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Keywords = multiple soliton solutions

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18 pages, 1843 KiB  
Article
The Compatibility of Some Integrability Methods and Related Solutions for the Variable Coefficients Geophysical KdV Model
by Rodica Cimpoiasu, Radu Constantinescu and Corina Nicoleta Babalic
Axioms 2025, 14(8), 557; https://doi.org/10.3390/axioms14080557 - 23 Jul 2025
Viewed by 131
Abstract
This paper focuses on the variable coefficients geophysical KdV (VCGKdV) equation, which involves time-dependent perturbation, nonlinearity and dispersion parameters. It is a more realistic model than its constant coefficient counterpart and can be useful to, for instance, investigate the Coriolis effect on oceanic [...] Read more.
This paper focuses on the variable coefficients geophysical KdV (VCGKdV) equation, which involves time-dependent perturbation, nonlinearity and dispersion parameters. It is a more realistic model than its constant coefficient counterpart and can be useful to, for instance, investigate the Coriolis effect on oceanic flows. Firstly, we analyzed this model using three strong methods that allow the investigation of its integrability: the Lie symmetry approach, Painlevé property and Hirota formalism. The general constraints between the involved parameters under which the complete integrability in Lie, Painlevé or Hirota sense exists, as well as the largest class of this type of equations, which admits the same class of imposed symmetries are generated. Then, some new specific families of solutions for the model endowed with either Lie symmetry properties, Lie and Painlevé constraints or with Lie, Painlevé and Hirota constraints were generated and compared with solutions derived with other techniques. By numerical simulations, the dynamical behaviors of some Lie invariant solutions and nonautonomous multiple solitons are depicted. Full article
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16 pages, 1929 KiB  
Article
Dynamical Behavior of Solitary Waves for the Space-Fractional Stochastic Regularized Long Wave Equation via Two Distinct Approaches
by Muneerah Al Nuwairan, Bashayr Almutairi and Anwar Aldhafeeri
Mathematics 2025, 13(13), 2193; https://doi.org/10.3390/math13132193 - 4 Jul 2025
Viewed by 194
Abstract
This study investigates the influence of multiplicative noise—modeled by a Wiener process—and spatial-fractional derivatives on the dynamics of the space-fractional stochastic Regularized Long Wave equation. By employing a complete discriminant polynomial system, we derive novel classes of fractional stochastic solutions that capture the [...] Read more.
This study investigates the influence of multiplicative noise—modeled by a Wiener process—and spatial-fractional derivatives on the dynamics of the space-fractional stochastic Regularized Long Wave equation. By employing a complete discriminant polynomial system, we derive novel classes of fractional stochastic solutions that capture the complex interplay between stochasticity and nonlocality. Additionally, the variational principle, derived by He’s semi-inverse method, is utilized, yielding additional exact solutions that are bright solitons, bright-like solitons, kinky bright solitons, and periodic structures. Graphical analyses are presented to clarify how variations in the fractional order and noise intensity affect essential solution features, such as amplitude, width, and smoothness, offering deeper insight into the behavior of such nonlinear stochastic systems. Full article
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10 pages, 674 KiB  
Article
Abundant Exact Traveling-Wave Solutions for Stochastic Graphene Sheets Model
by Wael W. Mohammed, Taher S. Hassan, Rabeb Sidaoui, Hijyah Alshammary and Mohamed S. Algolam
Axioms 2025, 14(6), 477; https://doi.org/10.3390/axioms14060477 - 19 Jun 2025
Viewed by 247
Abstract
Here, we consider the stochastic graphene sheets model (SGSM) forced by multiplicative noise in the Itô sense. We show that the exact solution of the SGSM may be obtained by solving some deterministic counterparts of the graphene sheets model and combining the result [...] Read more.
Here, we consider the stochastic graphene sheets model (SGSM) forced by multiplicative noise in the Itô sense. We show that the exact solution of the SGSM may be obtained by solving some deterministic counterparts of the graphene sheets model and combining the result with a solution of stochastic ordinary differential equations. By applying the extended tanh function method, we obtain the soliton solutions for the deterministic counterparts of the graphene sheets model. Because graphene sheets are important in many fields, such as electronics, photonics, and energy storage, the solutions of the stochastic graphene sheets model are beneficial for understanding several fascinating scientific phenomena. Using the MATLAB program, we exhibit several 3D graphs that illustrate the impact of multiplicative noise on the exact solutions of the SGSM. By incorporating stochastic elements into the equations that govern the evolution of graphene sheets, researchers can gain insights into how these fluctuations impact the behavior of the material over time. Full article
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17 pages, 1944 KiB  
Article
Abundant Elliptic, Trigonometric, and Hyperbolic Stochastic Solutions for the Stochastic Wu–Zhang System in Quantum Mechanics
by Wael W. Mohammed, Ekram E. Ali, Athar I. Ahmed and Marwa Ennaceur
Mathematics 2025, 13(5), 714; https://doi.org/10.3390/math13050714 - 22 Feb 2025
Cited by 1 | Viewed by 736
Abstract
In this article, we look at the stochastic Wu–Zhang system (SWZS) forced by multiplicative Brownian motion in the Itô sense. The mapping method, which is an effective analytical method, is employed to investigate the exact wave solutions of the aforementioned equation. The proposed [...] Read more.
In this article, we look at the stochastic Wu–Zhang system (SWZS) forced by multiplicative Brownian motion in the Itô sense. The mapping method, which is an effective analytical method, is employed to investigate the exact wave solutions of the aforementioned equation. The proposed scheme provides new types of exact solutions including periodic solitons, kink solitons, singular solitons and so on, to describe the wave propagation in quantum mechanics and analyze a wide range of essential physical phenomena. In the absence of noise, we obtain some previously found solutions of SWZS. Additionally, using the MATLAB program, the impacts of the noise term on the analytical solution of the SWZS were demonstrated. Full article
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17 pages, 919 KiB  
Article
Exploring the Diversity of Kink Solitons in (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation
by Musawa Yahya Almusawa and Hassan Almusawa
Mathematics 2024, 12(21), 3340; https://doi.org/10.3390/math12213340 - 24 Oct 2024
Cited by 1 | Viewed by 812
Abstract
The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions [...] Read more.
The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves. Full article
(This article belongs to the Topic AI and Data-Driven Advancements in Industry 4.0)
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18 pages, 313 KiB  
Review
Progresses on Some Open Problems Related to Infinitely Many Symmetries
by Senyue Lou
Mathematics 2024, 12(20), 3224; https://doi.org/10.3390/math12203224 - 15 Oct 2024
Cited by 3 | Viewed by 1122
Abstract
The quest to reveal the physical essence of the infinitely many symmetries and/or conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related [...] Read more.
The quest to reveal the physical essence of the infinitely many symmetries and/or conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of n-wave solutions, such as the n-soliton solutions, multiple breathers, complexitons, and the n-periodic wave solutions (the algebro-geometric solutions with genus n), wherein n denotes an arbitrary integer that can potentially approach infinity. Each subwave comprising the n-wave solution may possess free parameters, including center parameters ci, width parameters (wave number) ki, and periodic parameters (the Riemann parameters) mi. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinitely many symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. This conjecture appears to hold true for all integrable systems with n-wave solutions. The conjecture intimates that the currently known infinitely many symmetries is not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned Korteweg–de Vries (KdV) equation and the Burgers equation as simple examples, the conjecture is substantiated for the n-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative, which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework. Notably, a ren-symmetric integrable Burgers hierarchy is explicitly derived. Both the supersymmetric and the classical integrable hierarchies are encompassed within the ren-symmetric integrable hierarchy. The results of this paper will make further progresses in nonlinear science: to find more infinitely many symmetries, to establish novel methods to solve nonlinear systems via symmetries, to find more novel exact solutions and new physics, and to open novel integrable theories such as the ren-symmetric integrable systems and the possible relations to fractional integrable systems. Full article
(This article belongs to the Special Issue Soliton Theory and Integrable Systems in Mathematical Physics)
20 pages, 1576 KiB  
Article
A New (3+1)-Dimensional Extension of the Kadomtsev–Petviashvili–Boussinesq-like Equation: Multiple-Soliton Solutions and Other Particular Solutions
by Xiaojian Li and Lianzhong Li
Symmetry 2024, 16(10), 1345; https://doi.org/10.3390/sym16101345 - 11 Oct 2024
Cited by 1 | Viewed by 1310
Abstract
In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). [...] Read more.
In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). Exact analytical solutions were derived through the power series method, with a comprehensive proof of solution convergence. Employing the (G/G2)-expansion method enabled the identification of trigonometric, exponential, and rational solutions of the equation. Furthermore, we established the auto-Bäcklund transformation of the equation. Multiple-soliton solutions were identified by utilizing Hirota’s bilinear method. The fundamental properties of these solutions were elucidated through graphical representations. Our results are of certain value to the interpretation of nonlinear problems. Full article
(This article belongs to the Section Mathematics)
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20 pages, 6930 KiB  
Article
Riemann–Hilbert Method Equipped with Mixed Spectrum for N-Soliton Solutions of New Three-Component Coupled Time-Varying Coefficient Complex mKdV Equations
by Sheng Zhang, Xianghui Wang and Bo Xu
Fractal Fract. 2024, 8(6), 355; https://doi.org/10.3390/fractalfract8060355 - 14 Jun 2024
Viewed by 1552
Abstract
This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating [...] Read more.
This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating the ccmKdVEs are proposed. Then, based on the msEs, a solvable RH problem related to the ccmKdVEs is constructed. By using the constructed RH problem with mixed spectrum, scattering data for the recovery of potential formulae are further determined. In the case of reflectionless coefficients, explicit N-soliton solutions of the ccmKdVEs are ultimately obtained. Taking N equal to 1 and 2 as examples, this paper reveals that the spatiotemporal solution structures with time-varying nonlinear dynamic characteristics localized in the ccmKdVEs is attributed to the multiple selectivity of mixed spectrum and time-varying coefficients. In addition, to further highlight the application of our work in fractional calculus, by appropriately selecting these time-varying coefficients, the ccmKdVEs are transformed into a conformable time-fractional order system of three-component coupled complex mKdV equations. Based on the obtained one-soliton solutions, a set of initial values are assigned to the transformed fractional order system, and the N-th iteration formulae of approximate solutions for this fractional order system are derived through the variational iteration method (VIM). Full article
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19 pages, 5495 KiB  
Article
Exploring Novel Soliton Solutions to the Time-Fractional Coupled Drinfel’d–Sokolov–Wilson Equation in Industrial Engineering Using Two Efficient Techniques
by Md Nur Hossain, M. Mamun Miah, Moataz Alosaimi, Faisal Alsharif and Mohammad Kanan
Fractal Fract. 2024, 8(6), 352; https://doi.org/10.3390/fractalfract8060352 - 13 Jun 2024
Cited by 7 | Viewed by 2375
Abstract
The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma [...] Read more.
The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the ΩΩ, 1Ω method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model’s behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation’s underlying dynamics and offer potential applications across various scientific fields. Full article
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22 pages, 1740 KiB  
Article
Analyzing Dynamics: Lie Symmetry Approach to Bifurcation, Chaos, Multistability, and Solitons in Extended (3 + 1)-Dimensional Wave Equation
by Muhammad Bilal Riaz, Adil Jhangeer, Faisal Z. Duraihem and Jan Martinovic
Symmetry 2024, 16(5), 608; https://doi.org/10.3390/sym16050608 - 14 May 2024
Cited by 10 | Viewed by 1635
Abstract
The examination of new (3 + 1)-dimensional wave equations is undertaken in this study. Initially, the identification of the Lie symmetries of the model is carried out through the utilization of the Lie symmetry approach. The commutator and adjoint table of the symmetries [...] Read more.
The examination of new (3 + 1)-dimensional wave equations is undertaken in this study. Initially, the identification of the Lie symmetries of the model is carried out through the utilization of the Lie symmetry approach. The commutator and adjoint table of the symmetries are presented. Subsequently, the model under discussion is transformed into an ordinary differential equation using these symmetries. The construction of several bright, kink, and dark solitons for the suggested equation is then achieved through the utilization of the new auxiliary equation method. Subsequently, an analysis of the dynamical nature of the model is conducted, encompassing various angles such as bifurcation, chaos, and sensitivity. Bifurcation occurs at critical points within a dynamical system, accompanied by the application of an outward force, which unveils the emergence of chaotic phenomena. Two-dimensional plots, time plots, multistability, and Lyapunov exponents are presented to illustrate these chaotic behaviors. Furthermore, the sensitivity of the investigated model is executed utilizing the Runge–Kutta method. This analysis confirms that the stability of the solution is minimally affected by small changes in initial conditions. The attained outcomes show the effectiveness of the presented methods in evaluating solitons of multiple nonlinear models. Full article
(This article belongs to the Special Issue Symmetry in the Soliton Theory)
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16 pages, 3262 KiB  
Article
Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative
by Md Ashik Iqbal, Abdul Hamid Ganie, Md Mamun Miah and Mohamed S. Osman
Fractal Fract. 2024, 8(4), 210; https://doi.org/10.3390/fractalfract8040210 - 3 Apr 2024
Cited by 22 | Viewed by 2673
Abstract
Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and [...] Read more.
Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts. Full article
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14 pages, 3182 KiB  
Article
Machine Learning Based Automatic Mode-Locking of a Dual-Wavelength Soliton Fiber Laser
by Qi Yan, Yiwei Tian, Tianqi Zhang, Changjian Lv, Fanchao Meng, Zhixu Jia, Weiping Qin and Guanshi Qin
Photonics 2024, 11(1), 47; https://doi.org/10.3390/photonics11010047 - 2 Jan 2024
Cited by 12 | Viewed by 3142
Abstract
Recent years have witnessed growing research interest in dual-wavelength mode-locked fiber lasers for their pivotal role in diverse applications and the exploration of nonlinear dynamics. Despite notable progress in their development, achieving reliable mode-locked dual-wavelength operation typically necessitates intricate manual adjustments of the [...] Read more.
Recent years have witnessed growing research interest in dual-wavelength mode-locked fiber lasers for their pivotal role in diverse applications and the exploration of nonlinear dynamics. Despite notable progress in their development, achieving reliable mode-locked dual-wavelength operation typically necessitates intricate manual adjustments of the cavity’s polarization components. In this article, we present the realization of automatic mode-locking in a dual-wavelength soliton fiber laser. To provide guidance for the algorithm design, we systematically investigated the impact of polarization configurations and initial states on the laser’s operation through numerical simulations and linear scan experiments. The results indicate that operational regimes can be finely adjusted around the wave plate position supporting the mode-locked dual-wavelength solution. Furthermore, the laser exhibits multiple stable states at the mode-locked dual-wavelength point, with critical dependence on the initial conditions. Accordingly, we developed a two-stage genetic algorithm that was demonstrated to be effective for realizing automatic dual-wavelength mode-locking. To further improve the performance of the algorithm, a feedforward neural network was trained and integrated into the algorithm, enabling accurate identification of the dual-wavelength states. This study provides valuable insights into understanding how polarization configurations and initial conditions impact the operational regimes of dual-wavelength mode-locked fiber lasers. The algorithm developed can be extended to optimize other systems with multiple stable states supported at the same parameter point. Full article
(This article belongs to the Special Issue Advances in Fiber Laser Mode Locking)
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16 pages, 5209 KiB  
Article
Multiple Soliton Solutions for Coupled Modified Korteweg–de Vries (mkdV) with a Time-Dependent Variable Coefficient
by Haroon D. S. Adam, Khalid I. A. Ahmed, Mukhtar Yagoub Youssif and Marin Marin
Symmetry 2023, 15(11), 1972; https://doi.org/10.3390/sym15111972 - 25 Oct 2023
Cited by 3 | Viewed by 1534
Abstract
In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for [...] Read more.
In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences. Full article
(This article belongs to the Special Issue Singular Distributions With Special Structures and Symmetries)
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17 pages, 1633 KiB  
Article
On the Dynamical Behavior of Solitary Waves for Coupled Stochastic Korteweg–De Vries Equations
by Wael W. Mohammed, Farah M. Al-Askar and Clemente Cesarano
Mathematics 2023, 11(16), 3506; https://doi.org/10.3390/math11163506 - 14 Aug 2023
Cited by 14 | Viewed by 1399
Abstract
In this paper, we take into account the coupled stochastic Korteweg–De Vries (CSKdV) equations in the Itô sense. Using the mapping method, new trigonometric, rational, hyperbolic, and elliptic stochastic solutions are obtained. These obtained solutions can be applied to the analysis of a [...] Read more.
In this paper, we take into account the coupled stochastic Korteweg–De Vries (CSKdV) equations in the Itô sense. Using the mapping method, new trigonometric, rational, hyperbolic, and elliptic stochastic solutions are obtained. These obtained solutions can be applied to the analysis of a wide variety of crucial physical phenomena because the coupled KdV equations have important applications in various fields of physics and engineering. Also, it is used in the design of optical fiber communication systems, which transmit information using soliton-like waves. The dynamic performance of the various obtained solutions are depicted using 3D and 2D curves in order to interpret the effects of multiplicative noise. We conclude that multiplicative noise influences the behavior of the solutions of CSKdV equations and stabilizes them. Full article
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11 pages, 1073 KiB  
Article
New (3+1)-Dimensional Kadomtsev–Petviashvili–Sawada– Kotera–Ramani Equation: Multiple-Soliton and Lump Solutions
by Abdul-Majid Wazwaz, Ma’mon Abu Hammad, Ali O. Al-Ghamdi, Mansoor H. Alshehri and Samir A. El-Tantawy
Mathematics 2023, 11(15), 3395; https://doi.org/10.3390/math11153395 - 3 Aug 2023
Cited by 10 | Viewed by 1811
Abstract
In this investigation, a novel (3+1)-dimensional Lax integrable Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation is constructed and analyzed analytically. The Painlevé integrability for the mentioned model is examined. The bilinear form is applied for investigating multiple-soliton solutions. Moreover, we employ the positive quadratic function method to create [...] Read more.
In this investigation, a novel (3+1)-dimensional Lax integrable Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation is constructed and analyzed analytically. The Painlevé integrability for the mentioned model is examined. The bilinear form is applied for investigating multiple-soliton solutions. Moreover, we employ the positive quadratic function method to create a class of lump solutions using distinct parameters values. The current study serves as a guide to explain many nonlinear phenomena that arise in numerous scientific domains, such as fluid mechanics; physics of plasmas, oceans, and seas; and so on. Full article
(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)
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