Parkes Equation Research Articles

Bifurcation, Traveling Wave Solutions and Dynamical Analysis in the (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-Dimensional Extended Vakhnenko–Parkes Equation

This paper is concerned with the bifurcation of the traveling wave solutions, as well as the dynamical behaviors and physical property of the soliton solutions of the (2+1)-dimensional extended Vakhnenko–Parkes (eVP) equation. Firstly, based on the traveling wave transformation, the planar dynamical system corresponding to the (2+1)-dimensional eVP equation is derived, and then the singularity type and trajectory map of this system are obtained and analyzed. Based on the bifurcation of this system, the analytical expression for the periodic wave solution is given and shown graphically. Secondly, the N-soliton solutions are obtained via the bilinear method, and some important physical quantities and asymptotic analysis of one-soliton and two-soliton solutions are discussed. The results obtained in this paper might be useful for understanding the propagation of high-frequency waves.

Implementation of optical soliton behavior of the space–time conformable fractional Vakhnenko–Parkes equation and its modified model

Implementation of optical soliton behavior of the space–time conformable fractional Vakhnenko–Parkes equation and its modified model

Multi solitons, bifurcations, high order breathers and hybrid breather solitons for the extended modified Vakhnenko–Parkes equation

This manuscript delves into a comprehensive investigation of significant exact solutions pertaining to the extended modified Vakhnenko–Parkes equation (emVPE), with a focus on novel solutions. By taking into account various kinds of auxiliary functions, we are able to generate a number of high order solitons, bifurcations, and breathers solutions for the given model. The study of multiple solitons, multiple bifurcation solitons, high order breathers, and hybrid breathers are all part of our current research. Our proposed results are validated through graphical visualizations.

Novel multiple solitons, their bifurcations and high order breathers for the novel extended Vakhnenko–Parkes equation

This paper presents a rigorous exploration of the extended Vakhnenko–Parkes-Equation (eVPE), unveiling a rich tapestry of multi solitons, bifurcations, and high-order breathers. Notably, it achieves an impressive feat by deriving solitons up to sixth order and breathers up to third order, pushing the boundaries of the equation’s solutions. These intricate solutions are not only derived but also vividly brought to life through numerical visualizations, underscoring the profound implications of the our work. Through insightful discussions, the manuscript effectively underscores the significance of these findings, opening new avenues for understanding nonlinear dynamics and their applications across various scientific domains.

An invariance analysis of the Vakhnenko–Parkes Equation

Lie symmetry analysis of differential equations is an effective technique to compute exact solutions of a differential equation, to reduce number of independent variables or to reduce the order and nonlinearity of the equation. The present article focuses on the symmetry analysis of Vakhnenko–Parkes Equation (VP Equation). At first, Lie point symmetries of the VP Equation that permit an optimal system of one-dimensional subalgebras are discussed and then using the commutator table we construct the discrete symmetries of the VP Equation using the modified Hydon’s method. Also, we compute the exact solutions using symmetry transformations.

Abundant explicit and exact solutions for the space–time fractional Vakhnenko–Parkes model in the relaxing medium with stability analysis

In this work, we have considered the beta-fractional derivative form of the Vakhnenko–Parkes equation (VPE) to study its novel analytical solutions. To achieve the required new distinct traveling wave solutions, the new sub-equation method is used for the space–time fractional VPE. The dynamics of these solutions are analyzed under different parametric and fractional conditions by the graphical view. The produced wave patterns include combined bright–dark, w-shape, u-shape, v-shape, bright, propagation of two solitons, combined dark–bright, bell shape, a dark, periodic wave, breather, v-shape with two bright spots and combined anti-peakon–bright solutions. In recognizing the physical significance of the obtained wave solutions at each fraction of an interval, these solutions are highly commendable. In addition, to show the stability of the considered model, the stability analysis of the governing model is discussed by the linear stability analysis method which states the behavior of the model in the corresponding medium. These solutions are useful to study the proliferation of high-frequency waves in the relaxing medium.

Extraction of the gravitational potential and high‐frequency wave perturbation properties of nonlinear (3 + 1)‐dimensional Vakhnenko–Parkes equation via novel approach

The (3 + 1)‐dimensional Vakhnenko–Parkes mathematical model has a wide range of applications in science and engineering. In this paper, the model studied is investigated and analyzed by using two effective schemes such as sine‐Gordon expansion method and its newly developed version, rational SGEM. Moreover, many novel properties of model studied are extracted in detail. Furthermore, their wave distributions properties and graphs are also plotted under the strain conditions. Interactions of the gravitational potential and high‐frequency wave perturbation properties are also reported in a detailed manner.

Novel multiple soliton and front wave solutions for the 3D-Vakhnenko–Parkes equation

The multiple exp-function technique, as an extension to Hirota’s perturbation algorithm, is employed to find N-soliton solutions of the 3D Vakhnenko–Parkes (3D-VP) equation. This is a straightforward and effective alternative method. Consequently, one- two- and three-soliton solutions besides the front wave solutions and their wave numbers are obtained, which are completely distinct from the other published literature. 3D and density plots were investigated via the help of the Maple package program by suitably choosing the free parameters in the solutions. Further, the physical meaning for these solutions is clarified. We mention that the propagation of high-frequency waves in a relaxing medium can be investigated by discussing the dynamical behavior of the obtained solutions for the 3D-VP equation.

Various forms of lumps and interaction solutions to generalized Vakhnenko Parkes equation arising from high-frequency wave propagation in electromagnetic physics

This article retrieves multiple lump, periodic cross kink, rogue wave, multiwave, breathers, M-shaped and interactional solutions for generalized Vakhnenko Parkes (gVP) equation via appropriate transformations. This leads to a transformed equation for which it is straightforward to evaluate the exact solutions. The gVP equation gives a description of high-frequency waves in the relaxation medium possessing periodic and solitary traveling wave solutions some of which are loop-like in nature. These types of solutions have been observed in wave propagations, rogue waves and many other physical systems. We also interpret some of our solutions in distinct dimensions via 3D, 2D, contours and density profiles.

Novel soliton wave solutions of the Vakhnenko–Parkes equation arising in the relaxation medium

This study checks the solitary wave solutions of the nonlinear Vakhnenko–Parkes (VP) equation that was derived by Parkes in 1998 to describe the high-frequency waves’ dynamical and physical behavior in the relaxation medium. The solitary wave solutions of the nonlinear VP model are computationally investigated by implementing the Khater II method. Many soliton solutions are constructed and represented in some distinct graphs. The stability property of these solutions is also investigated. All obtained solutions’ accuracy is checked by putting them back into the original model by employing Mathematica 12. The method's performance shows its effectiveness, power, and ability to handle some nonlinear evolution equations.

New Breather and Multiple-Wave Soliton Dynamics for Generalized Vakhnenko–Parkes Equation With Variable Coefficients

Abstract In investigation is the generalized Vakhnenko–Parkes equation with time-dependent coefficients, which is a new nonlinear model connecting to high-frequency wave propagation in relaxing media with variable perturbations. An extended Hirota bilinear method is proposed to construct soliton, breather, and multiple-wave soliton solutions for the equation. Our research shows that the soliton solutions can degenerate into existing single soliton solutions while the breather and multiple-wave soliton solutions are first obtained. By utilizing the two free functions involved in the solutions, the dynamics of some novel excited breathers and multiple-wave solitons are demonstrated. Our results confirm that the generalized Vakhnenko–Parkes equation possesses rich solution structures and interesting dynamical features, which may be depict various nonlinear wave behaviors of high-frequency waves.

Abundant breather and semi-analytical investigation: On high-frequency waves’ dynamics in the relaxation medium

This paper investigates the high-frequency waves’ dynamical behavior in the relaxation medium through two recent analytical schemes. This study depends on the Vakhnenko–Parkes (VP) equation that has been reduced from the well-known Ostrovsky equation. The modified Khater (MKhat) and the extended simplest equation (ESE) methods are used to handle the considered model. As a result, many novel solitary wave solutions have been obtained to construct the initial and boundary conditions. These conditions allow employing the variational iteration (VI) method to study the semi-analytical solutions of the considered model. The accuracy of solutions is explained along with showing the matching between analytical and semi-analytical solutions and comparing our obtained solutions with the previous results that have been obtained in published research papers. Moreover, the high-frequency waves’ behavior relaxation medium is illustrated through some distinct sketches. The methods’ performance shows their effectiveness, direct, easy, and consequential for studying many nonlinear evolution equations.

Multi-wave, breather and interaction solutions to (3+1) dimensional Vakhnenko–Parkes equation arising at propagation of high-frequency waves in a relaxing medium

In this study, based on the Hirota bilinear form, the exact analytic solutions of the (3 + 1) dimensional Vakhnenko–Parkes equation with various physical properties were constructed with the help of the Maple package program and symbolic computation. These solutions are the type of multi-waves, breather wave, lump–kink, lump–periodic solutions and interaction solutions (between lump and hyperbolic wave solutions). The constructed solutions have expanded and enriched the solution forms of this new model existing in the literature. By means of Maple package program, 3D and 2D graphs were drawn for the special choices of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed. The solutions obtained can be used in the explanation of physical phenomena occurring in the propagation of high-frequency waves in a relaxing medium.

Higher dimensional integrable Vakhnenko–Parkes equation: multiple soliton solutions

PurposeThis study aims to develop a new (3 + 1)-dimensional Painlevé-integrable extended Vakhnenko–Parkes equation. The author formally derives multiple soliton solutions for this developed model.Design/methodology/approachThe study used the simplified Hirota’s method for deriving multiple soliton solutions.FindingsThe study finds that the developed (3 + 1)-dimensional Vakhnenko–Parkes model exhibits complete integrability in analogy with the standard Vakhnenko–Parkes equation.Research limitations/implicationsThis study addresses the integrability features of this model via using the Painlevé analysis. The study also reports multiple soliton solutions for this equation by using the simplified Hirota’s method.Practical implicationsThe work reports extension of the (1 + 1)-dimensional standard equation to a (3 + 1)-dimensional model.Social implicationsThe work presents useful algorithms for constructing new integrable equations and for handling these equations.Originality/valueThe paper presents an original work with newly developed integrable equation and shows useful findings.

Modified Vakhnenko–Parkes equation with power law nonlinearity: Painlevé analysis, analytic solutions and conservation laws

The Vakhnenko equation governs the propagation of high-frequency waves in a relaxation medium. The Painleve analysis of modified Vakhnenko–Parkes equation is performed by using the Kruskal approach. The Lie symmetry formalism is also applied to derive symmetries of generalised modified Vakhnenko–Parkes equation with power law nonlinearity. The exact solutions of this equation are obtained in the form of power series; and other explicit exact solutions in the form of trigonometric, hyperbolic and Jacobi elliptic functions are obtained for modified Vakhnenko–Parkes equation. The bright soliton solutions of generalised modified Vakhnenko–Parkes equation are also obtained. The conservation laws are obtained with the help of multiplier approach. The 3D plots and contour plots are shown by considering the possibilities of arbitrary function occurring in the solutions.

Excited solitons, breathers and periodic waves for a short pulse equation with time-dependent coefficients in relaxing media

In this study, the attention is focused on the excitation of Vakhnenko–Parkes (VP) equation with time-dependent coefficients, which can describe the short pulse wave propagation in relaxing media with variable perturbations. Hirota bilinear method is employed to construct three types of analytical solutions, namely, soliton, breather and periodic wave solutions. By utilizing the two free functions involved in the solutions, abundant excited patterns are revealed for the equation. Several examples are taken to demonstrate the novel effects of excited solitons, breathers and periodic waves.

Application of the modified exponential function method to Vakhnenko-Parkes equation

In this paper, we submit some new travelling wave solutions for the Vakhnenko--Parkes equation via the modified exponential function method. The obtained solutions include hyperbolic, exponential, trigonometric function solutions. Regarding these solutions, the 2D and 3D graphs and contour simulations are presented.

The integrable Vakhnenko–Parkes (VP) and the modified Vakhnenko–Parkes (MVP) equations: Multiple real and complex soliton solutions

In this work, we show that the integrable Vakhnenko–Parkes (VP) equation passes the Painlevé test and admits multiple real and multiple complex soliton solutions. We also present, for the first time, the modified Vakhnenko-Parkes (MVP) equation, show its complete integrability, and formally derive its multiple real and multiple complex soliton solutions. To achieve the goal set for this work, we introduce two complex forms of the simplified Hirota’s method, the first works effectively for the VP equation, and the other form is nicely applicable for the MVP equation. We believe that establishing the complex forms will shed light on complex solitons of other integrable equations.

Application of the ITEM for solving three nonlinear evolution equations arising in fluid mechanics

In this paper, by introducing new approach, the improved $$\tan \left( \phi (\xi )/2\right) $$ -expansion method (ITEM) is further extended into the Vakhnenko–Parkes (VP) equation, the generalized regularized-long-wave (GRLW) equation and the symmetric regularized-long-wave (SRLW) equation in fluid mechanic. We extended the ITEM proposed by Manafian et al. (Int J Appl Comput Math 2:243–268, 2016) to construct new types of soliton wave solutions of nonlinear partial differential equations (NPDEs). The merit of the presented method is finding the further solutions of the considering problems including soliton, periodic, kink and kink-singular wave solutions. Comparing our new results with other results shows that our results give the further solutions. The results of applying this procedure (Figs. 1, 2, 3, 4, 5, 6) to the studied cases show the high efficiency of the new technique. Finally, these solutions might play an important role in engineering, physics and applied mathematics fields.

New analytic solutions of the fractional Vakhnenko–Parkes equation

In the present paper, new analytical solutions for the fractional Vakhnenko–Parkes (VP) equation in the sense of the conformable derivative are obtained using the $$\exp (-\phi (\xi ))$$ expansion method. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Simulation of the obtained solutions are given at the end of the paper.