Kink Solutions Research Articles

Certain analytical solutions of the concatenation model with a multiplicative white noise in optical fibers

In the presence of spatio-temporal dispersion, perturbation terms of the Hamiltonian type as well as multiplicative white noise, analytical investigation of the concatenation model having the Kerr law of nonlinearity is carried out in this work. The Cole–Hopf transformation and direct assumptions with arbitrary functions are utilized to determine several analytic solutions to the governing equation, including multi-wave, two solitary wave, breather, periodic cross kink, Peregrine-like rational, and three-wave solutions. The parameter constraints that serve as the requisite condition for the existence of these wave solutions are also identified. In order to explore and illustrate the propagation and dynamical behaviors of some solutions reported in this research, 3D graphics and their corresponding contour plots are included. Results of this paper may be useful for the experimental realization of certain nonlinear waves in optical fibers and further understanding of their propagation dynamics.

Bifurcations and Exact Solutions of Optical Soliton Models in Fifth-Order Weakly Nonlocal Nonlinear Media

For the optical soliton model in fifth-order weakly nonlocal nonlinear media, to find its exact explicit solutions, the corresponding traveling wave system is formulated as a planar dynamical system with a singular straight line. Then, by using techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze the planar system and find the corresponding phase portraits, the dynamical behavior of the amplitude component can be assessed. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink, and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons are found with precise formulations.

Asymptotic stability of the fourth-order $\phi^{4}$ kink for general perturbations in the energy space

The fourth-order \phi^{4} model extends the classical \phi^{4} model of quantum field theory to the fourth-order case, but sharing the same kink solution. It is also the dispersive counterpart of the well-known parabolic Cahn–Hilliard equation. Mathematically speaking, the kink is characterized by a fourth-order nonnegative linear operator with a simple kernel at the origin but no spectral gap. In this paper, we consider the kink of this theory, and prove orbital and asymptotic stability for any perturbation in the energy space.

Lump–kink and hybrid solutions of the extended (3+1)-dimensional potential KP equation in fluid mechanics

In this paper, the extended (3+1)-dimensional potential KP equation in fluid mechanics is studied through Hirota bilinear method. Many types of hybrid solutions, such as the lump–kink solution, lump-two kink solution and periodic lump solution are obtained by assuming different functions in the bilinear equation. The interaction solution between lump and triangular periodic wave is also derived by combining sine and cosine functions with quadratic functions. Dynamical structures of these exact solutions are depicted by presenting the corresponding three-dimensional, two-dimensional structures and density graphs. These diverse interaction solutions could be helpful for understanding physical phenomena in fluid mechanics.

Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions.

Geometrically constrained multifield models with BNRT solutions

In this paper, we investigate multifield models in which the two-field BNRT model is coupled to a third field through mediator functions in the Lagrangian density. To conduct the investigation, we obtain the equations of motion and develop a first-order formalism based on energy minimization. Two possibilities are considered: (i) the third field acting in the mediator functions to modify the BNRT solutions; (ii) the BNRT fields feeding the mediator function to produce effects in the kink solution of the third field. In the case (i), the results show that the solutions may be related to the standard ones with the coordinate redefined in terms of the mediator functions if they are equal. This allows to induce effects similar to geometric constrictions in the core, or to compactify the tail of the BNRT solutions. If the mediator functions differ one from another, we show that the effects are distinct, with the compactification of just one of the two-field solutions. In the case (ii), the orbit parameter of the model plays an important role, modifying the mediator function that induces changes in the profile of the kink associated to the third field.

Families of propagating soliton solutions for (3+1)-fractional Wazwaz-BenjaminBona-Mahony equation through a novel modification of modified extended direct algebraic method

The article presents a new modification to the modified Extended Direct Algebraic Method (mEDAM) namely r+mEDAM to effectively and precisely acquire propagating soliton and other travelling wave solutions to the Fractional Wazwaz-Benjamin-Bona-Mahony (FWBBM) equation. By using this updated approach, we are able to find more and new families of propagating soliton solutions for the FWBBM problem, such as soliton, kink, lump-like singular, trigonometric, hyperbolic, periodic, shock, singular & non-singular wave solutions. We also provide 3D and 2D graphs that visually illustrate the obtained solutions. By obtaining accurate propagating soliton solutions, our r+mEDAM proves to be practical while also revealing important details about the dynamics of the equation and suggesting possible applications in the fields of optics, materials research, and water waves.

On the Painlevé integrability and nonlinear structures to a (3 + 1)-dimensional Boussinesq-type equation in fluid mediums: Lumps and multiple soliton/shock solutions

This work examines the Painlevé integrability of a (3 + 1)-dimensional Boussinesq-type equation. Using the Mathematica program, we rigorously establish Painlevé's integrability for the suggested problem. By utilizing Hirota's bilinear technique, we obtain the dispersion relations and phase shifts, which enable us to derive multiple soliton solutions. In addition, we systematically derive a wide range of lump solutions using the Maple symbolic computation. The investigation extends to encompass a variety of exact solutions with distinct structural features, including kink, periodic, singular, and rational solutions. This comprehensive analysis illustrates the profound richness of the model's dynamics and its potential to elucidate diverse nonlinear wave phenomena across various physical contexts. Therefore, the results that we will obtain play a vital role in understanding the mechanism of generation and propagation of many mysterious phenomena that arise in various scientific fields, including plasma physics, fluid mechanics, and the propagation of waves on the surfaces of seas and oceans to optical fibers.

On the optical soliton solutions to the fractional complex structured (1+1)-dimensional perturbed gerdjikov-ivanov equation

In this work, we examine the complex structured Fractional Perturbed Gerdjikov-Ivanov equation (FPGIE), which describes the propagation of optical pulses with perturbation effects. This model finds applications in optical fibers, especially in photonic crystal fibers. We are discovered novel and unique optical soliton solutions using the modified Extended Direct Algebraic Method (mEDAM), which has never been used with this model previously. As a result, a hierarchy of traveling wave solutions including singular kink, periodic, solitary kink, and rogue-shaped soliton solutions, etc., are derived. Some obtained solutions are discussed graphically based on numerical values of some parameters related to the solution. The results add new and unique soliton types to the model and demonstrate how they interact and impact the system’s overall dynamics.

Painlevé analysis, Painlevé–Bäcklund, multiple regular and singular kink solutions of dynamical thermopherotic equation drafting wrinkle propagation

The thermophoretic motion (TM) system with a variable heat transmission factor, based on the Korteweg-de Vries (KdV) equation, is used to model soliton-like thermophoresis of creases in graphene sheets. Painlevé test is employed to discover that the equation is Painlevé integrable. Then an auto-Bäcklund transformation using the truncated Painlevé expansion is obtained. Concerning the additional variables, the auto-Bäcklund transformations convert the nonlinear model to a set of linear partial differential equations. Finally, various explicit precise solutions based on the acquired auto-Bäcklund transformations are investigated and the researched solutions are illustrated in 3D, 2D and contour plots. Furthermore, the Cole-Hopf transformation is used in conjunction with Hirota’s bilinear technique to get multiple regular and singular kink solutions.

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.

Abundant interaction solutions of the integrable (1 + 1)‐dimensional coupled KdV system

Based on the Hirota bilinear method and the consistent tanh expansion method, we study different interaction excitations of the integrable ‐dimensional coupled KdV system, which is a typical and essential water wave model in the two‐layer fluid. Abundant new interaction solutions including lumps, lump–kink solution, breathers, two‐kink solution, resonant soliton, and soliton–cnoidal solution are obtained. All these exact solutions are shown both analytically and graphically with proper constant selection.

Collisions between kinks with long-range tails: a simple and efficient method

We construct initial configurations for the scattering between kinks with long-range tails. For this purpose, we exploit kink solutions in the presence of Bogomol’nyi-Prasad-Sommerfield (BPS)-preserving impurities. This approach offers a highly efficient method and effortless implementation with a negligible computational cost. Our algorithm has a much smaller complexity than the usual minimization method, becoming more than a hundred times faster in some scenarios. Consequently, conducting kink-antikink simulations becomes remarkably straightforward.

Novel kink and multi wave soliton solutions to the stochastic Phi-4 equation driven by the Weiner process

Novel kink and multi wave soliton solutions to the stochastic Phi-4 equation driven by the Weiner process

The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3D representation and a 2D representation, as well as a 2D contour plot.

Kink solutions in generalized 2D dilaton gravity

We study static kink solutions in a generalized two-dimensional dilaton gravity model, where the kinetic term of the dilaton is generalized to be an arbitrary function of the canonical one X=−12(∇φ)2, say F(X), and the kink is generated by a canonical scalar matter field ϕ. It is found that for arbitrary F(X), the background field equations have a simple first-order formalism, and the linear perturbation equation can always be written as a Schrödinger-like equation with factorizable Hamiltonian operator. After choosing appropriate F(X) and superpotential, we obtain a sine-Gordon type kink solution with pure AdS2 metric. The linear perturbation issue of this solution becomes an exactly solvable conformal quantum mechanics problem, if one of the model parameter takes a critical value.

Lie symmetry analysis, optimal system and exact solutions for variable-coefficients Boiti-Leon-Manna-Pempinelli equation

In this paper, the constant coefficients are extended to variable coefficients in the Boiti-Leon-Manna-Pempinelli(BLMP) equation, and its optimal system, exact solutions, conservation laws are studied. First, the infinitesimal generators for independent variables of the variable-coefficient Boiti-Leon-Manna-Pempinelli(vcBLMP) equation are solved by using the Lie symmetry analysis method. Subsequently, the optimal system of the vcBLMP equation is solved, and then the equations are subjected to similarity reduction, the partial differential equations(PDEs) are gained. Based on these PDEs, the representative equations are converted into ordinary differential equations(ODEs) by traveling wave transformation, and some new exact solutions of vcBLMP equation is obtained by using G′/G− expansion method and tanh-function method, such as the kink solutions, periodic solutions and soliton solutions. According to the obtained solutions, the corresponding images are plotted in order to facilitate the presentation of structural features of the solutions. At the end, the vcBLMP equation is proved to be nonlinear self-adjoint equation and the conservation laws of vcBLMP are solved.

Lump and kink soliton phenomena of Vakhnenko equation

Understanding natural processes often involves intricate nonlinear dynamics. Nonlinear evolution equations are crucial for examining the behavior and possible solutions of specific nonlinear systems. The Vakhnenko equation is a typical example, considering that this equation demonstrates kink and lump soliton solutions. These solitons are possible waves with several intriguing features and have been characterized in other naturalistic nonlinear systems. The solution of nonlinear equations demands advanced analytical techniques. This work ultimately sought to find and study the kink and lump soliton solutions using the Riccati–Bernoulli sub-ode method for the Vakhnenko equation (VE). The results obtained in this work are lump and kink soliton solutions presented in hyperbolic trigonometric and rational functions. This work reveals the effectiveness and future of our method for solving complex solitary wave problems.

Kink phenomena of the time-space fractional Oskolkov equation

<p>In this study, we applied the Riccati-Bernoulli sub-ODE method and Bäcklund transformation to analyze the time-space fractional Oskolkov equation for kink solutions by matching the coefficients and optimal series parameters. The time-space fractional Oskolkov equation is used to analyze the behavior of solitons for different applications such as fluid dynamics and viscoelastic flow. The kink solutions derived have important consequences for stability analysis and interaction dynamic in these systems, and these are useful in controlling the physical behaviour of systems described by this equation. Such effects are illustrated by 2D and 3D plots, showing that the proposed model can handle both fractional and integer-order solitons with different but equally efficient outcomes. This research contributes to a valuable analytical method that can determine and manage processes in diversified systems based on fractional differential equations. This work provides a basis for subsequent analysis in other branches of science and technology in which the fractional Oskolkov model is used.</p>

EXACT SOLUTIONS AND BIFURCATION OF A MODIFIED GENERALIZED MULTIDIMENSIONAL FRACTIONAL KADOMTSEV–PETVIASHVILI EQUATION

In this paper, we investigate the exact solutions of a modified generalized multidimensional fractional Kadomtsev–Petviashvili (KP) equation by the bifurcation method. First, the equation is converted into a planar dynamical system through fractional complex wave transformation. The phase portraits of the equation and qualitative analysis are presented under different bifurcation conditions. Then, the bounded and unbounded traveling wave solutions, including periodic, kink, anti-kink, dark-solitary, bright-solitary and breaking wave solutions, are acquired by integrating along different orbits. Finally, numerical simulations of the dynamic behaviors of the solutions obtained are graphically illustrated by choosing appropriate parameters.