Kink Waves Research Articles

Exact solutions for the Bogoyavlensky-Konopelchenko equation with variable coefficients with an efficient technique

In this investigation, a different form of extended generalized G′G2-expansion technique with variable coefficient is proposed. The advantage of the extended generalized G′G2–expansion technique is that it can be used to solve nonlinear evolution equations with both constant and variable coefficients, whereas the basic G′G2 method can only be used with constant coefficients. The proposed technique is used to solve the Bogoyavlensky–Konopelchenko (BK) equation with variable coefficients, which depicts the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis in a fluid. Further, the BK equation with variable coefficients is applied for stratified internal waves, shallow-water waves, ion-acoustic waves, and water propagation in a liquid. The hyperbolic, trigonometric, and rational form solutions of the BK equation are dynamically represented as the annihilation of three-dimensional kink waves, multi-soliton waves, single solitons, and so on. Furthermore, the derived solutions of the considered equation, which comprise arbitrary functional parameters and other constant parameters, can be used to enhance the advanced behaviours of physical situations.

On the dynamics of exact solutions to a (3+1)-dimensional YTSF equation emerging in shallow sea waves: Lie symmetry analysis and generalized Kudryashov method

This paper focuses on obtaining a large number of exact solutions to the highly nonlinear (3+1)-dimensional Yu–Toda–Sassa–Fukuyama (YTSF) equation, which is important in fluid dynamics, plasma physics, nonlinear sciences, and weakly dispersive media, among other things. The primary goal of this research is to obtain a variety of exact soliton solutions and different dynamical wave-forms of the YTSF equation using two different techniques: Lie symmetry and generalized Kudryashov (GK) methods. These techniques reduce the considered partial differential equation (PDE) into ordinary differential equations (ODEs). Hereafter, a variety of exact solutions are derived with the help of the associated subalgebras and wave transformation by the Lie symmetry technique and GK technique, respectively. The obtained exact solutions involve various arbitrary constants, parameters, and arbitrary functionals that enhance the dynamic representation of the derived solutions in the research area. We use numerical simulations to generate 3-dimensional graphics and contour plots to determine the dynamical behavior of the obtained exact solutions. The physical interpretation of interaction soliton solutions demonstrates a rich diversity of attained solutions, such as solitons, kink waves, kink wave interaction with solitons, and others.

Cutoff of transverse waves through the solar transition region

Context. Transverse oscillations are ubiquitously observed in the solar corona, both in coronal loops and in open magnetic flux tubes. Numerical simulations suggest that their dissipation could heat coronal loops, thus counterbalancing radiative losses. These models rely on a continuous driver at the footpoint of the loops. However, analytical works predict that transverse waves are subject to a cutoff in the transition region. It is thus unclear whether they can reach the corona and indeed heat coronal loops. Aims. Our aims are to determine how the cutoff of kink waves affects their propagation into the corona and to characterize the variation of the cutoff frequency with altitude. Methods. Using 3D magnetohydrodynamic simulations, we modelled the propagation of kink waves in a magnetic flux tube, embedded in a realistic atmosphere with thermal conduction, which starts in the chromosphere and extends into the corona. We drove kink waves at four different frequencies and determined whether they experienced a cutoff. We then calculated the altitude at which the waves were cut off and compared it to the prediction of several analytical models. Results. We show that kink waves indeed experience a cutoff in the transition region, and we identified the analytical model that gives the best predictions. In addition, we show that waves with periods shorter than approximately 500 s can still reach the corona by tunnelling through the transition region with little to no attenuation of their amplitude. This means that such waves can still propagate from the footpoints of loop and result in heating in the corona.

Similarity reduction and multiple novel travelling and solitary wave solutions for the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients.

Persistence of Kink and Periodic Waves to Singularly Perturbed Two-Component Drinfel’d–Sokolov–Wilson System

This paper concerns the persistence of kink and periodic waves to singularly perturbed two-component Drinfel’d-Sokolov-Wilson system. Geometric singular perturbation theory is first employed to reduce the high-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we then are able to find the sufficient conditions about the wave speed to guarantee the existence of heteroclinic orbit and periodic orbits, which indicates the existence of kink and periodic waves. Furthermore, we also show that the limit wave speed c_0(k) is increasing.

Study of Stochastic–Fractional Drinfel’d–Sokolov–Wilson Equation for M-Shaped Rational, Homoclinic Breather, Periodic and Kink-Cross Rational Solutions

We explore stochastic–fractional Drinfel’d–Sokolov–Wilson (SFDSW) equations for some wave solutions such as the cross-kink rational wave solution, periodic cross-rational wave solution and homoclinic breather wave solution. We also examine some M-shaped solutions such as the M-shaped rational solution, M-shaped rational solution with one and two kink waves. We also derive the M-shaped interaction with rogue and kink waves and the M-shaped interaction with periodic and kink waves. This model is used in mathematical physics, surface physics, plasma physics, population dynamics and applied sciences. Moreover, we also show our results graphically in different dimensions. We obtain these solutions under some constraint conditions.

Bifurcations and Exact Bounded Solutions of Some Traveling Wave Systems Determined by Integrable Nonlinear Oscillators withq-Degree Damping

For a class of nonlinear diffusion–convection–reaction equations, the corresponding traveling wave systems are well-known nonlinear oscillation type of systems. Under some parameter conditions, the first integrals of these nonlinear oscillators can be obtained. In this paper, the bifurcations, exact solutions and dynamical behavior of these nonlinear oscillators are studied by using methods of dynamical systems. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink and anti-kink wave solutions, as well as limit cycles, are obtained. Most important and interestingly, a new global bifurcation phenomenon of limit bifurcation is found: as a key parameter is varied, so that singular points (except the origin) disappear, a planar dynamical system can create a stable limit cycle.

Numerical Investigation of Time-Fractional Phi-Four Equation via Novel Transform

This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article’s results possess complimentary properties that relate to the symmetry of partial differential equations.

Bifurcation of Some Novel Wave Solutions for Modified Nonlinear Schrödinger Equation with Time M-Fractional Derivative

In this paper, we investigate the time M-fractional modified nonlinear Schrödinger equation that describes the propagation of rogue waves in deep water. Periodic, solitary, and kink (or anti-kink) wave solutions are discussed using the bifurcation theory for planar integrable systems. Some new wave solutions are constructed using the first integral for the traveling wave system. The degeneracy of the obtained solutions is investigated by using the transition between orbits. We visually explore some of the solutions using graphical representations for different values of the fractional order.

The Sun at millimeter wavelengths

Aims. We used solar observations of a plage-enhanced network with the Atacama Large Millimeter/sub-millimeter Array (ALMA) in Band 3 and Band 6, together with synthetic continuum maps from numerical simulations with Bifrost in the same bands, to carry out a detailed study of bright small-scale magnetic features. Methods. We made use of an algorithm to automatically identify and trace bright features within the field of view (FoV) of the ALMA observations and the simulation. In particular, the algorithm recovers information of the time evolution of the shape, motion of the centre of gravity, temperature, and size for each feature. These quantities are used to determine the oscillatory properties of each feature utilising wavelets analysis. Results. We found 193 and 293 features in the Bands 3 and 6 observations, respectively. In the degraded simulation, the total number of features were 24 for Band 3 and 204 for Band 6. In the original simulation, the total number of features were 36 for Band 3 and 392 for Band 6. Based on the simulation, we confirm the magnetic nature of the features. We have obtained average oscillation periods of 30–99 s for the temperature, 37–92 s for size, and 37–78 s for horizontal velocity. There are indications for the possible presence of transverse (kink) waves with average amplitude velocities of 2.1–5.0 km s−1. We find a predominant anti-phase behaviour between temperature and size oscillations suggesting that the variations of the bright features are caused by compressible fast-sausage magnetohydrodynamics (MHD) modes. For the first time to our knowledge, we estimated the flux of energy of the fast-sausage waves at the chromospheric heights sampled by ALMA as 453–1838 W m−2 for Band 3 and 3640–5485 W m−2 for Band 6. Conclusions. We have identified MHD waves, both transverse (kink) and compressible sausage modes, in small-scale (magnetic) structures, independently, in both ALMA Band 3 and Band 6 observations, along with their corresponding synthetic images from simulations. The decrease of wave energy-flux with height (from Band 6 to Band 3) could possibly suggest energy dissipation at chromospheric heights, namely, wave heating, with the assumptions that the identified small-scale waves are typical at each band and they propagate upward through the chromosphere.

Novel rational solitons and generalized breathers for (1+1)-dimensional longitudinal wave equation

This paper deals with the evaluation of multiwave, homoclinic breather, M-shaped rational solitons and interaction of M-shaped rational solitons with single and double kinks for [Formula: see text]-dimensional longitudinal wave equation via the logarithmic transformation and symbolic computation with ansatz functions method. We will obtain two types of M-shaped rational solitons and their dynamics are revealed in figures by choosing the distinct values of involved parameters. In addition, two forms of interaction of M-shaped rational soliton with kink wave will also be evaluated. Furthermore, we will compute kink cross-rational solutions, periodic cross-rational solutions, generalized breathers and Akhmediev breathers for governing model. The solitons behaviors via different values of parameters in the produced solutions will be analyzed.

Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M-shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots.

Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, which is different from the previous works, we obtain the phase portraits of the the corresponding traveling wave systems. In addition, we derive the exact parametric representations of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, peakon solutions, periodic peakon solutions and compacton solutions under different parameter conditions.

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

In this article, (3+1)-dimensional generalized Shallow Water-like (gSWl) equation is discussed. The infinitesimal generators of the equation are derived by using the Lie symmetry analysis method. The optimal system is obtained based on the adjoint table of the generators of the equation. Exact solutions of the equation are constructed by applying symmetry reduction, Exp−ϕ(ξ) expansion method, Exp-function expansion method, Riccati equation method, and G′/G expansion method. For analyzing the dynamical behavior of the solutions, we derive the physical structures of dark soliton, kink wave, and periodic solutions via numerical simulations.

Persistence of kink and anti-kink wave solutions for the perturbed double sine-Gordon equation

Based on the geometric singular perturbation theory and the Melnikov method, we study the persistence of kink and anti-kink wave solutions for the perturbed double sine-Gordon equation. The explicit expression of the Melnikov function is given. Moreover, the monotonicity of the period function for unperturbed double sine-Gordon equation is investigated.

Bifurcation and Analytical Solutions of the Space-Fractional Stochastic Schrödinger Equation with White Noise

The qualitative theory for planar dynamical systems is used to study the bifurcation of the wave solutions for the space-fractional nonlinear Schrödinger equation with multiplicative white noise. Employing the first integral, we introduce some new wave solutions, assorted into periodic, solitary, and kink wave solutions. The dependence of the solutions on the initial conditions is investigated. Some solutions are clarified by the display of their 2D and 3D representations with varying levels of noise to show the influence of multiplicative white noise on the solutions.

Some general bright soliton solutions and interactions for a (2＋1)-dimensional nonlocal nonlinear Schrödinger equation

In this paper, a novel application of Darboux transformation is presented for (2＋1)-dimensional nonlocal nonlinear Schrödinger equation. The Darboux transformation is a power method to solve some (1＋1)-dimensional classical nonlinear Schrödinger (NLS) equations, however, there is less work of (2＋1)-dimensional ((2＋1)-D) nonlocal nonlinear Schrödinger(NNLS) equation with Darboux transformation. With the development of science, the NNLS equation gradually appears, where the nonlocality is the reverse spatial field or reverse time field. We focus on how to solve the (2＋1)-D NNLS equation with reverse time field q(x,y,−t). Using the Darboux transformation, some novel (2＋1)-D nonlocal soliton solutions are derived on a background of kink waves, including 1-soliton solution, 2-soliton solution, bright soliton and soliton interaction on kink wave backgrounds. This method is a novel application of Darboux transformation, which can be extend to some other nonlocal nonlinear or higher-dimensional soliton equations on kink wave backgrounds.

Kink oscillations of magnetic flux tubes in presence of resistivity

ABSTRACTKink oscillations in the solar coronal flux tubes are among the best seismological tools for estimating the unknown parameters of the solar corona. Here, we study the effect of resistivity on the linear evolution of magnetohydrodynamic (MHD) kink waves in magnetic flux tubes with transverse inhomogeneity in the plasma density. In thin tube thin boundary approximation, we obtain the well-known resistive spectrum of Alfvén continuum modes with a semi-analytical method and study the relative importance of the eigenmodes in the spectrum. As shown in previous studies, the quasi-mode solution of ideal MHD appears in the spectrum. Results show that while the internal and external homogeneous regions of the flux tube oscillate dominantly in quasi-mode in the inhomogeneous region in each magnetic surface a different combination of resistive eigenmodes contributes to the perturbations. Comparing the results obtained here with those obtained previously in the same configuration for a viscous plasma shows that resistivity is less efficient than viscosity in transforming the energy of the wave to heat.

Traveling Wave Solutions for Time-Fractional mKdV-ZK Equation of Weakly Nonlinear Ion-Acoustic Waves in Magnetized Electron–Positron Plasma

In this paper, we discuss the time-fractional mKdV-ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on the properties of certain employed truncated M-fractional derivatives, we reduce the time-fractional mKdV-ZK equation to an integer-order ordinary differential equation utilizing an adequate traveling wave transformation. Further, we derive a dynamical system to present bifurcation of the equation equilibria and show existence of solitary and kink singular wave solutions for the time-fractional mKdV-ZK equation. Furthermore, we establish symmetric solitary, kink, and singular wave solutions for the governing model by using the ansatz method. Moreover, we depict desired results at different physical parameter values to provide physical interpolations for the aforementioned equation. Finally, we introduce applications of the governing model in detail.

Exact Solution of ( 4 + 1 )-Dimensional Boiti–Leon–Manna–Pempinelli Equation

Based on the Hirota bilinear method, using the heuristic function method and mathematical symbolic computation system, various exact solutions of the ( 4 + 1 )-dimensional Boiti–Leon–Manna–Pempinelli equation including the block kink wave solution, block soliton solution, periodic block solution, and new composite solution are obtained. Upon selection of the appropriate parameters, three-dimensional and contour diagrams of the exact solution were generated to illustrate their properties.