Topological Soliton Solutions Research Articles

Skyrmions and pion stars in the gauged U(1) Einstein-Skyrme model

We consider topological and nontopological regular soliton solutions in the Einstein-Maxwell-Skyrme theory. We analyze the properties of these solutions and determine their domains of existence. The dependence of the solutions on the gauge coupling and on the strength of the effective gravitational coupling are examined. Topologically trivial localized field configurations, , are shown to exist, as nonlinear gravitational bound states of the Skyrme field. Both spherically symmetric and axially symmetric pion stars are considered. We find that these solutions share many features with the usual (mini)boson stars. In particular they also exhibit a spiraling behavior and do not possess a flat space limit. Published by the American Physical Society 2024

Traveling Wave Solutions and Conservation Laws of a Generalized Chaffee–Infante Equation in (1+3) Dimensions

This paper aims to analyze a generalized Chaffee–Infante equation with power-law nonlinearity in (1+3) dimensions. Ansatz methods are utilized to provide topological and non-topological soliton solutions. Soliton solutions to nonlinear evolution equations have several practical applications, including plasma physics and the diffusion process, which is why they are becoming important. Additionally, it is shown that for certain values of the parameters, the power-law nonlinearity Chaffee–Infante equation allows solitons solutions. The requirements and restrictions for soliton solutions are also mentioned. Conservation laws are derived for the aforementioned equation. In order to comprehend the dynamics of the underlying model, we graphically show the secured findings. Hirota’s perturbation method is included in the multiple exp-function technique that results in multiple wave solutions that contain new general wave frequencies and phase shifts.

Non-topological, topological and rogue wave Soliton solutions for Sharma Tasso Olver equation

This paper proposes three types of envelope solitons for the Sharma-Tasso-Olver equation with power-law nonlinearity. The ansatz approach is used to investigate the presence of these solitons identified during the solution derivation. As a result, we acquired the bright, singular, and dark solitons, which have a wide range of applications in applied science and engineering. Furthermore, the physical structures of the obtained solitons are investigated using two-dimensional, three-dimensional, and three-dimensional revolution plots with varying parameter values.

On the optical soliton solutions of Kundu–Mukherjee–Naskar equation via two different analytical methods

We introduce optical soliton solutions of the Kundu–Mukherjee–Naskar (KMN) equation by using the Sardar subequation (SSM) and the new Kudryashov methods (nKM). The KMN equation plays an important role in modeling the fiber pulse in optics, the rogue waves in the oceans and the bending of the light beam. In order to motivate and contribute to researchers working in these fields, we focus on new solutions for the KMN model in fiber pulse transmission by utilizing the Sardar subequation and recently presented New Kudryashov technique. To reach our goal, firstly, the KMN equation has been converted to a nonlinear ordinary differential equation (NODE), then in accordance with the definitions of the proposed methods the solution sets and the solution functions have been obtained. To simulate the new solutions we have obtained and make them more understandable, the topological, anti-topological, periodic,singular and periodic soliton solutions and their properties are presented by 3D and 2D graphics. According to our literature research, these two methods that we are working on have not been applied to the KMN equation before, and we believe that the new solutions we have obtained will be useful to researchers working in modeling in this field.

New topological and non-topological unidirectional-wave solutions for the modified-mixed KdV equation and bidirectional-waves solutions for the Benjamin Ono equation using recent techniques

In this paper, new explicit unidirectional wave solutions for the modified-mixed KdV equation and bidirectional waves for the Benjamin Ono equation are studied. New extension of the rational sine-cosine and sinh-cosh methods are used. In addition, an extension of the recent polynomial function method is applied as well. Different types of topological and non-topological soliton solutions are extracted to the proposed models and categorized by providing 2D and 3D graphs. Finally, some physical properties of the new bidirectional waves solutions to the Benjamin Ono model are discussed.

Solitons in lattice field theories via tight-binding supersymmetry

Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), and closely related solitonic solutions to the Bogoliubov-de Gennes equations in the theory of superconductivity. These solutions rely on the inverse scattering method, which reduces these seemingly unrelated problems to identifying reflectionless potentials of an auxiliary one-dimensional quantum scattering problem. There are several ways of constructing these potentials, one of which is quantum mechanical supersymmetry (SUSY). In this paper, motivated by recent experimental platforms, we generalize this framework to develop a theory of lattice solitons. We first briefly review the classical inverse scattering method in the continuum limit, focusing on the Korteweg-de Vries (KdV) equation and SU(N) Gross-Neveu model in the large N limit. We then generalize this methodology to lattice versions of interacting field theories. Our analysis hinges on the use of trace identities, which are relations connecting the potential of an equation of motion to the scattering data. For a discrete Schrödinger operator, such trace identities had been known as far back as Toda; however, we derive a new set of identities for the discrete Dirac operator. We then use these identities in a lattice Gross-Neveu and chiral Gross-Neveu (Nambu-Jona-Lasinio) model to show that lattice solitons correspond to reflectionless potentials associated with the discrete scattering problem. These models are of significance as they are equivalent to a mean-field theory of a lattice superconductor. To explicitly construct these solitons, we generalize supersymmetric quantum mechanics to tight-binding models. We show that a matrix transformation exists that maps a tight-binding model to an isospectral one which shares the same structure and scattering properties. The corresponding soliton solutions have both modulated hopping and onsite potential, the former of which has no analogue in the continuum limit. We explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.

The analytical solutions of Zoomeron equation via extended rational sin-cos and sinh-cosh methods

In this study, we investigate the nonlinear Zoomeron equation by using the extended rational sin − cos and sinh − cosh methods. We successfully constructed some important solutions such as singular periodic wave, periodic wave, topological, and singular soliton solutions. Using suitable parameter values, we present the numerical simulations to some of the obtained results via the 2- and 3-dimensional graphs. The methods used in this study are efficient and powerful mathematical tools that can be applied in reporting various wave solutions to different nonlinear physical models.

Transforming kinks into compactons in the [formula omitted]-sigma model

In this work, we investigate the solutions of vortices in the O(3)-sigma model with the gauge field governed by the Chern–Simons term and subject to a hyperbolic self-dual potential. We show that this model admits both topological and nontopological soliton solutions. Employing numerical analysis, we realize that the topological solutions of the model can be transformed into compacton-like solutions. On the other hand, after modifying the model by the introduction of a dielectric constant, an exciting feature appears; namely, the nontopological solutions can be transformed into kink-like solutions through the numerical variation of the dielectric constant. Furthermore, we discuss the degeneracy for the topological solitons in a given sector. Finally, we present the numerical solutions of the first model.

Solitons and other solutions to (n+1)-dimensional modified Zakharov–Kuznetsov equation by exp-function method

In this paper, the exp-function method is applied to construct exact solutions and soliton solutions of (n+1)-dimensional modified Zakharov–Kuznetsov (mZK) equation. Solitary and topological soliton solutions are obtained. These solutions are significant importance for studying waves in plasma physics.

Novel curved lump and topological solitons of integrable (2+1) dimensional KMN equation

In this article, an exact lump solution, snoidal wave solution and one topological soliton solution having a special feature, are obtained for the integrable (2+1) dimensional Kundu–Mukherjee–Naskar (KMN) equation. These solutions have an unusual property that they can get curved in the x–y plane arbitrarily due to the presence of an arbitrary function of space (x) and time (t) in their analytic forms. Due to this special feature, the solutions can be used to model the bending of optical solitonic beam in the field of nonlinear optics. This novel feature, which is an unusual property for a constant coefficient completely integrable equation, arises due to the Galilean co-variance property and “current like” nonlinearity present in the KMN equation.

New wave behaviors and stability analysis of the Gilson–Pickering equation in plasma physics

This study investigates the Gilson–Pickering equation by using the sine-Gordon expansion method. Sine-Gordon expansion method is one of the most powerful methods for solving the nonlinear partial differential equations. We successfully construct various exact solitary wave solutions to the governing equation, such as shock wave, topological, non-topological, compound topological, and non-topological soliton wave solutions. In addition, the stability of the studied nonlinear equation is analyzed via the linear stability analysis. The 2D, 3D, and contour surfaces are also plotted for all obtained solutions.

Zn modified XY and Goldstone models and vortex confinement transition

The modified XY model is a modification of the XY model by addition of a half-periodic term. The modified Goldstone model is a regular and continuum version of the modified XY model. The former admits a vortex molecule, that is, two half-quantized vortices connected by a domain wall, as a regular topological soliton solution to the equation of motion while the latter admits it as a singular configuration. Here we define the ${\mathbb Z}_n$ modified XY and Goldstone models as the $n=2$ case to be the modified XY and Goldstone models, respectively. We exhaust all stable and metastalble vortex solutions for $n=2,3$ and find a vortex confinement transition from an integer vortex to a vortex molecule of $n$ $1/n$-quantized vortices, depending on the ratio between the term of the XY model and the modified term. We find for the case of $n=3$, a rod-shaped molecule is the most stable while a Y-shaped molecule is metastable. We also construct some solutions for the case of $n=4$.The vortex confinement transition can be understood in terms of the ${\mathbb C}/{\mathbb Z}_n$ orbifold geometry.

Ground state of the Frenkel–Kontorova model with a globally deformable substrate potential

The classical ground state of the Frenkel–Kontorova model is investigated for the case of a globally deformable substrate potential, whose amplitude is a function of the position of all particles. This system serves as an idealised description of a one-dimensional crystal of trapped ions in an optical cavity, where the deformable potential originates from the cavity back action due to the dynamical Stark shift. In the limit of a low relative substrate depth, a sine-Gordon field theory is obtained with coefficients that are functionals of the chain configuration. It is shown that the back action modifies the phase diagram, where the order and scaling of the phase-transition can be controlled by the cavity parameters. Due to the back action, the kinks, which constitute topological soliton solutions in the sine-Gordon model, are shown to exhibit two stable profiles with qualitatively different properties. The choice between the stable configurations is made collectively, all kinks taking either one form or the other.

Invariant Subspace and Classification of Soliton Solutions of the Coupled Nonlinear Fokas-Liu System

In this work, the coupled nonlinear Fokas–Liu system which is a special type of KdV equation is studied using the invariant subspace method (ISM). The method determines an invariant subspace and construct the exact solutions of the nonlinear partial differential equations (NPDEs) by reducing them to ordinary differential equations (ODEs). As a result of the calculations, polynomial and logarithmic function solutions of the equation are derived. Further more, the ansatz approached is utilized to derive the topological, non-topological and other singular soliton solutions of the system. Numerical simulation off the obtained results are shown.

Exact Topological Soliton Solutions to the Strongly Perturbed Family of Sine-Gordon Type Equations

This paper uses the Ansatz method to solve for exact topological soliton solutions to sine-Gordon type equations. Single, double, and triple sine-Gordon and sine-cosine-Gordon equations are investigated along with dispersive and highly dispersive variations. After these solutions are found, strong perturbations are added to each equation and the new solutions are found. In solving both the perturbed and unperturbed sine-Gordon type equations, constraints are imposed on the parameters. The novel contributions of the authors are the soliton solutions to the strongly perturbed sine-Gordon equation and its variations. These results are important to the study of Josephson junctions, crystal dislocations, ultra-short optical pulses, relativistic field theory, and elementary particles.

The affective factors on the uncertainty in the collisions of the soliton solutions of the double field sine-Gordon system

Inspired by the well known sine-Gordon equation, we present a symmetric coupled system of two real scalar fields in 1+1 dimensions. There are three different topological soliton solutions which be labelled according to their topological charges. These solitons can absorb some localized non-dispersive wave packets in collision processes. It will be shown numerically, during collisions between solitons, there will be an uncertainty which originates from the amount of the maximum amplitude and arbitrary initial phases of the trapped wave packets.

On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering

The studies of the dynamic behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. In this study, we investigate the coupled Boussinesq equation which arises in the shallow water waves for two-layered fluid flow. The modified exp $$(-\varphi (\zeta ))$$ -expansion function method is utilized in reaching the solutions to this equation such as the topological kink-type soliton and singular soliton solutions. The interesting 2D and 3D graphics of the obtained analytical solutions in this study are presented. Via one of the reported analytical solutions, the finite forward difference method is used in obtaining the approximate numerical and exact solutions to this equation. The Fourier–Von Neumann analysis is used in checking the stability of the used numerical method with the studied model. The $$L_{2}$$ and $$L_{\infty }$$ error norms are computed. We finally present a comprehensive conclusion to this study.

The skyrmion-bubble transition in a ferromagnetic thin film

Magnetic skyrmions and bubbles, observed in ferromagnetic thin films with perpendicular magnetic anisotropy, are topological solitons which differ by their characteristic size and the balance in the energies at the origin of their stabilisation. However, these two spin textures have the same topology and a continuous transformation between them is allowed. In the present work, we derive an analytical model to explore the skyrmion-bubble transition. We evidence a region in the parameter space where both topological soliton solutions coexist and close to which transformations between skyrmion and bubbles are observed as a function of the magnetic field. Above a critical point, at which the energy barrier separating both solutions vanishes, only one topological soliton solution remains, which size can be continuously tuned from micrometer to nanometer with applied magnetic field.

On the solitary wave solutions to the longitudinal wave equation in MEE circular rod

This study investigates the longitudinal wave equation in a magneto-electro-elastic circular rod by using the extended sinh-Gordon equation expansion method. Topological, non-topological and singular soliton solutions are extracted. To illustrate the physical appearance of the obtained solutions, 2D, 3D and the contour graphs to some of the obtained solutions are plotted. The reported results may be useful in explaining the physical meaning of the studied models and other nonlinear physical models arising in nonlinear sciences.

Soliton solution of the generalized modified BBM equation and the generalized Boussinesq equation

In this paper, we obtained the topological soliton solution of the (1+1)-dimensional generalized modified Benjamin–Bona–Mahony equation and shock wave solution of the generalized Boussinesq equation. We get that solutions by using solitary wave ansatz in terms of tanh p functions. The velocity and the free parameters are the physical parameters in the soliton solutions. They can be obtained as functions of the dependent model coefficients. The domain restriction were also identified in the process. we hope that in nonlinear dynamical system these solutions will be explain some nonlinear physical phenomena.