the relative positions of three solitons confirms that they form a soliton molecule. Multi-order lumps are obtained by applying the long-wave limit method in the multi-soliton. By analyzing the dynamics of one-order and two-order lumps, the energy concentration and localization property for lump waves are displayed. In the meanwhile, a multi-soliton can transform into multi-order breathers by the complex conjugation relations of parameters. The interaction among lumps, breathers and soliton molecules can be constructed by combining the above comprehensive analysis. The interaction between a one-order lump and a soliton molecule is an elastic collision, which can be observed through investigating evolutionary processes. The results obtained in this paper are useful for explaining certain nonlinear phenomena in fluid dynamics.

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49

Devnath, Sujoy, Maha M. Helmi, and M. Ali Akbar. "Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative." Computation 12, no. 9 (2024): 187. http://dx.doi.org/10.3390/computation12090187.

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The fractional regularized long wave equation and the fractional nonlinear shallow-water wave equation are the noteworthy models in the domains of fluid dynamics, ocean engineering, plasma physics, and microtubules in living cells. In this study, a reliable and efficient improved F-expansion technique, along with the fractional beta derivative, has been utilized to explore novel soliton solutions to the stated wave equations. Consequently, the study establishes a variety of reliable and novel soliton solutions involving trigonometric, hyperbolic, rational, and algebraic functions. By setting appropriate values for the parameters, we obtained peakons, anti-peakon, kink, bell, anti-bell, singular periodic, and flat kink solitons. The physical behavior of these solitons is demonstrated in detail through three-dimensional, two-dimensional, and contour representations. The impact of the fractional-order derivative on the wave profile is notable and is illustrated through two-dimensional graphs. It can be stated that the newly established solutions might be further useful for the aforementioned domains.

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50

Zheng, Fengshan, Nikolai S. Kiselev, Filipp N. Rybakov, et al. "Hopfion rings in a cubic chiral magnet." Nature 623, no. 7988 (2023): 718–23. http://dx.doi.org/10.1038/s41586-023-06658-5.

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AbstractMagnetic skyrmions and hopfions are topological solitons1—well-localized field configurations that have gained considerable attention over the past decade owing to their unique particle-like properties, which make them promising objects for spintronic applications. Skyrmions2,3 are two-dimensional solitons resembling vortex-like string structures that can penetrate an entire sample. Hopfions4–9 are three-dimensional solitons confined within a magnetic sample volume and can be considered as closed twisted skyrmion strings that take the shape of a ring in the simplest case. Despite extensive research on magnetic skyrmions, the direct observation of magnetic hopfions is challenging10 and has only been reported in a synthetic material11. Here we present direct observations of hopfions in crystals. In our experiment, we use transmission electron microscopy to observe hopfions forming coupled states with skyrmion strings in B20-type FeGe plates. We provide a protocol for nucleating such hopfion rings, which we verify using Lorentz imaging and electron holography. Our results are highly reproducible and in full agreement with micromagnetic simulations. We provide a unified skyrmion–hopfion homotopy classification and offer insight into the diversity of topological solitons in three-dimensional chiral magnets.

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