Soliton Solutions Research Articles

Schrödinger-Hirota equation in birefringent fibers with cubic-quantic nonlinearity and multiplicative white noise in the ito sense: Nucci’s reductions and soliton solutions

This paper explores innovative solutions for the Stochastic Schrödinger-Hirota equation within the context of birefringent fibers with cubic-quintic nonlinearity, emphasizing incorporating multiplicative white noise in the Itô sense. Leveraging the Nucci reduction method, the study focuses on obtaining exact solutions, shedding light on the intricate interplay between quantum mechanics and stochastic processes. The Nucci reduction method is a powerful tool to facilitate the derivation of precise solutions, showcasing its efficacy in unravelling complex mathematical structures and providing valuable insights into the behaviour of quantum systems under the influence of diverse parameters. In addition, two effective and convenient procedures are employed to extract bright, dark, and unique soliton solutions, as well as their combination. Exploring these solutions contributes to a deeper understanding of the equation’s dynamics, particularly in real-world applications such as quantum optics and condensed matter physics. Additionally, this study incorporates graphical depictions of specific solutions to demonstrate the effect of white noise on solitons visually.

Pioneering the plethora of soliton for the (3+1)-dimensional fractional heisenberg ferromagnetic spin chain equation

Abstract In this scholarly article, we investigate the complex structured (3+1)-dimensional Fractional Heisenberg Ferromagnetic Spin Chain equation (FHFSCE) with conformable fractional derivatives. We develop a diverse glut of soliton solutions using an improved version of the G ′ G -expansion method, namely the r + G ′ G -expansion method. The constraints for the existence of these solutions are painstakingly explained. Our findings are clearly communicated through a number of 3D and 2D graphical representations displaying periodic, multiple periodic, kink, and shock soliton solutions. These soliton solutions are expressed in several mathematical function forms, such as hyperbolic, trigonometric, and rational functions. Our findings support the suggested method’s efficacy as a powerful symbolic algorithm for discovering innovative soliton solutions within nonlinear evolution systems.

Resonant optical soliton solutions for time-fractional nonlinear Schrödinger equation in optical fibers

In this paper, several new optical soliton solutions of the time-fractional generalized resonant nonlinear Schrödinger equation, relevant to pulse propagation in optical fibers, are constructed using the extended simplest equation approach. The new exact solutions are expressed in terms of hyperbolic functions, trigonometric functions, and rational functions. Further, two-dimensional, three-dimensional, and contour graphs of king-type, wave, dark–bright, bright, and bell-shaped optical soliton solutions are plotted to elucidate the significance of the time-fractional generalized resonant nonlinear Schrödinger equation using appropriate values of physical parameters. The dynamic behavior of the present solutions, incorporating the effect of the conformable fractional derivative, is depicted through two-dimensional graphs. The derived optical solutions are considered novel and have presumably not been previously reported in the literature.

Dynamics of resonant soliton, novel hybrid interaction, complex N-soliton and the abundant wave solutions to the (2+1)-dimensional Boussinesq equation

The presented work concerns with some novel solutions of the (2+1)-dimensional Boussinesq equation (BE), which acts as an important model for shallow water wave. Some resonant soliton solutions such as the X-shape soliton (XSS) and Y-shape soliton (YSS) solutions are developed via imposing the resonant conditions on the N-soliton solutions (N-SSs) that developed by the Hirota bilinear approach(HBA). Based on the XSS and YSS solutions, the novel hybrid interactions including the interaction between the 1-soliton and the YSS, the interaction between two YSS solutions are extracted. In addition, the complex N-SSs are also explored and discussed. Finally, the travelling wave solutions including the bright solitary and kinky solitary wave solutions are studied by employing the Bernoulli sub-equation function method (BSEFM). The graphs of the attained solutions are drawn to show the physical properties. The findings of this study are expected to help us apprehend the dynamics of the (2+1)-dimensional BE better.

Stability and instability nature of solitons in an optical fiber with four wave mixing effect

The investigation into modulational instability (MI) within the Kundu-Eckhaus (KE) equation, governing optical solitons, involves a thorough examination of the effects of self-phase modulation, cross-phase modulation, and intermodal dispersion. Special attention is given to understanding the influence of the four-wave mixing effect. The KE equation, which models birefringent fiber and includes terms related to intermodal dispersion, cross-phase modulation, and self-phase modulation, serves as the fundamental framework for this analytical study. Employing conventional linear stability analysis, the gain within the KE equation is determined. To shed light on the role of four-wave mixing in various scenarios, the gain spectrum is utilized as a tool to analyze the behavior of the KE equation under different conditions. This methodology seeks to provide insightful information about the intricate interactions that impact the modulational instability of solitonic pulses in an optical systems. After that, we have investigated the soliton solution by implementing the Jacobian elliptical function approach. Finally, our focus here is on linear stability analysis, which employs eigenvalue spectra to study solitons’ stability via direct numerical simulation.

Dynamical behavior of analytical soliton solutions to the Kuralay equations via symbolic computation

Dynamical behavior of analytical soliton solutions to the Kuralay equations via symbolic computation

Bifurcation analysis and solitary wave solution of fractional longitudinal wave equation in magneto-electro-elastic (MEE) circular rod

In this study, we investigate the longitudinal wave equation (LWE) with the M−fractional derivative, which describes the propagation of longitudinal waves along a rod while incorporating interactions between mechanical, electrical, and magnetic fields within the material. Initially, we apply bifurcation analysis to examine the critical points or phase portraits where the system transitions to new behaviors, such as stability shifts or the emergence of chaos, and observe the mechanism of static soliton formation through a saddle-node bifurcation. Subsequently, we utilize a modified simple equation (MSE) technique to find solitary wave solutions. Depending on the relationships between free parameters, the solutions are expressed as hyperbolic, trigonometric, and exponential functions. The numerical form of the obtained solutions reveals complex phenomena, including dark and bright bell waves, kink periodic lump waves, kink periodic waves, periodic lump waves, interaction waves between kink and periodic waves, linked lump waves, and interactions of periodic and lump waves. Additionally, we compare our results with previously published work, demonstrating that the discussed methods are valuable tools for providing distinct, accurate soliton solutions relevant to nonlinear science and technology applications.

Novel highly dispersive soliton solutions in couplers for optical metamaterials: leveraging generalized Kudryashov’s Law of refractive index with eighth-order dispersion and multiplicative white noise

This article represents a significant advancement in the understanding of highly dispersive optical solitons within the context of optical metamaterials, leveraging a generalized form of Kudryashov’s law of refractive index. By integrating eighth-order dispersion and multiplicative white noise into the analysis, crucial elements in the development and optimization of sophisticated optical metamaterials are accounted for in this current paper for the first time. Through an improved direct algebraic method, a diverse range of soliton solutions are derived, encompassing bright, dark, singular, and straddled solitons. Moreover, the study goes beyond mere derivation by presenting exact solutions expressed using Jacobi and Weierstrass’s elliptic functions. This mathematical framework offers deeper insights into the dynamics of solitons within the investigated context. These findings substantially expand the theoretical underpinnings governing optical solitons in metamaterials, with direct implications for the design, and implementation of next-generation optical devices. The bridging of theoretical advancements with practical applications underscores the significance of this work. By elucidating precise control over soliton properties, it lays the groundwork for innovative solutions in optical communications and beyond. Also, this research serves as a crucial stepping stone towards realizing the full potential of optical metamaterials in shaping the future of optical technologies.

Diverse soliton solutions to the nonlinear partial differential equations related to electrical transmission line

This paper introduces novel traveling wave solutions for the (1+1)-dimensional nonlinear telegraph equation (NLTE) and the (2+1)-dimensional nonlinear electrical transmission line equation (NETLE). These equations are pivotal in the transmission and propagation of electrical signals, with applications in telegraph lines, digital image processing, telecommunications, and network engineering. We applied the improved tanh technique combined with the Riccati equation to derive new solutions, showcasing various solitary wave patterns through 3D surface and 2D contour plots. These results provide more comprehensive solutions than previous studies and offer practical applications in communication systems utilizing solitons for data transmission. The proposed method demonstrates an efficient calculation process, aiding researchers in analyzing nonlinear partial differential equations in applied mathematics, physics, and engineering

Characterizing stochastic solitons behavior in (3+1)-dimensional Schrödinger equation with Cubic–Quintic nonlinearity using improved modified extended tanh-function scheme

An extended version of (3+1)-dimensional non-linear Schrödinger equation that has a cubic–quintic nonlinear component under the stochastic effects is examined in this investigation. Several stochastic exact solutions of this model is acquired through the application of the improved modified extended tanh-function scheme (IMETFS). This method offers a practical and effective approach to finding precise solutions to several kinds of nonlinear partial differential equations. In addition, these solutions include stochastic soliton solutions (bright, singular, combo dark-singular), and exact solution such as singular periodic, Jacobi elliptic function, Weierstrass elliptic doubly periodic solution, rational, and exponential functions. Since it is the first study of its sort to examine multiplicative white noise’s impacts in this particular setting, it offers fresh insights and innovative research approaches for the field’s future studies. The work adds much to our understanding of soliton theory and how it relates to optical fiber technology while illuminating hitherto unknown facets of multiplicative white noise. To illustrate the impact of the noise, a few recovered solutions with varying noise strengths are given graphically as examples.

Soliton, breather, lump, interaction solutions and chaotic behavior for the (2+1)-dimensional KPSKR equation

Nonlinear evolution equations are widely applied in various fields, and understanding their solutions is crucial for predicting and controlling the behavior of complex systems. In this paper, the (2+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani (KPSKR) equation is investigated, which has rich physical significance in nonlinear waves. Making use of Hirota’s bilinear form, soliton, breather and lump solutions of the (2+1)-dimensional KPSKR equation are derived. The interactions between lump solutions and exponential function, as well as between lump solutions and hyperbolic cosine function, are explored. Furthermore, the chaotic behavior of 1-soliton, 2-soliton, lump and interaction solutions are studied via applying the Duffing chaotic system. The physical structure and characteristics of begotten results are illustrated through 3D plots and corresponding two-dimensional profiles. These results indicate that the strategies utilized are more direct and effective, enriching the study of dynamics in high-dimensional nonlinear differential equations.

Exploring chaotic behavior of optical solitons in complex structured Conformable Perturbed Radhakrishnan-Kundu-Lakshmanan Model

In this research, we aim to construct and examine optical soliton solutions for the complex structured Conformable Perturbed Radhakrishnan-Kundu-Lakshmanan Model (CPRKLM) using the Generalized-Kudryashov-Auxiliry Jacobian Method (GKAJM). The current study is notable for its thorough examination and for shedding insight on the chaotic behavior of families of localized optical soliton. Through the creation of 3D and contour visualizations that effectively capture the chaotic behaviors shown by these solitons, we are able to demonstrate that the optical solitons exhibit two distinct forms of perturbations: axial and periodic. Our research stimulates improvements in data processing tools and optical equipment, with consequences for communication networks and nonlinear fiber optics. Through a deeper understanding of optical solitons and their applications, this work also makes a substantial contribution to the discipline of nonlinear optics.

Lie group analysis, solitary wave solutions and conservation laws of Schamel Burger’s equation

This paper presents a Lie group analysis of the Schamel Burger’s equation, notable for producing shock-type traveling waves in distinctive physical contexts. We determine the infinitesimal generators for this equation using the Lie group theory of differential equations. By applying Lie point symmetries, we establish commutation relations, the adjoint representation, and identify the optimal system of sub-algebras. Using elements from this optimal system, we perform symmetry reductions, resulting in various nonlinear ordinary differential equations (ODEs). Some of these reductions yield exact explicit solutions, while others necessitate the use of the new auxiliary equation method to obtain optical soliton solutions. We illustrate the dynamics of these soliton solutions graphically through both two and three-dimensional representations of wave structures. Additionally, we compute the conservation laws for the Schamel Burger’s equation by applying Ibragimov’s theorem, deriving conserved quantities corresponding to its point Lie symmetries. This analysis underscores our novel contribution, offering insights not previously explored in the literature.

Investigation of the complex dynamical structure of bifurcation and dark soliton solutions to fractional generalized double [formula omitted]-Gordon equation

In the current research work, the classical wave equation is combined with a nonlinear sinh source term in the sinh-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

Construction of periodic wave soliton solutions for the nonlinear Zakharov–Kuznetsov modified equal width dynamical equation

Construction of periodic wave soliton solutions for the nonlinear Zakharov–Kuznetsov modified equal width dynamical equation

Exact solutions, wave dynamics and conservation laws of a generalized geophysical Korteweg de Vries equation in ocean physics using Lie symmetry analysis

An evolution equation is a partial diferential equation that describes the time evolution of a physical system starting from given initial data. Evolution equations arise from many areas of applied and engineering sciences. To this end, this article investigates the analytical studies of a generalized geophysical Korteweg-de Vries equation in ocean physics. The examination of this model is conducted via the Lie group theory of diferential equations. In the first place, point symmetries, which are constituent elements of a four-dimensional Lie algebra, are systematically computed. Thereafter, one-parameter transformation groups for the algebra are calculated. Besides, going forward, a one-dimensional optimal system of subalgebras is derived in a procedural manner. Sequel to this, the subalgebras and combination of the achieved symmetries are invoked in the reduction proces which enables the derivation of nonlinear ordinary diferential equations associated with the generalized geophysical Korteweg-de Vries equation under study. Most of the achieved nonlinear ordinary diferential equations are further solved either via direct integration or using a power series approach. Furthermore, travelling wave solutions are initially obtained. This is attained via direct integration and the use of Jacobi elliptic function approach. These techniques enable the attainment of various exact soliton solutions, including non-topological soliton solutions as well as general periodic function solutions of note, such as cosine amplitude, sine amplitude, and delta amplitude solutions of the model. Furthermore, numerical simulations of the solutions are invoked to gain a gross knowledge of the physical phenomena represented by the under-study generalized geophysical Korteweg-de Vries equation in ocean physics. In the end, the investigation further gives attention to the calculation of conserved vectors for the model using Ibragimov's theorem for conservation laws, as well as Noether's theorem.

Construction of bilinear Bäcklund transformation and complexitons for a newer form of Boussinesq equation describing shallow water waves

This research investigates the characteristics and attributes of a new (3+ 1)-dimensional Boussinesq equation that describes shallow water waves in higher dimensions. By utilizing the Hirota bilinear representation, a bilinear Bäcklund transformation is provided for the proposed model to get soliton solutions. Then, the extended transform rational function method is applied to calculate the complexitons type solutions. The results demonstrate various exact solutions with different structures, including periodic, singular, and bright solitons. Comprehensive graphical representations in 2D, 3D, and density plots are provided to highlight the physical properties of these solutions. Our approach is distinguished by the unique nature of the problem and the use of previously untested methods in this context, leading to many new and original optical soliton solutions. These results highlight the effectiveness of the proposed method in tackling nonlinear problems in engineering and the natural sciences, exceeding previous work found in the literature.

Systematic soliton shape modulation by engineering superposed plane wave and soliton parameters.

We investigate the linear interference of a plane wave with different localized waves using the coupled Fokas-Lenells equation (FLE) with four-wave mixing term. We obtain the localized wave solution of the coupled FLE by linear superposition of two distinctly independent wave solutions, namely, the plane wave and one soliton solution and the plane wave and two soliton solution. We obtain several nonlinear profiles depending on the relative phase induced by soliton parameters. We present a systematic analysis of the linear interference profile under four different conditions on the spatial and temporal phase coefficients of interfering waves. We further investigate the interaction of two soliton solution and a plane wave. In this case, we find that, asymptotically, two soliton profiles may be similar or different from each other depending on the choices of soliton parameters in the two cases. The present analysis may also be applied to study the linear interference pattern of other localized waves. We believe that the results obtained by us shall be useful in soliton control, all-optical switching, and optical computing.

Resonant Y-type soliton, interaction wave and other wave solutions to the (3+1)-dimensional shallow water wave equation

The current study aims to develop some novel wave solutions of the (3+1)-dimensional shallow water wave equation (SWWE). First, the Y-type soliton (YTS) solutions are developed via imposing the certain resonant condition to the N-solitons solutions (NSSs) that extracted by the Hirota bilinear approach. Then, the symbolic computation and ansatz function scheme are employed to explore the interaction wave solutions. Eventually, the sub-equation approach is utilized to find the diverse wave solutions which include the kink wave, singular wave and the singular periodic wave solutions. The dynamics of the obtained solutions are presented graphically for revealing the nonlinear physical characteristics. The investigative solutions in this article have not been probed elsewhere and can help us comprehend the nonlinear behaviors of the (3+1)-dimensional SWWE better.

Multi-lump, resonant Y-shape soliton, complex multi kink solitons and the solitary wave solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for incompressible fluid

The major contribution in this paper is to inquire into some new exact solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE) which plays a major role in area of the incompressible fluid. Taking advantage of the Cole-Hopf transform, we extract its bilinear form. Then two different kinds of the multi-lump solutions are probed by applying the new homoclinic approach. Secondly, the Y-shape soliton solutions are explored via assigning the resonance conditions to the N-soliton solutions. Additionally, the complex multi kink soliton solutions (CMKSSs) are investigated through the Hirota bilinear method. Lastly, some other wave solutions including the kink and anti-kink solitary wave solutions are developed with the aid of two efficacious approaches, namely the variational method and Kudryashov method. In the meantime, the profiles of the accomplished solutions are displayed graphically via Maple.