Exact Soliton Research Articles

Dynamics of optical solitons and sensitivity analysis in fiber optics

The nonlinear Schrödinger equation (NLSE) and its various forms have significant applications in the field of soliton theory. The Fokas-Lenells (FL) equation stands as a cornerstone in deepening our understanding of nonlinear wave dynamics within optical systems, particularly concerning the behavior of ultrashort pulses across different media. Its significance lies in providing a comprehensive framework to study and analyze complex phenomena, ultimately contributing to advancements in optical technology and applications. The FL equation is an integrable extension of the NLSE that provides a description of the nonlinear propagation of pulses in optical fiber. This paper seeks to discover optical soliton solutions for the FL equation by employing a modified sub-equation method. Additionally, the sensitivity analysis is described by using the various initial conditions. The main novelty of this paper lies in conducting a sensitivity analysis of the FL equation by examining the effects of various initial conditions, providing deeper insights into how these conditions influence the behavior of soliton solutions. For the physical behavior of the models, some solutions are graphically shown in 2D, 3D, and contour graphs by assigning specific values to the parameters under the provided situation at each solution. As a result, we discovered several new families of exact traveling wave solutions, such as bright solitons, dark solitons, and combined bright and dark solitons. This research opens numerous avenues for further exploration in the field of nonlinear wave dynamics and optical soliton theory. The discovery of exact soliton solutions for the FL equation through a modified sub-equation method paves the way for deeper investigations for newcomer researchers. The results of this study will contribute further to the field of mathematical physics, particularly in enhancing the understanding of nonlinear wave propagation and soliton theory in optical and other physical systems.

The Construction of Analytical Exact Soliton Waves of Kuralay Equation

Abstract The primary objective of this work is to examine the Kuralay equation, which is a complex integrable coupled system, in order to investigate the integrable motion of induced curves. The soliton solutions derived from the Kuralay equation are thought to be the supremacy study of numerous significant phenomena and extensive applications across a wide range of domains, including optical fibres, nonlinear optics and ferromagnetic materials. The inverse scattering transform is unable to resolve the Cauchy problem for this equation, so the analytical method is used to produce exact travelling wave solutions. The modified auxiliary equation and Sardar sub-equation approaches are used to find solitary wave solutions. As a result, singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane and combined bright–dark soliton solution can be obtained. The derived solutions are graphically displayed in 2-D and 3-D glances to demonstrate how the fitting values of the system parameters can be used to predict the behavioural responses to pulse propagation. This study also provides a rich platform for further investigation.

PT-symmetric solitons in networks: a metric graph based approach

Abstract We propose a model for networks approving PT-symmetric standing solitons. The model is described in terms of the nonlinear Schrödinger equation on metric graphs, with complex PT-symmetric potential given on each bond. Exact soliton solutions of the problem are obtained. The stability analysis of the PT-symmetric standing solitons on the star graph is studied. Extension for other graph topologies is demonstrated by considering a metric tree graph.

Dynamical analysis and soliton solutions of a variety of quantum nonlinear Zakharov–Kuznetsov models via three analytical techniques

Some new types of truncated M-fractional exact soliton solutions of the two important quantum plasma physics models, extended quantum Zakharov–Kuznetsov and extended quantum nonlinear Zakharov–Kuznetsov, are successfully achieved by applying the expa function technique, the improved (G′/G)-expansion technique, and the Sardar sub-equation technique. These two models have many useful applications when explaining the waves in the quantum electron-positron-ion magnetoplasmas as well as weakly nonlinear ion-acoustic waves in plasma. The obtained results are in the form of dark, bright, periodic, and other soliton solutions. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs. The results are newer than the existing results in the literature due to the use of fractional derivatives. Hence, the solutions will be fruitful in future studies on these models. The solutions obtained are useful in the areas of applied physics, applied mathematics, dynamical systems, and nonlinear waves in plasmas and in dense space plasma. The applied techniques are simple, fruitful, and reliable for solving other models in mathematical physics.

Optical dromions for M-fractional Kuralay equation via complete discrimination system approach along with sensitivity analysis and quasi-periodic behavior

This paper investigates new analytical wave solutions for the fractional Kuralay-IIB equations (FK-IIBE), including a new definition of the fractional derivative. This model works in disciplines such as ferromagnetic materials, optical fibers, wave mixing, and nonlinear optics. The integrable motion of space curves can be determined by our governing model. This research holds great importance in the area of optical fibers, exact and analytical solitons solution. The polynomial method’s complete discrimination system (CDSPM) is used to identify a variety of exact and analytical soliton solutions such as periodic, hyperbolic, rational, Jacobian elliptic (JE), and trigonometric functions. Additionally, we also convert the JE function into a solitary wave (SW) solution. To provide a comprehensive understanding of the dynamic aspects of the system, we also address bifurcation points, quasi-periodic behavior, sensitivity analysis, Poincarè maps, time series profile, and critical solution conditions. The research offers straightforward, efficient algorithms that can consistently and effectively address FK-IIBE. Machine learning technologies are also utilized to satisfy the criteria defined by dynamic analysis. Python regression learner tools are used for reverse engineering to analyze the mathematical model’s equilibrium based on parameter intervals and thresholds determined by dynamical analysis.

Soliton wave profiles and dynamical analysis of fractional Ivancevic option pricing model

This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions.

Dynamic Solitary Wave Solutions Arising in Nonlinear Chains of Atoms Model

This article consists of several travelling wave soliton solutions for solving a complex nonlinear chains of atoms model with long-range interactions via the generalized auxiliary equation method. We have converted considered PDE into nonlinear ODE by utilizing the specific wave transformation. The solutions attained in the form of periodic, bright, singular, W and M types. The adapted method seems a very impressive and competent avenue to obtain the exact soliton solutions of several advanced distinct models. Moreover, to understand the physical nature of the newly attained solutions 2D, 3D and contour-type graphs are presented with distinct dimensions. Additionally, sensitivity and stability through a dynamical system and with the use of different initial conditions are successfully analyzed.

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems.

Phase trajectories, chaotic behavior, and solitary wave solutions for (3+1)-dimensional integrable Kadomtsev–Petviashvili equation in fluid dynamics

This paper focuses on finding exact soliton solutions and further examining the qualitative characteristics of these solutions for a nonlinear partial differential equation known as the (3+1)-dimensional integrable Kadomtsev–Petviashvili equation that portrays a unique dispersion effect. The (3+1)-dimensional integrable Kadomtsev–Petviashvili equation models the evolution of weakly nonlinear dispersive waves in three spatial dimensions “x,y,z” and time “t”. It balances nonlinearity and dispersion, leading to solitary wave solutions that are stable and localized. This equation is crucial in understanding complex wave phenomena in fields like fluid dynamics, plasma physics, and optics. Initially, the traveling wave transformation is employed to convert the nonlinear partial differential equations into the non-linear ordinary differential equations. Then, we use the Riccati equation approach to find solitary wave solutions for the resulting ordinary differential equation. To better comprehend the physical significance of these derived solutions, we illustrate them using various visual representations. Diverse novel collections of solitary wave solutions are obtained such as bright solitons, dark solitons, and dark singular solitons. Second, we convert the nonlinear ordinary differential equation into a linear dynamical system using the planar dynamical transformation. Then, we undertake a qualitative analysis of the dynamical system to study its chaotic characteristics and bifurcation phenomena. Phase portraits of bifurcation are observed at the equilibrium points of the planner dynamical system. We discuss various tools to identify the chaos (random and unpredictable behavior) in dynamical systems. Graphical visualization of qualitative analysis of the system is also depicted in 3-D phase portraits, 2-D phase portraits, Poincaré mapping, and time series. Additionally, the Runge–Kutta technique is used to do a thorough sensitivity analysis of the dynamical system. Using this analytical procedure, it is verified that the stability of the solution has negligible impact while there is little change in the beginning circumstances. This paper aims to provide vital views to scientists and researchers seeking to advance their experimental operations. Furthermore, this research might be expanded by incorporating solitons and situations and investigating wave dynamics. The methods employed provide a direct and uncomplicated approximation of all solutions when compared to the previously established techniques. Various combinations and values of the physical parameters are utilized to explore the optical soliton solutions of the resultant system.

Stochastic analysis and soliton solutions of the Chaffee–Infante equation in nonlinear optical media

The Chaffee–Infante (CI) equation is a nonlinear equation that may be used to predict the complex dynamics of soliton propagation in a nonlinear optical medium. In this study, we elucidate soliton solutions of the CI equation by stochastic differential equations (SDEs) with the Wiener process. In excess of the modified auxiliary equation (MAE) method, we obtain new exact soliton solutions. By combining stochastic differential equations with the Wiener process, we explain the stochastic processes directing the magnitude of solitons within the basis of the CI equation, providing dynamic views on their behavior. The acquired solitary waves are depicted via 3D and 2D graphs to show their dynamics with and without Brownian motion.

Analysis of Lie symmetry, bifurcations with phase portraits, sensitivity and diverse W − M-shape soliton solutions for the (2 + 1)-dimensional evolution equation

This research investigates the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation, a key model in nonlinear wave dynamics, particularly for shallow water waves. By integrating the features of the KP and BBM equations, it offers a comprehensive framework for studying the propagation, stability, and interactions of long waves. Using Lie group analysis for symmetry reductions and bifurcation with phase portraits to explore dynamic behaviour, the study also applies chaos theory to examine system features. Additionally, the new extended direct algebraic method (NEDAM) is employed to derive novel soliton solutions, including dark, bright-dark, W-shape, singular, periodic, and mixed forms. Sensitivity analysis is performed, and 3D, 2D, contour, and density plots visually demonstrate the results. These findings show that NEDAM effectively extracts exact solitons, providing a basis for further studies in nonlinear wave theory and its applications in engineering and physics.

Newly formed solitary wave solutions and other solitons to the (3+1)-dimensional mKdV–ZK equation utilizing a new modified Sardar sub-equation approach

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg–de Vries-Zakharov–Kuznetsov (mKdV–ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron–positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV–ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark–bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV–ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the [Formula: see text]-dimensional mKdV–ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method.

The Discovery of Truncated M-Fractional Exact Solitons and a Qualitative Analysis of the Generalized Bretherton Model

This paper is concerned with the novel exact solitons for the truncated M-fractional (1+1)-dimensional nonlinear generalized Bretherton model with arbitrary constants. This model is used to explain the resonant nonlinear interaction between the waves in different phenomena, including fluid dynamics, plasma physics, ocean waves, and many others. A series of exact solitons, including bright, dark, periodic, singular, singular–bright, singular–dark, and other solitons are obtained by applying the extended sinh-Gordon equation expansion (EShGEE) and the modified (G′/G2)-expansion techniques. A novel definition of fractional derivative provides solutions that are distinct from previous solutions. Mathematica software was used to obtain and verify the solutions. The solutions are shown through 2D, 3D, and density plots. A stability process was conducted to verify that the solutions are exact and accurate. Modulation instability was used to determine the steady-state results for the corresponding equation.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Study of complex dynamics and novel soliton solutions of the Kraenkel-Manna-Merle model describing saturated ferromagnetic materials

This paper introduces a framework for nonlinear short wave propagation in an external field for saturated ferromagnetic materials with zero conductivity: the Kraenkel-Manna-Merle system. New solutions are produced via the G′/(bG′+G+a)-expansion technique. The suggested method is easier to understand, more precise, and simpler to compute. Specifically, a set of singular-periodic and kink-shaped exact soliton solutions is produced. Contour plots, 3D, and 2D visualizations with suitable parametric values are employed to present the computed solutions, which are derived through constraint conditions. For the development of certain novel soliton structures, the arbitrary functions in the solutions are selected. Moreover, bifurcation and chaos theory are applied to the primary dynamical system in order to provide a qualitative analysis of the system under study. Phase portraits of bifurcation are presented at fixed locations to illustrate different situations about parameter values in the dynamic system. However, adopting techniques for identifying chaos in a dynamical system and applying an external force verifies the occurrence of chaotic behavior in system. These results offer new perspectives that can enhance understanding of the dynamics of the KMM system.

Exact wave solutions of truncated M-fractional Boussinesq-Burgers system via an effective method

Abstract In this paper, we present distinct types of exact wave soliton solutions of an important fluid flow dynamic system called the truncated M-fractional (1+1)-dimensional nonlinear Boussinesq-Burgers system (BBS). This model is used to explain ocean waves, matter-wave pulses, waves in ferromagnetic media, the proliferation of waves in shallow water, etc. We transform the nonlinear fractional system into a nonlinear ordinary differential equation by using a fractional transformation to obtain dark, bright, singular, dark-bright, dark-singular, bright-singular and periodic type solitons solutions by employing the modified extended tanh function method (METhFM). The use of fractional derivatives makes the solutions different from the existing solutions. The obtained results are useful in the optical fibers, fluid dynamics, ocean engineering and other related fields. To visualize the system’s behavior, some of the solutions are represented by two- and three-dimensional graphs which are obtained and verified with the help of Mathematica. The achieved results provide a better understanding of the behavior of the nonlinear fractional partial differential equations and the dynamics of BBS, which are not present in the literature and are helpful in future studies of the concerned system.

Uncovering the stochastic dynamics of solitons of the Chaffee–Infante equation

In this paper, we apply stochastic differential equations with the Wiener process to investigate the soliton solutions of the Chaffee–Infante (CI) equation. The CI equation, a fundamental model in mathematical physics, explains concepts such as wave propagation and diffusion processes. Exact soliton solutions are obtained through the application of the modified extended tanh (MET) method. The obtained wave figures in 3D, 2D, and contour are highly localized and determine an individual frequency shift under the behavior of sharp peak, periodic wave, and singular soliton. The MET method shows to be a valuable analytical tool for obtaining soliton solutions, essential for understanding the dynamics of nonlinear wave phenomena. Numerical simulations enable us to explore soliton solutions in two and three dimensions, shedding light on their properties over time. Our results have wide applications in various domains, including stochastic processes and nonlinear dynamics, impacting advancements in physics, engineering, finance, biology, and beyond.

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.

Analysis of Truncated M-Fractional Mathematical and Physical (2+1)-Dimensional Nonlinear Kadomtsev–Petviashvili-Modified Equal-Width Model

This study focuses on the mathematical and physical analysis of a truncated M-fractional (2+1)-dimensional nonlinear Kadomtsev–Petviashvili-modified equal-width model. The distinct types of the exact wave solitons of an important real-world equation called the truncated M-fractional (2+1)-dimensional nonlinear Kadomtsev–Petviashvili-modified equal-width (KP-mEW) model are achieved. This model is used to explain ocean waves, matter-wave pulses, waves in ferromagnetic media, and long-wavelength water waves. The diverse patterns of waves on the oceans are yielded by the Kadomtsev–Petviashvili-modified equal-width (KP-mEW) equation. We obtain kink-, bright-, and periodic-type soliton solutions by using the expa function and modified extended tanh function methods. The solutions are more valuable than the existing results due to the use of a truncated M-fractional derivative. These solutions may be useful in different areas of science and engineering. The methods applied are simple and useful.

Stability Analysis, Modulation Instability, and Beta-Time Fractional Exact Soliton Solutions to the Van der Waals Equation

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.