Processing math: 100%
 
 
Sign in to use this feature.

Years

Between: -

Subjects

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline

Journals

Article Types

Countries / Regions

Search Results (139)

Search Parameters:
Keywords = two-soliton solution

Order results
Result details
Results per page
Select all
Export citation of selected articles as:
19 pages, 4040 KiB  
Article
Fractional Solitons in Optical Twin-Core Couplers with Kerr Law Nonlinearity and Local M-Derivative Using Modified Extended Mapping Method
by Noorah Mshary, Hamdy M. Ahmed and Wafaa B. Rabie
Fractal Fract. 2024, 8(12), 755; https://doi.org/10.3390/fractalfract8120755 - 23 Dec 2024
Viewed by 480
Abstract
This study focuses on optical twin-core couplers, which facilitate light transmission between two closely aligned optical fibers. These couplers operate based on the principle of coupling, allowing signals in one core to interact with those in the other. The Kerr effect, which describes [...] Read more.
This study focuses on optical twin-core couplers, which facilitate light transmission between two closely aligned optical fibers. These couplers operate based on the principle of coupling, allowing signals in one core to interact with those in the other. The Kerr effect, which describes how a material’s refractive index changes in response to the intensity of light, induces the nonlinear behavior essential for generating solitons—self-sustaining wave packets that preserve their shape and speed. In our research, we employ fractional derivatives to investigate how fractional-order variations influence wave propagation and soliton dynamics. By utilizing the modified extended mapping method (MEMM), we derive solitary wave solutions for the equations governing the behavior of optical twin-core couplers under Kerr nonlinearity. This methodology produces novel fractional traveling wave solutions, including dark, bright, singular, and combined bright–dark solitons, as well as hyperbolic, Jacobi elliptic function (JEF), periodic, and singular periodic solutions. To enhance understanding, we present physical interpretations through contour plots and include both 2D and 3D graphical representations of the results. Full article
Show Figures

Figure 1

14 pages, 2003 KiB  
Article
Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods
by Svetislav Savović, Miloš Ivanović, Branko Drljača and Ana Simović
Axioms 2024, 13(12), 872; https://doi.org/10.3390/axioms13120872 - 15 Dec 2024
Viewed by 501
Abstract
This study employs a novel physics-informed neural network (PINN) approach, the standard explicit finite difference method (EFDM) and unconditionally positivity preserving FDM to tackle the one-dimensional Sine–Gordon equation (SGE). Two test problems with known analytical solutions are investigated to demonstrate the effectiveness of [...] Read more.
This study employs a novel physics-informed neural network (PINN) approach, the standard explicit finite difference method (EFDM) and unconditionally positivity preserving FDM to tackle the one-dimensional Sine–Gordon equation (SGE). Two test problems with known analytical solutions are investigated to demonstrate the effectiveness of these techniques. While the three employed approaches demonstrate strong agreement, our analysis reveals that the EFDM results are in the best agreement with the analytical solutions. Given the consistent agreement between the numerical results from the EFDM, unconditionally positivity preserving FDM and PINN approach and the analytical solutions, all three methods are recommended as competitive options. The solution techniques employed in this study can be a valuable asset for present and future model developers engaged in various nonlinear physical wave phenomena, such as propagation of solitons in optical fibers. Full article
Show Figures

Figure 1

24 pages, 4367 KiB  
Article
New Abundant Analytical Solitons to the Fractional Mathematical Physics Model via Three Distinct Schemes
by Abdulrahman Alomair, Abdulaziz S. Al Naim and Ahmet Bekir
Mathematics 2024, 12(23), 3691; https://doi.org/10.3390/math12233691 - 25 Nov 2024
Viewed by 397
Abstract
New types of truncated M-fractional wave solitons to the simplified Modified Camassa–Holm model, a mathematical physics model, are obtained. This model is used to explain the unidirectional propagation of shallow water waves. The required solutions are obtained by utilizing the simplest equation, the [...] Read more.
New types of truncated M-fractional wave solitons to the simplified Modified Camassa–Holm model, a mathematical physics model, are obtained. This model is used to explain the unidirectional propagation of shallow water waves. The required solutions are obtained by utilizing the simplest equation, the Sardar subequation, and the generalized Kudryashov schemes. The obtained results consist of the dark, singular, periodic, dark-bright, and many other analytical solitons. Dynamical behaviors of some obtained solutions are represented by two-dimensional (2D), three-dimensional (3D), and Contour graphs. An effect of fractional derivative is shown graphically. The results are newer than the existing results of the governing equation. Obtained solutions have much importance in the various areas of applied science as well as engineering. We concluded that the utilized methods are helpful and applicable for other partial fractional equations in applied science and engineering. Full article
Show Figures

Figure 1

13 pages, 6271 KiB  
Article
Bound States and Particle Production by Breather-Type Background Field Configurations
by Abhishek Rout and Brett Altschul
Symmetry 2024, 16(12), 1571; https://doi.org/10.3390/sym16121571 - 24 Nov 2024
Cited by 1 | Viewed by 391
Abstract
We investigate the interaction of fermion fields with oscillating domain walls, inspired by breather-type solutions of the sine-Gordon equation, a nonlinear system of fundamental importance. Our study focuses on the fermionic bound states and particle production induced by a time-dependent scalar background field. [...] Read more.
We investigate the interaction of fermion fields with oscillating domain walls, inspired by breather-type solutions of the sine-Gordon equation, a nonlinear system of fundamental importance. Our study focuses on the fermionic bound states and particle production induced by a time-dependent scalar background field. The fermions couple to two domain walls undergoing harmonic motion, and we explore the resulting dynamics of the fermionic wave functions. We demonstrate that while fermions initially form bound states around the domain walls, the energy provided by the oscillatory motion of the scalar field induces an outward flux of fermions and antifermions, leading to particle production and eventual flux propagation toward spatial infinity. Through numerical simulations, we observe that the fermion density exhibits quasiperiodic behavior, with partial recurrences of the bound state configurations after each oscillation period. However, the fermion wave functions do not remain localized, and over time, the density decreases as more particles escape the vicinity of the domain walls. Our results highlight that the sine-Gordon-like breather background, when coupled non-supersymmetrically to fermions, does not preserve integrability or stability, with the oscillations driving a continuous energy transfer into the fermionic modes. This study sheds light on the challenges of maintaining steady-state fermion solutions in time-dependent topological backgrounds and offers insights into particle production mechanisms in nonlinear dynamical systems with oscillating solitons. Full article
(This article belongs to the Section Physics)
Show Figures

Figure 1

15 pages, 269 KiB  
Article
The Extension of Noncommutative Modified KP Hierarchy and Its Quasideterminant Solutions
by Hongxia Wu, Chunxia Li and Haifeng Wang
Axioms 2024, 13(12), 816; https://doi.org/10.3390/axioms13120816 - 22 Nov 2024
Viewed by 398
Abstract
The extended noncommutative modified KP (exncmKP) hierarchy is firstly constructed, which gives rise to two types of the ncmKP equation with self-consistent sources (ncmKPESCSs). Then, the noncommutative (NC) Miura transformation between the extended noncommutative KP (exncKP) hierarchy and the exncmKP hierarchy is presented, [...] Read more.
The extended noncommutative modified KP (exncmKP) hierarchy is firstly constructed, which gives rise to two types of the ncmKP equation with self-consistent sources (ncmKPESCSs). Then, the noncommutative (NC) Miura transformation between the extended noncommutative KP (exncKP) hierarchy and the exncmKP hierarchy is presented, and the quasideterminant solutions of the exncmKP hierarchy are also given. As its byproduct, the quasideterminant solutions of two types of ncmKPESCSs are obtained. The matrix solutions of two types of ncmKPESCSs are finally investigated, and the impact of the source terms on the NC soliton is analyzed. Full article
(This article belongs to the Section Mathematical Physics)
15 pages, 289 KiB  
Article
Soliton Solutions to Sasa–Satsuma-Type Modified Korteweg–De Vries Equations by Binary Darboux Transformations
by Wen-Xiu Ma
Mathematics 2024, 12(23), 3643; https://doi.org/10.3390/math12233643 - 21 Nov 2024
Viewed by 505
Abstract
Sasa–Satsuma (SS)-type integrable matrix modified Korteweg–de Vries (mKdV) equations are derived from two group constraints, involving the replacement of the spectral matrix in the Ablowitz–Kaup–Newell–Segur matrix eigenproblems with its matrix transpose and its Hermitian transpose. Using the Lax pairs and dual Lax pairs [...] Read more.
Sasa–Satsuma (SS)-type integrable matrix modified Korteweg–de Vries (mKdV) equations are derived from two group constraints, involving the replacement of the spectral matrix in the Ablowitz–Kaup–Newell–Segur matrix eigenproblems with its matrix transpose and its Hermitian transpose. Using the Lax pairs and dual Lax pairs of matrix eigenproblems as a foundation, binary Darboux transformations are constructed. These transformations, initiated with a zero seed solution, facilitate the generation of soliton solutions for the SS-type integrable matrix mKdV equations presented. Full article
20 pages, 6607 KiB  
Article
Investigating the Dynamics of a Unidirectional Wave Model: Soliton Solutions, Bifurcation, and Chaos Analysis
by Tariq Alraqad, Muntasir Suhail, Hicham Saber, Khaled Aldwoah, Nidal Eljaneid, Amer Alsulami and Blgys Muflh
Fractal Fract. 2024, 8(11), 672; https://doi.org/10.3390/fractalfract8110672 - 18 Nov 2024
Viewed by 610
Abstract
The current work investigates a recently introduced unidirectional wave model, applicable in science and engineering to understand complex systems and phenomena. This investigation has two primary aims. First, it employs a novel modified Sardar sub-equation method, not yet explored in the literature, to [...] Read more.
The current work investigates a recently introduced unidirectional wave model, applicable in science and engineering to understand complex systems and phenomena. This investigation has two primary aims. First, it employs a novel modified Sardar sub-equation method, not yet explored in the literature, to derive new solutions for the governing model. Second, it analyzes the complex dynamical structure of the governing model using bifurcation, chaos, and sensitivity analyses. To provide a more accurate depiction of the underlying dynamics, they use quantum mechanics to explain the intricate behavior of the system. To illustrate the physical behavior of the obtained solutions, 2D and 3D plots, along with a phase plane analysis, are presented using appropriate parameter values. These results validate the effectiveness of the employed method, providing thorough and consistent solutions with significant computational efficiency. The investigated soliton solutions will be valuable in understanding complex physical structures in various scientific fields, including ferromagnetic dynamics, nonlinear optics, soliton wave theory, and fiber optics. This approach proves highly effective in handling the complexities inherent in engineering and mathematical problems, especially those involving fractional-order systems. Full article
Show Figures

Figure 1

11 pages, 2794 KiB  
Article
Multilinear Variable Separation Approach in (4+1)-Dimensional Boiti–Leon–Manna–Pempinelli Equation
by Jia-Rong Zhu and Bo Ren
Symmetry 2024, 16(11), 1529; https://doi.org/10.3390/sym16111529 - 15 Nov 2024
Viewed by 654
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Schrödinger Equations)
Show Figures

Figure 1

17 pages, 3536 KiB  
Article
Exploring Soliton Solutions and Chaotic Dynamics in the (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation: A Generalized Rational Exponential Function Approach
by Amjad E. Hamza, Muntasir Suhail, Amer Alsulami, Alaa Mustafa, Khaled Aldwoah and Hicham Saber
Fractal Fract. 2024, 8(10), 592; https://doi.org/10.3390/fractalfract8100592 - 9 Oct 2024
Cited by 1 | Viewed by 972
Abstract
This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals [...] Read more.
This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations. Full article
(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
Show Figures

Figure 1

29 pages, 3713 KiB  
Article
New Coupled Optical Solitons to Birefringent Fibers for Complex Ginzburg–Landau Equations with Hamiltonian Perturbations and Kerr Law Nonlinearity
by Emmanuel Yomba and Poonam Ramchandra Nair
Mathematics 2024, 12(19), 3073; https://doi.org/10.3390/math12193073 - 30 Sep 2024
Viewed by 531
Abstract
In this study, we use an analytical method tailored for the in-depth exploration of coupled nonlinear partial differential equations (NLPDEs), with a primary focus on the dynamics of solitons. Traditional methods are quite effective for solving individual nonlinear partial differential equations (NLPDEs). However, [...] Read more.
In this study, we use an analytical method tailored for the in-depth exploration of coupled nonlinear partial differential equations (NLPDEs), with a primary focus on the dynamics of solitons. Traditional methods are quite effective for solving individual nonlinear partial differential equations (NLPDEs). However, their performance diminishes notably when addressing systems of coupled NLPDEs. This decline in effectiveness is mainly due to the complex interaction terms that arise in these coupled systems. Commonly, researchers have attempted to simplify coupled NLPDEs into single equations by imposing proportional relationships between various solutions. Unfortunately, this simplification often leads to a significant deviation from the true physical phenomena that these equations aim to describe. Our approach is distinctively advantageous in its straightforwardness and precision, offering a clearer and more insightful analytical perspective for examining coupled NLPDEs. It is capable of concurrently facilitating the propagation of different soliton types in two distinct systems through a single process. It also supports the spontaneous emergence of similar solitons in both systems with minimal restrictions. It has been extensively used to investigate a wide array of new coupled progressive solitons in birefringent fibers, specifically for complex Ginzburg–Landau Equations (CGLEs) involving Hamiltonian perturbations and Kerr law nonlinearity. The resulting solitons, with comprehensive 2D and 3D visualizations, showcase a variety of coupled soliton configurations, including several that are unprecedented in the field. This innovative approach not only addresses a significant gap in existing methodologies but also broadens the horizons for future research in optical communications and related disciplines. Full article
(This article belongs to the Section Mathematical Physics)
Show Figures

Figure 1

14 pages, 4664 KiB  
Article
Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative
by Sujoy Devnath, Maha M. Helmi and M. Ali Akbar
Computation 2024, 12(9), 187; https://doi.org/10.3390/computation12090187 - 11 Sep 2024
Cited by 1 | Viewed by 823
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section Computational Engineering)
Show Figures

Figure 1

24 pages, 12404 KiB  
Article
Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation
by Sheng Zhang, Hongwei Li and Bo Xu
Fractal Fract. 2024, 8(9), 520; https://doi.org/10.3390/fractalfract8090520 - 31 Aug 2024
Viewed by 1052
Abstract
In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order [...] Read more.
In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media. Full article
Show Figures

Figure 1

23 pages, 6260 KiB  
Article
Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension
by Lihua Zhang, Zitong Zheng, Bo Shen, Gangwei Wang and Zhenli Wang
Fractal Fract. 2024, 8(9), 517; https://doi.org/10.3390/fractalfract8090517 - 30 Aug 2024
Viewed by 531
Abstract
We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite [...] Read more.
We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry group method, and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton and elliptical function interplaying solution for the fractional KdVSKR model in (1+1)-dimension has been derived for the first time. For the fractional KdVSKR model in (2+1)-dimension, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. The effect of the order γ on the dynamical behaviors of the solutions has been illustrated by figures. The three-wave interaction solution has not been studied in the current references. The novelty of this paper is that the finite symmetry group method is adopted to construct interaction solutions of fractional nonlinear systems. This research idea can be applied to other fractional differential equations. Full article
(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
Show Figures

Figure 1

16 pages, 906 KiB  
Article
mKdV Equation on Time Scales: Darboux Transformation and N-Soliton Solutions
by Baojian Jin, Yong Fang and Xue Sang
Axioms 2024, 13(9), 578; https://doi.org/10.3390/axioms13090578 - 25 Aug 2024
Viewed by 932
Abstract
In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When xR and  [...] Read more.
In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When xR and tR, the equation degenerates to the classical mKdV equation. Then, the single-soliton, two-soliton, and N-soliton solutions of the mKdV equation under the zero boundary condition on time scales are presented via employing the Darboux transformation (DT). Particularly, we obtain the corresponding single-soliton solutions expressed using the Cayley exponential function on four different time scales (RZ, q-discrete, C). Full article
(This article belongs to the Section Mathematical Physics)
Show Figures

Figure 1

23 pages, 3421 KiB  
Article
Stability Analysis, Modulation Instability, and Beta-Time Fractional Exact Soliton Solutions to the Van der Waals Equation
by Haitham Qawaqneh, Jalil Manafian, Mohammed Alharthi and Yasser Alrashedi
Mathematics 2024, 12(14), 2257; https://doi.org/10.3390/math12142257 - 19 Jul 2024
Cited by 9 | Viewed by 1081
Abstract
The study consists of the distinct types of the exact soliton solutions to an important model called the beta-time fractional (1 + 1)-dimensional non-linear Van der Waals equation. This model is used to explain the motion of molecules and materials. The Van der [...] Read more.
The study consists of the distinct types of the exact soliton solutions to an important model called the beta-time fractional (1 + 1)-dimensional non-linear Van der Waals equation. This model is used to explain the motion of molecules and materials. The Van der Waals equation explains the phase separation phenomenon. Noncovalent Van der Waals or dispersion forces usually have an effect on the structure, dynamics, stability, and function of molecules and materials in different branches of science, including biology, chemistry, materials science, and physics. Solutions are obtained, including dark, dark-singular, periodic wave, singular wave, and many more exact wave solutions by using the modified extended tanh function method. Using the fractional derivatives makes different solutions different from the existing solutions. The gained results will be of high importance in the interaction of quantum-mechanical fluctuations, granular matters, and other applications of the Van der Waals equation. The solutions may be useful in distinct fields of science and civil engineering, as well as some basic physical ones like those studied in geophysics. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs by using Mathematica software. The obtained results are newer than the existing results. Stability analysis is also performed to check the stability of the concerned model. Furthermore, modulation instability is studied to study the stationary solutions of the concerned model. The results will be helpful in future studies of the concerned system. In the end, we can say that the method used is straightforward and dynamic, and it will be a useful tool for debating tough issues in a wide range of fields. Full article
Show Figures

Figure 1

Back to TopTop