Antikink Soliton Research Articles

Interaction of lump, periodic, bright and kink soliton solutions of the (1+1)-dimensional Boussinesq equation using Hirota-bilinear approach

In this paper, we explore the characteristics of lump and interaction solutions for a (1+1) dimensional Boussinesq equation. By employing the Hirota bilinear method, we derive and analyze the exact solutions of this equation. Specifically, we achieve the lump with bright-bright soliton solution, 1-lump,2-lumps and 3-lumps with single bright soliton solution, lump with periodic, kink, and anti-kink soliton solutions. Alongside deriving these solutions, we also illustrate their dynamic properties through graphical simulations. The Boussinesq equation holds significant importance due to its applications in various domains, such as water wave modeling, coastal engineering, and the numerical simulation of water wave dynamics in harbors and shallow seas. Our research shows that the employed method is straightforward, easy to understand, and highly efficient, providing valuable insights into the equation’s nature and its practical applications.

A comprehensive study on geometric shape optical soliton solutions to the time-factional nonlinear Schrödinger-Hirota equation

In this study, we investigate the analytical soliton solutions of a fundamental model, namely the nonlinear Schrödinger-Hirota equation, in the context of beta time-fractional derivative. We adopt the (ω′/ω, 1/ω)-expansion method, which is a reliable and straightforward approach to extract fresh and general soliton solutions in terms of hyperbolic, trigonometric, and rational functions. The solitons include anti-kink, anti-bell-shaped, bell-shaped, and periodic solitons. These solitons have significant applications in various scientific fields, such as optical fiber communications, signal processing, plasma physics, and trans-oceanic data transfer. This study demonstrates the significance of fractional-order differentiation in revealing new solitons. We also provide a comprehensive comparison with existing literature in normal and anomalous dispersion regions, highlighting the uniqueness of the solutions. Moreover, the graphical representations are used to illustrate the properties and potential applications of these solitons. This research might contribute to the advancement of nonlinear optical research and technology.

Dynamical investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative

Abstract This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering.

Investigating pseudo parabolic dynamics through phase portraits, sensitivity, chaos and soliton behavior

This research examines pseudoparabolic nonlinear Oskolkov-Benjamin-Bona-Mahony-Burgers (OBBMB) equation, widely applicable in fields like optical fiber, soil consolidation, thermodynamics, nonlinear networks, wave propagation, and fluid flow in rock discontinuities. Wave transformation and the generalized Kudryashov method is utilized to derive ordinary differential equations (ODE) and obtain analytical solutions, including bright, anti-kink, dark, and kink solitons. The system of ODE, has been then examined by means of bifurcation analysis at the equilibrium points taking parameter variation into account. Furthermore, in order to get insight into the influence of some external force perturbation theory has been employed. For this purpose, a variety of chaos detecting techniques, for instance poincaré diagram, time series profile, 3D phase portraits, multistability investigation, lyapounov exponents and bifurcation diagram are implemented to identify the quasi periodic and chaotic motions of the perturbed dynamical model. These techniques enabled to analyze how perturbed dynamical system behaves chaotically and departs from regular patterns. Moreover, it is observed that the underlying model is quite sensitivity, as it changing dramatically even with slight changes to the initial condition. The findings are intriguing, novel and theoretically useful in mathematical and physical models. These provide a valuable mechanism to scientists and researchers to investigate how these perturbations influence the system’s behavior and the extent to which it deviates from the unperturbed case.

Effect of the nonlinear dispersive coefficient on time-dependent variable coefficient soliton solutions of the Kolmogorov-Petrovsky-Piskunov model arising in biological and chemical science

In this article, we study the soliton solutions with a time-dependent variable coefficient to the Kolmogorov-Petrovsky-Piskunov (KPP) model. At first, this model was used as the genetics model for the spread of an advantageous gene through a population, but it has also been used as a number of physics, biological, and chemical models. The enhanced modified simple equation technique applies to get the time-dependent variable coefficient soliton solutions from the KPP model. The obtained solutions provide diverse, exact solutions for the different functions of the time-dependent variable coefficient. For the special value of the constants, we get the kink, anti-kink shape, the interaction of kink, anti-kink, and singularities, the interaction of instanton and kink shape, instanton shape, kink, and bell interaction, anti-kink and bell interaction, kink and singular solitons, anti-kink and singular solitons, the interaction of kink and singular, and the interaction of anti-kink and singular solutions to diverse nature wave functions as time-dependent variable coefficients. The presented phenomena are clarified in three-dimension, contour, and two-dimension plots. The obtained wave patterns are powerfully exaggerated by the variable coefficient wave transformation and connected variable parameters. The effect of second-order and third-order nonlinear dispersive coefficients is also explored in 2D plots.

Abundant soliton solutions on fractional Kraenkel Manna Merle model (FKMM) via new extended of generalized exponential rational function approach (GERFA)

In this paper, we use the generalized exponential rational function approach (GERFA) for constructing new solitary wave solutions for the fractional Kraenkel Manna Merle (FKMM) model, which is saturated ferromagnetic materials (MF) with a field outside that has very little conductivity, reflects the nonlinear of an ultrashort pulse. The proportional behavior of the suggested model is examined using the beta-derivative (BD). Through the use of this computational method, multiple types of solitons, such as kink, dark, anti-kink, periodic, bright, kink dark, kink bright, anti-kink dark, and anti-kink bright solitons, were obtained for (FKMM). Some of the revealed solutions’ 3D graphs are also employed in the numerical simulations. This investigation demonstrates the efficacy and simplicity of the offered strategy and the simple way to create many new solutions for different kinds of nonlinear partial differential equations, which have significant applications in the engineering and applied sciences. The findings show that the model theoretically has extraordinarily rich soliton structures.

Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

Dynamical analysis of exact optical soliton structures of the complex nonlinear Kuralay-II equation through computational simulation

In this paper, we successfully extracted the various types of soliton solutions for the complex nonlinear Kuralay-II equation through the improved F-expansion method with symbolic computational software Mathematica. The extracted soliton solutions for the Kuralay-II equation are interesting, novel and more general such as anti-kink wave solitons, dark solitons, kink wave solitons, bright solitons, periodic wave solitons, mixed solitons in bright-dark soliton shape, peakon solitons, and solitary wave structures. The graphical structure of some extracted solutions is visualized in 2D, 3D and contour plottings with imaginary, real, and absolute values of the functions by using the numerical simulation. The proposed research will contribute to advancing our knowledge about the complex nonlinear Kuralay-II equation and demonstrating the applicability to the proposed approach to investigate other higher-order complex nonlinear equations. The successful investigation demonstrated that the proposed method is effective, simple, more powerful, efficient and can be utilized on a variety of other nonlinear equations. The explored solitary waves and optical solitons will play an important role in the investigation of nonlinear phenomena in various domains of science and engineering.

Physical constructions of kink, anti-kink optical solitons and other solitary wave solutions for the generalized nonlinear Schrödinger equation with cubic–quintic nonlinearity

Physical constructions of kink, anti-kink optical solitons and other solitary wave solutions for the generalized nonlinear Schrödinger equation with cubic–quintic nonlinearity

Weakly restoring forces and shallow water waves with dynamical analysis of periodic singular solitons structures to the nonlinear Kadomtsev–Petviashvili-modified equal width equation

In this study, the nonlinear two-dimensional Kadomtsev–Petviashvili-modified equal width (KP-MEW) equation is under investigation, which is described as a nonlinear wave model in weakly restoring forces and shallow water waves in the way of ferromagnetic, solitary waves in two dimensions with short amplitude and long wavelength and are also helpful in the investigation of various behaviors in nonlinear sciences. We successfully found the interesting and novel solitary wave results in bright solitons, kink solitons, dark solitons, anti-kink solitons, periodic singular solitons for nonlinear two-dimensional KP-MEW equation on the base of extension of simple equation technique under symbolic computational software Mathematica. The created solitons play an important role in nonlinear engineering and physics such as nonlinear optics, nonlinear dynamics, soliton wave theory, optical fiber and communication system. We are sure that these constructed results are novel and interesting which does not exist in the previous literature works. The graphical representation of constructed results is shown by 2D, 3D and contour graphs. The investigated research work proved that utilized technique is more concise, straightaway and efficient for study of other nonlinear partial differential equations.

Nonlinear behavior of dust acoustic periodic soliton structures of nonlinear damped modified Korteweg–de Vries equation in dusty plasma

In this article, the nonlinear plasma model known the damped modified Korteweg–de Vries equation under consideration on the base of analytical technique to know the novel behavior of this nonlinear model. We formulated the nonlinear plasma model by utilizing the RP (reductive perturbation) technique with basic set of fluid system which consisted on cold negative charge dust fluid, immobile background neutral charges, positive charge ion fluid, fast (ionized) electrons and thermally electrons. We also constructed various kinds of novel soliton solutions including kink wave solitons, periodic solitary waves, anti-kink wave solitons for the damped modified KdV equation. The determined soliton solutions are novel, interested and did not determined in the previous research. The physical interpretation of demonstrated solutions represent in contour, two-dim and three-dim with computational software. The study prove that our proposed approach is powerful, efficient and can also be fruitful for study of other nonlinear models.

Exploration of unexpected optical mixed, singular, periodic and other soliton structure to the complex nonlinear Kuralay-IIA equation

In this research work, we explored novel soliton solutions to the complex nonlinear Kuralay-II (K-IIA) equation on the bases of computational simulation. This complex nonlinear equation is used in many fields such as, optical fibers, ferromagnetic materials and nonlinear optics. We utilized the extended simple equation method on complex nonlinear Kuralay-IIA equation and extracted the unexpected soliton solutions. The extracted soliton solutions having various kinds of physical structure such as bright solitons, periodic wave solitons, kink wave solitons, peakon solitons, dark solitons, anti-kink wave solitons, mixed bright and dark solitons, periodic travelling waves, singular solitons and solitary waves. The graphically analysis of obtained solutions demonstrated with absolute, real and imaginary values of the function visualizing in three dimensional, two dimensional and contour graphics under the numerical simulation by utilizing the Mathematica software. The extracted soliton solutions in this research provide valuable insights into this complex nonlinear equation. The results offer a framework for further analysis, making the solutions effective, easy to apply, and reliable for future complex nonlinear problems development, making them valuable for future. The method used in this work effective, reliable, powerful, easy to apply for other nonlinear partial differential equations.

Noval soliton solution, sensitivity and stability analysis to the fractional gKdV-ZK equation

This work examines the fractional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation (gKdV-ZKe) by utilizing three well-known analytical methods, the modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{G^{'}}{G^2}\\right)$$\\end{document}-expansion method, 1G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{1}{G^{'}}\\right)$$\\end{document}-expansion method and the Kudryashov method. The gKdV-ZK equation is a nonlinear model describing the influence of magnetic field on weak ion-acoustic waves in plasma made up of cool and hot electrons. The kink, singular, anti-kink, periodic, and bright soliton solutions are observed. The effect of the fractional parameter on wave shapes have been analyzed by displaying various graphs for fractional-order values of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}. In addition, we utilize the Hamiltonian property to observe the stability of the attained solution and Galilean transformation for sensitivity analysis. The suggested methods can also be utilized to evaluate the nonlinear models that are being developed in a variety of scientific and technological fields, such as plasma physics. Findings show the effectiveness simplicity, and generalizability of the chosen computational approach, even when applied to complex models.

New investigation of the analytical behaviors for some nonlinear PDEs in mathematical physics and modern engineering

Numerous complex situations arising from science and engineering are modeled by integral-differential equations (IDEs). Examples of these situations include mathematical modeling of infectious diseases, epidemics, circuit analysis, voltage drop equations, computational neuroscience, intergalactic modeling, and many more. To explain the mathematical structure and the behavior of various phenomena in nature, two equations with simple mathematical structure namely: the (1 + 1)-dimensional integro-differential Ito equation (IDIE) and the (2 + 1)-dimensional integro-differential Sawada-Kotera equation (IDSKE) are considered. In this analytical investigation, we have conducted an in-depth enquiry to trace the distinctive closed form progressive wave solutions of these proposed models using the (G′G′+G+A)-expansion method. The graphical forms of the acquired solutions have been displayed in the bell-shaped soliton, singular periodic, and anti-kink shape soliton solutions after the values for the free parameters were specified. The two non-algebraic solutions, named, exponential and trigonometric function solutions, have emerged by applying our proposed method. Based on the general findings of our investigation for different parametric values, the attained wave solutions revealed can keep a vital role for natural balances and can be engaged in modern physical applications. The parameters have a visible effect on the wave amplitude and the swiftness of the traveling wave. The executed outcomes are innovative and have monumental applications in the present research, especially in the fields of theoretical physics, astrophysics, plasma physics, particle physics, theory of relativity, advance quantum mechanics, string theory, and geotechnical engineering.

Dynamical characteristics and physical structure of cusp-like singular solitons in birefringent fibers

In this paper, we explore optical nonlinear waves in a birefringent fiber by investigating the dynamics of a coupled nonlinear Schrödinger system with four-wave mixing factors. Utilizing the Jacobi elliptic function approach, we achieve this goal by building cusp-like singular soliton solutions in this optical system. Depending on the characteristics of the focusing and defocusing optical medium, we investigate four different physical possibilities. We identify numerous cusp-type singular soliton profiles in these settings, namely cusp-like bright, dark, kink, and antikink solitons. Group velocity dispersion, spatio-temporal dispersion, and four-wave mixing effects are further evaluated in terms of their influence on the constructed singular soliton solutions. We show that the intensity of the singular soliton profiles can be suppressed or enhanced by altering the dispersion values. We also witness interesting transitions: when four-wave mixing parameters are tuned, cusp-like kink-type solitons can become cusp-like antikink solitons and vice versa. These novel findings about singular solitons in birefringent fibers have great promise for experimental uses in the realm of optics, as well as for advancing our knowledge of four wave mixing and nonlinear propagation.

A bookshelf layer model for anti-kink and kink pair solitons in the ferroelectric liquid crystal

A bookshelf layer model for anti-kink and kink pair solitons in the ferroelectric liquid crystal

Physical structure and multiple solitary wave solutions for the nonlinear Jaulent–Miodek hierarchy equation

In this paper, under the observation of extended modified rational expansion method based on symbolic computation, the multiple solitary wave solutions for nonlinear two-dimensional Jaulent–Miodek Hierarchy (JMH) equation are constructed. In this investigation, we use the computer software Mathematica for the construction of multiple solitary wave solutions. The interested and important things in this work are the multiple solitary wave solutions which have various kinds of physical structures such as kink soliton, periodic traveling wave, bright soliton, anti-kink soliton, dark soliton, combined bright and dark solitons, topological soliton and peakon soliton. We are sure that the various kinds of soliton solutions are found first time by using one method in the existing literature works. On the basis of this research, we can say that the applied technique is very efficient, reliable, fruitful and powerful. The constructed soliton solutions for nonlinear JMH equation will play an important role in the investigation of different physical phenomena in nonlinear sciences.

Computational approach and dynamical analysis of multiple solitary wave solutions for nonlinear coupled Drinfeld–Sokolov–Wilson equation

In this article, the coupled Drinfeld–Sokolov–Wilson equation under investigation base on computational approach named extension of modified rational expansion approach. By this computational approach secured various kinds of solitons named kink solitons, dark solitons, anti kink solitons, periodic solitons, bright solitons, kink dark solitons, kink bright solitons, anti kink dark solitons, anti kink bright solitons for coupled DSW system. The calculated results are interesting, precise, novel, and different which have not been secured in the previous research. We represent the graphical behaviour of determined solutions by 2-D, 3-D and contour through computational software. The determined results may helpful and play significant role in the study of nonlinear behaviour in various branches of physical sciences such as laser optics, communication system, fluid dynamics, transfer of heat, shallow water waves, acoustics, optics and many others. The constructed results prove that proposed computational approach is concise, powerful, and straightforward for securing the solitons results of other nonlinear partial differential equations.

The Painlevé integrability and abundant analytical solutions for the potential Kadomtsev–Petviashvili (pKP) type coupled system with variable coefficients arising in nonlinear physics

Acknowledging the inhomogeneities of media and non-uniformity of boundaries, the nonlinear evolution equations with variable coefficients may show more realistic situations dealing with time-varying environment, inhomogeneous medium, etc. In this order, this article deals with the variable-coefficients (2+1)-dimensional potential Kadomtsev–Petviashvili (pKP) type coupled equations. The Painlevé analysis approach is being used to assess the integrability of the governing model. The Laurent series, which appears when using the Painlevé analysis technique, has been truncated in order to build an auto-Bäcklund transformation (ABT). Three different analytical solutions in the form of hyperbolic, exponential, periodic and rational have been evaluated using the ABT approach. The multi-soliton solutions are also derived by using Hirota bilinear method for the aforesaid equation. All the solutions are displayed as 3D plots in which the variable coefficients and parameters can be varied to see the effects. In these plots, one can see the many different behaviors of the examined coupled system, such as periodic waves, kink-soliton, anti-kink soliton, kink anti-kink wave, complex periodic waves, complex kink anti-kink wave, and multi-soliton surfaces.

Bilinear representation, bilinear Bäcklund transformation, Lax pair and analytical solutions for the fourth-order potential Ito equation describing water waves via Bell polynomials

This paper focuses on the fourth-order nonlinear potential Ito equation, which describes wave patterns in shallow waters. To reveal the integrable characteristics of the considered equation, such as bilinear BT and Lax pair, the Bell polynomials method is used. By employing this technique, the bilinear representation in the form of classical Hirota operators is derived. Moreover, with the help of bilinear form, the bilinear Bäcklund transformation and the Lax pair of the considered equation are obtained successfully. The three wave method in the form of test function is adopted to generate the analytical solutions of the considered equation. By applying this approach, ten analytical solutions are obtained successfully. For each of the obtained solutions, a 3D graph has been plotted by varying free parametric values. These graphs show the different kinds of wave behaviour, including kink-soliton, anti-kink soliton, periodic wave, dark-soliton, bright-soliton, and some complex kink and periodic-type wave solutions.