Kink Solutions Research Articles

Symmetry and asymptotic solutions for a magnetohydrodynamics Darcy–Forchheimer flow with a p-Laplacian operator

Fluid flows under a p-Laplacian operator formulation have been considered recently in connection with the modeling of non-Newtonian fluid processes. To a certain extent, the main reason behind the interest in p-Laplacian operators is the possibility of determining experimental values for the constant p appearing in them. The goal of the present study is to introduce the analysis of solutions of a one-dimensional porous media flow arising in magnetohydrodynamics with generalized initial data under a Lebesgue integrability condition. We present a weak formulation of this problem, and we consider boundedness and uniqueness properties of solutions and also its asymptotic relation with the standard parabolic p-Laplacian equation. Then, we explore solutions arising from classical symmetries (including an explicit kink solution in the p = 3 case) along with asymptotic stationary and non-stationary solutions. The search for stationary solutions is based on a Hamiltonian approach. Finally, non-stationary solutions are obtained by using an exponential scaling resulting in a Hamilton–Jacobi type of equation.

A Variational Approach to Excited Fermions on Kinks

We study the back-reaction of fermion fields on the kink solution in one space and one time dimension. We employ a variational procedure to determine an upper limit for the minimum of the total energy. This energy has three contributions: the classical kink energy, the energy of valence fermions and the fermion vacuum polarization energy. The latter arises from the interaction of the kink with the Dirac sea and is required for consistency of the semi-classical expansion for the fermions. Earlier studies only considered the valence part and observed a substantial back-reaction. This was reflected by a sizable distortion of the kink profile. We find that this distortion is strongly mitigated when the Dirac sea is properly accounted for. As a result, the back-reaction merely produces a slight squeeze or stretch of the kink profile.

Multi wave, kink, breather, Peregrine-like rational and interaction solutions for the concatenation model

Multi wave, kink, breather, Peregrine-like rational and interaction solutions for the concatenation model

Rational solutions and some interactions phenomena of a (3+1)-dimensional BLMP equation in incompressible fluids: A Hirota bilinear method and dimensionally reduction approach

This manuscript addresses the rational solutions of a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation, emphasizing key aspects in response to specific questions. The study begins by elucidating the problem’s significance, highlighting the pursuit of rational solutions for the mentioned equation. The methodology involves obtaining the Hirota bilinear (HB) form through a transformative process. Furthermore, the analysis extends to the dimensionally reduced BLMP equation, specifically exploring lump and rogue wave-like (Rwl) solutions, along with the interaction between kink and lump solutions. The HB forms play a crucial role in this examination. Importantly, the solutions presented in this study are characterized as non-singular and localized. Visualizations of these solutions are instrumental in understanding the equation’s behavior, particularly how the solutions evolve under varying physical conditions. The outcomes are visually conveyed through 3D and surface graphs, offering a comprehensive representation for specific parameter values. In conclusion, this work contributes novel insights beyond previous efforts in the literature, showcasing the importance and advancements in the study of the (3+1)-dimensional BLMP equation.

Physical multiple shock solutions to the integrability of linear structures of Burgers hierarchy

This work is concerned with the formation of linear structures of components of Burgers' hierarchy. The integrability of some linear systems of two or more parts is investigated. A linear arrangement of five elements is studied by using the Painlevé analysis. The simplified Hirota's method provides a trustworthy cure for the proposed linear design of the five components, where multiple kink solutions are formally derived for this constructed system. Our findings can help many researchers interested in fluid mechanics and plasma physics.

The Lie symmetry analysis, optimal system and exact solutions of the (2+1)-dimensional variable coefficients integrable coupled Burgers equations

The variable coefficients integrable coupled Burgers equations (vcICBs) are researched by the Lie symmetry analysis method. The infinitesimal generators, commutative relations and adjoint relations are given. Optimal system of vcICBs are studied by Olver’s method. Based on the optimal system, the (2+1)-dimensional vcICB equations are reduced to (1+1)-dimensional equations. Some new exact solutions of vcICB equations are derived. Some kink solutions, breathe wave solutions and soliton solutions are got by utilising -expansion method, Riccati equation method, Tanh-expansion method and Painlevé analysis method. By altering different coefficient functions, many different types of exact solutions can be derived. The effect of the coefficient functions on the solution is analysed. By studying the evolution of the solution with time t, the dynamic behaviors of the solutions are analysed.

Coherent structures and sequences of exact kink and anti-kink solutions to the complex cubic–quintic Ginzburg-Landau equation perturbed by intrapulse Raman scattering

Coherent structures and sequences of exact kink and anti-kink solutions to the complex cubic–quintic Ginzburg-Landau equation perturbed by intrapulse Raman scattering

Bifurcation and Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation

This paper focuses on the construction of traveling wave solutions to the nonlinear Klein--Gordon equation by employing the qualitative theory of planar dynamical systems. Based on this theory, we analytically study the existence of periodic, kink (anti-kink), and solitary wave solutions. We then attempt to construct such solutions. For this purpose, we apply a well-known traveling wave solution to convert the nonlinear Klein--Gordon equation into an ordinary differential equation that can be written as a one-dimensional Hamiltonian system. The qualitative theory is applied to investigate and describe phase portraits of the Hamiltonian system. Based on the bifurcation constraints on the system parameters, we integrate the conserved quantities to build new wave solutions that can be classified into periodic, kink (anti-kink), and solitary wave solutions. Some of the obtained solutions are clarified graphically and their connection with the phase orbits is derived.

Exact solutions and bifurcations of the time-fractional coupled Boussinesq-Burgers equation

In this paper, the time-fractional coupled Boussinesq-Burgers equations are studied to construct exact solutions by utilizing the modified extended tanh-function method, the improved method, the two variables -expansion method and bifurcation analysis. Firstly, based on the modified Riemann-Liouville derivative and traveling wave transformation, the equations are rewritten as an ordinary differential equation. Some exact solutions, including singular, periodic, kink, anti-kink, soliton and periodic-singular solutions, are derived. Then, we graphically describe the 3D, density plots and solution profiles of some solutions by setting suitable parameters. Finally, the effectiveness of the methods applied is verified via bifurcation theory.

Kink, periodic and solitary solutions for coupled Benjamin–Bona–Mahony–KdV system

In this paper, exp ( − ϕ ( τ ) ) -expansion method, modified extended tanh-function method and Kudryashov method are used to investigate the exact solutions of the coupled Benjamin–Bona–Mahony–KdV (BBM-KdV) system. By the travelling wave transformation, the coupled BBM-KdV system is reduced to ordinary differential equations. Solving the nonlinear ordinary differential equations, kink, periodic and solitary solutions of the system are obtained. To further understand the natures of the solutions more intuitively, the dynamical behaviours of the solutions are given. The physical structures of the solutions are analysed in combination with 2D and 3D plots. These results enrich the available types of travelling wave solutions for the BBM-KdV model and are also useful for wave studies in complex engineering regions near the coast. It also proves the correctness and validity of our method.

Multiple Soliton Solutions for Coupled Modified Korteweg–de Vries (mkdV) with a Time-Dependent Variable Coefficient

In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences.

Exact traveling wave solutions of the generalized fifth‐order dispersive equation by the improved Fan subequation method

In this paper, the improved Fan subequation method is employed to construct exact traveling wave solutions for a generalized fifth‐order dispersive equation. By making use of bifurcation theory of dynamical systems, we have succeeded to obtain the phase portraits of the subequations involving all possible parameter conditions, and many different types of traveling wave solutions as well, which include more general soliton solutions, kink solutions, triangular function solutions, rational solutions, and Jacobian elliptic function solutions with double periods and so on. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors and could be applicable to explain diversity in qualitative features of wave phenomena.

On the modified versions of G′G-expansion technique for analyzing the fractional coupled Higgs system

In this research, two modified forms of the Ḡ≡G′G-expansion method are employed to investigate various kinds of solitary wave solutions that include kink, lump, periodic, and rogue wave solutions within the framework of the fractional coupled Higgs system. The underlying patterns in the targeted model are revealed by using extended and generalized Ḡ-expansion methods. The first step involves converting the model into nonlinear ordinary differential equations via a fractional complex transformation. Following that, the suggested improved versions of the Ḡ-expansion approach are used to provide numerous solitary wave solutions. Some solitary wave solutions are represented by two- and three-dimensional graphs, demonstrating their typical propagating behavior. This research finishes by summarizing the vast findings and exploring their implications for high-energy physics.

Study on (3+1)-dimensional nonlocal Boussinesq equation: multiple soliton solutions

PurposeThis study aims to explore novel solitary wave solutions of a new (3 + 1)-dimensional nonlocal Boussinesq equation that illustrates nonlinear water dynamics.Design/methodology/approachThe authors use the Painlevé analysis to study its complete integrability in the Painlevé sense.FindingsThe Painlevé analysis demonstrates the compatibility condition for the model integrability with the addition of new extra terms.Research limitations/implicationsThe phase shifts, phase variables and Hirota’s bilinear algorithm are used to furnish multiple soliton solutions.Practical implicationsThe authors also furnish a variety of numerous periodic solutions, kink solutions and singular solutions.Social implicationsThe work formally furnishes algorithms for investigating several physical systems, including plasma physics, optical communications and oceans and seas, among others.Originality/valueThis paper presents an original work using a newly developed Painlevé integrable model, as well as novel and insightful findings.

Analytical study on two new (3 + 1)-dimensional Painlevé integrable equations: Kink, lump, and multiple soliton solutions in fluid mediums

In this work, two new (3 + 1)-dimensional integrable wave equations are investigated. The complete Painlevé integrability of the two suggested equations will be investigated using Mathematica. We employ the method of Hirota to formally derive two sets of multiple soliton solutions for the two suggested models. Additionally, using symbolic computation with Maple, we provide a variety of lump solutions for the two suggested models. Other exact solutions of distinct structures, such as periodic, singular, and many other physical nonlinear structures, will be determined. We should mention here that the proposed two new models will assist many authors that are working in the field of fluids and plasma physics, in understanding the scenarios of the nonlinear waves that arise in different physical systems. Also, this study will contribute to understanding the nature of nonlinear waves that arise in the seas and oceans.

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.

Stability of kink, anti kink and dark soliton solution of nonlocal Kundu–Eckhaus equation

We study the modulational instability and soliton solution for nonlocal Kundu–Eckhaus (KE) equation which is distinctly different from KE equation. KE equation can describe the dynamical behavior of optical pulse envelopes in birefringent fiber. Although, the present equation has novel space, which induce new physical effects and inspire novel physical applications. Therefore, through the Jacobi elliptic function method, the elliptic function solution is derived for the nonlocal KE. Deng et al. (2022) found the N-soliton solution for nonlocal KE equation by the use of Hirota bilinear method. Though, the present problem of the solution is new which is Jacobi elliptic function solution. It is found that the solution exhibits a variety of forms of soliton as dark, kink and antikink soliton due to distinct choices of values of focusing and defocusing parameters. Importantly, with the use of the Jacobi elliptic solution, the role of focusing and defocusing parameters of cubic, quintic nonlinearity and self frequency shift on optical soliton in birefringent fiber for nonlocal KE equation is discussed. The linear stability analysis is used to construct the gain of nonlocal Kundu–Eckhaus equation and using it to analysis the stability/instability of various forms of soliton.

Bilinear auto-Bäcklund transformations, breather and mixed lump–kink solutions for a (3+1)-dimensional integrable fourth-order nonlinear equation in a fluid

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Bäcklund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump–kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids.

New exact solutions of optical metamaterial model with the Kerr law nonlinearity

This paper deals with the investigation of the newly obtained solitary solutions of perturbed optical metamaterial with Kerr law nonlinearity. Two novel analytical techniques, namely, the extended Riccati equation rational expansion (ERERE) method and the extended rational [Formula: see text]-expansion (EREE) methods are applied to develop new solitary solutions. The resultant wave solutions are periodic wave solution, singular kink, anti-peakon, dark, singular anti-kink, anti-kink, gray and singular soliton solutions. The detailed dynamics of the retrieved solutions are the most potent influence of the proposed methods which shows the efficacy and fruitfulness of both the methods.

Soliton solutions for the Zoomeron model applying three analytical techniques.

The Zoomeron equation is used in various categories of soliton with unique characteristics that arise in different physical phenomena, such as fluid dynamics, laser physics, and nonlinear optics. To achieve soliton solutions for the Zoomeron nonlinear structure, we apply the unified, the Kudryashov, and the improved Kudryashov techniques. We find periodic, breather, kink, anti-kink, and dark-bell soliton solutions from the derived optical soliton solutions. Bright, dark, and bright-dark breather waves are also established. Finally, some dynamic properties of the acquired findings are displayed in 3D, density, and 2D views.