KdV Equation Research Articles

Some Exact Solutions for the Fifth-Order Kdv Equation with Constant Coefficients by the Lie Symmetries

Some Exact Solutions for the Fifth-Order Kdv Equation with Constant Coefficients by the Lie Symmetries

Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with $ p $-power

Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with $ p $-power

Study of Fixed Points and Chaos in Wave Propagation for the Generalized Damped Forced Korteweg-de Vries Equation using Bifurcation Analysis

In this article, we consider the Generalized Damped Forced Korteweg-de Vries (GDFKdV) equation. The forcing term considered is of the form $F(U)=U(U-v_1)(U-v_2)$, where $v_1$ and $v_2$ are free parameters. We investigate the behaviour of fixed points evaluated for the corresponding dynamical system of our model problem. With respect to these fixed points, we investigate the effects of a few significant parameters involved in the model, namely, the free parameters $v_1$ and $v_2$, the nonlinear, dispersion and damping coefficients using the tools from bifurcation analysis. We also obtain the wave plots for the critical values of the nonlinear and dispersion coefficients for which the system becomes unstable and exhibit chaotic behaviour. We confirm the chaos in our dynamical system under various conditions with the help of Lyapunov exponents.

Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials

Abstract In this article, the (2+1)-dimensional KdV equation by Hirota’s bilinear scheme is studied. Besides, the binary bell polynomials and then the bilinear form is created. In addition, an interaction lump with k k -soliton solutions of the addressed system with known coefficients is presented. With the assistance of the stated methodology, a cloaked form of an analytical solution is discovered in expressions of lump-soliton rational functions with a few lovable parameters. Solutions to this study’s problems are identified specifically as belonging to the lump-one, two, three, and four soliton solutions. By defining the specific advantages of the epitomized parameters by the depiction of figures and by interpreting the physical occurrences are established acceptable soliton arrangements and dealt with the physical importance of the obtained arrangements. Finally, under certain conditions, the physical behavior of solutions is analyzed by using the mentioned method. Moreover, the graphs with high resolutions including three-dimensional plots, density plots, and two-dimensional plots to determine a deep understanding of plotted solutions that will arise in the applied mathematics and nonlinear physics are employed.

Bright and dark breathers of the Benjamin–Ono equation on the traveling periodic background

The Benjamin–Ono (BO) equation describes long internal waves of small amplitude in deep fluids. Compared to its counterpart for shallow fluids, the Korteweg–de Vries (KdV) equation, the BO equation admits exact solutions for the traveling periodic and solitary waves as well as their interactions expressed in elementary (trigonometric and polynomial) functions. Motivated by a recent progress for the KdV equation, we discover here two scenarios of the soliton-periodic wave interactions which result in the propagation of either elevation (bright) or depression (dark) breathers (periodic in time coherent structures). The existence of two different breathers is related to the band-gap spectrum of the Lax operator associated with the traveling periodic wave. Given a simple structure of the exact solutions in the BO equation, we obtain a closed-form expression for multi-solitons interacting with the traveling periodic wave.

Double-Periodic Soliton Solutions of the (2+1)-Dimensional Ito Equation

In this work, a (2 + 1)-dimensional Ito equation is investigated, which represents the generalization of the bilinear KdV equation. Abundant double-periodic soliton solutions to the (2 + 1)-dimensional Ito equation are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. The dynamic properties are described through some 3D graphics and contour graphics.

Generalized KdV and Burger Equations and Various Vanishing Limits

In this analytical and numerical study, we look at the generalized KdV equation by letting different coefficients go to zero. Numerical study is carried out by the pseudospectral method. Our study shows that there is no difference between the behavior of the solutions in the limiting case and the solutions in the case when the corresponding coefficients are exactly zero.

Vehicle-to-vehicle cooperative driving model considering end-to-end delay of communication network

To explore the mechanism of the end-to-end transmission delay of the communication network on the collaborative driving process for traffic flow in the vehicle-to-vehicle communication environment, based on the idea of the car-following model, this paper introduces characteristic parameters characterizing the end-to-end transmission delay of the network into Newell's following model and proposes a CD and OV model by considering the time delay characteristics of the collaborative driving process from information transmission to control decision and then to physical execution. To determine the cooperative driving system's stability criterion, the stability analysis of the new model is examined. By using the reductive perturbation approach, the spatiotemporal evolution mechanism of the traffic flow around the critical stability point under the influence of various transmission delays is analyzed. The resulting modified Korteweg-de Vries (mKdV) equations and density wave solutions are derived. The results show that the end-to-end transmission delay of the network has a significant shock effect on the stability of the vehicle-vehicle cooperative driving system, and the stability of the traffic flow and the ability to suppress traffic congestion becomes worse with the increase in the end-to-end transmission delay.

Optical solutions to the truncated M-fractional Schrödinger–KdV equation via an analytical method

Optical solutions to the truncated M-fractional Schrödinger–KdV equation via an analytical method

Overtaking interactions in a strongly coupled inhomogeneous dusty plasma: the role of polarization force

The nonlinear propagation of dust-acoustic waves (DAWs) is considered in a collisionless, inhomogeneous, weakly and strongly coupled dusty plasma composed of negatively charged dust, electrons, and ions. The reductive perturbation method is used to derive a Korteweg–de Vries equation (KdV). It is found that the KdV solitons are affected by the plasma parameters, whereas only rarefactive DAWs are obtained. Hirota’s bilinear approach is used to investigate the overtaking collision of two and three-soliton solutions. Physical parameters such as polarization, and the ions and electrons density gradient scale lengths have a significant impact and cause alterations in the behaviour of the solitons. Notably, as the polarization and ion density gradient scale length increase, the amplitude and width of the solitons decrease. Furthermore, the system parameters also affect the phase shifts of the solitons. The findings presented here have potential applications in the study of acoustic waves in compact astrophysical systems, where quantum polarization and inhomogeneity effects cannot be ignored, as well as in laboratory plasmas.

Some finite difference methods for solving linear fractional KdV equation

The time-fractional Korteweg de Vries equation can be viewed as a generalization of the classical KdV equation. The KdV equations can be applied in modeling tsunami propagation, coastal wave dynamics, and oceanic wave interactions. In this study, we construct two standard finite difference methods using finite difference methods with conformable and Caputo approximations to solve a time-fractional Korteweg-de Vries (KdV) equation. These two methods are named as FDMCA and FDMCO. FDMCA utilizes Caputo's derivative and a finite-forward difference approach for discretization, while FDMCO employs conformable discretization. To study the stability, we use the Von Neumann Stability Analysis for some fractional parameter values. We perform error analysis using L1 & L∞ norms and relative errors, and we present results through graphical representations and tables. Our obtained results demonstrate strong agreement between numerical and exact solutions when the fractional operator is close to 1.0 for both methods. Generally, this study enhances our comprehension of the capabilities and constraints of FDMCO and FDMCA when used to solve such types of partial differential equations laying some ground for further research.

Soliton molecules in Fermi–Pasta–Ulam–Tsingou lattice: Gardner equation approach

We revisit the Fermi–Pasta–Ulam–Tsingou lattice (FPUT) with quadratic and cubic nonlinear interactions in the continuous limit by deducing the Gardner equation. Through the Hirota bilinear method, the fundamental as well as multi-soliton, particularly two-, three- and four-soliton, solutions are obtained for the Gardner equation. Based on these multi-soliton solutions, we show the excitation of an interesting class of table-top soliton molecules in the FPUT lattice through the velocity resonance mechanism. Depending on the condition on the free parameters, which determine the bond length between the soliton atoms, we classify them as dissociated and synthetic type molecules. The main feature of the table-top soliton molecules is that they do not exhibit oscillations in the coalescence region. This property ensures that they are distinct from the soliton molecules, having retrieval force, of the nonlinear Schrödinger family of systems. Further, to study the stability of the soliton molecule we allow it to interact with a single (or multi) soliton(s). The asymptotic analysis shows that their structures remain constant, though the bond length varies, throughout the collision process. Then, from the relative separation distance calculation we also observe that the bond length of each of the soliton molecules is not maintained during soliton molecule–molecule interaction while their structures are preserved. In addition, we consider the FPUT lattice with quadratic nonlinear interaction and FPUT lattice with cubic nonlinearity as sub-cases and point out the nature of the soliton molecules for these cases also systematically. We achieve this based on the interconnections between the solutions of the Gardner, modified Korteweg–de Vries and Korteweg–de Vries equations. Finally, in order to validate the existence of all the soliton structures, especially reported for the Gardner equation in the continuous limit, we simulate the corresponding FPUT chain and verified these various structures numerically and analytically as well. We believe that the present study can be extended to other integrable and non-integrable systems with applications in fluid dynamics, Bose–Einstein condensates, nonlinear optics, and plasma physics.

Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval

Abstract We have established the existence and uniqueness of the local solution for (0.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 , \left\{\begin{array}{ll}{\partial }_{t}u+{\partial }_{x}^{5}u-u{\partial }_{x}u=0,& 0\lt x\lt 1,\hspace{1.0em}t\gt 0,\\ u\left(x,0)=\varphi \left(x),& 0\lt x\lt 1,\\ u\left(0,t)={h}_{1}\left(t),u\left(1,t)={h}_{2}\left(t),\hspace{0.33em}{\partial }_{x}u\left(1,t)={h}_{3}\left(t),& \\ {\partial }_{x}u\left(0,t)={h}_{4}\left(t),\hspace{0.33em}{\partial }_{x}^{2}u\left(1,t)={h}_{5}\left(t),& t\gt 0,\end{array}\right. in the study of Zhao and Zhang [Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval, J. Math. Anal. Appl. 470 (2019), 251–278]. A question arises naturally: Can the local solution be extended to a global one? This article will address this question. First, through a series of logical deductions, a global a priori estimate is established, and then the local solution is naturally extended to a global solution.

Zadacha Koshi dlya nagruzhennogo uravneniya Kortevega–de Friza v klasse periodicheskikh funktsiy

The inverse spectral problem method is applied to finding a solution of the Cauchyproblem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions.A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded termsand a derivation of an analog of Dubrovin’s system of differential equations are proposed. Itis shown that the sum of a uniformly convergent function series constructed by solving theDubrovin system of equations and the first trace formula actually satisfies the loaded nonlinearKorteweg–de Vries equation. In addition, we prove that if the initial function is a real π-periodicanalytic function, then the solution of the Cauchy problem is a real analytic function in thevariable x as well, and also that if the number π/n, n ∈ N, n ≥ 2, is the period of the initialfunction, then the number π/n is the period for solving the Cauchy problem with respect to thevariable x.

On the persistence of spatial analyticity for generalized KdV equation with higher order dispersion

AbstractPersistence of spatial analyticity is studied for solutions of the generalized Korteweg‐de Vries (KdV) equation with higher order dispersion where , are integers. For a class of analytic initial data with a fixed radius of analyticity , we show that the uniform radius of spatial analyticity of solutions at time cannot decay faster than as . In particular, this improves a recent result due to Petronilho and Silva [Math. Nachr. 292 (2019), no. 9, 2032–2047] for the modified Kawahara equation (, ), where they obtained a decay rate of order . Our proof relies on an approximate conservation law in a modified Gevrey spaces, local smoothing, and maximal function estimates.

Analysis and simulation of vehicle following behavior with consideration of multiple time delays

With the application of intelligent transportation system, drivers can receive more traffic status information in the process of driving, but they may have multiple time delays when they are presented with different information. As a result, to further research the influence of multiple time delays on vehicle following behavior, this study comes up with a modified car following model, which takes into account multiple time delays in sensing both distance and velocity information, based on the backward looking and velocity difference model. The linear stability theory is used to explore the influence of multiple time delays on the stability of the extended model. Different nonlinear equations are derived using nonlinear theory, which include Burgers, Korteweg–de Vries, and Modified Korteweg–de Vries equations and are used to depict triangular wave, soliton wave and kink–antikink wave corresponding to stable, metastable and unstable regions respectively. In the end, numerical simulation is used to further study the improved model to depict the dynamic evolution of traffic stream. The results display that the improvement of time delay in sensing distance can induce traffic congestion, conversely, the time delay in sensing velocity offsets the negative effect of time delay in sensing distance. The theoretical results and numerical simulation results are consistent with each other.

Various types of discrete exact localized wave solutions and dynamical analysis for the discrete complex modified Korteweg-de Vries equation

Under consideration is the second-order integrable discretization of complex modified Korteweg-de Vries (mKdV) equation which is regarded as the discrete counterpart of the mKdV equation having an essential role in describing the propagation behavior of water waves and acoustic waves in nonlinear media. First of all, based on the known linear spectral problem, the discrete generalized (n,N−n)-fold Darboux transformation is constructed to derive various types of discrete exact localized wave solutions, including soliton and semi-rational soliton solutions on vanishing background, breather, rogue wave and hybrid interaction solutions on plane wave background, and rational soliton solution on constant background, and the relevant evolution structures are studied graphically. Secondly, the asymptotic analysis is used to discuss the elastic interaction for two-soliton solutions and limit states for rational soliton solutions. Finally, the numerical simulation is utilized to investigate the dynamical behavior and propagation stability of some exact solutions. The findings presented in this paper may contribute to explaining physical phenomena described by the mKdV equation.

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

Dust acoustic inertial Alfvénic nonlinear structures in an electron depleted dusty plasma

In this investigation, the propagation properties of dust inertial Alfvén wave (DIAW) solitons, rogons, and breather structures have been studied in a magnetised electron-depleted two fluids plasma consisting of negatively charged dust and positively charged ions. The Korteweg–de Vries (KdV) equation has been derived by using the reductive perturbation method (RPM) to study the properties of solitons. Further, the nonlinear Schrödinger equation (NLSE) has also been derived by employing a suitable transformation in the KdV equation. The evolution of various kinds of nonlinear structures such as rogons, Akhmediev breathers (AB), and Kuznetsov–Ma breathers (KM) has been elucidated from the different rational solutions of the nonlinear Schrödinger equation. The impact of various physical parameters, such as the dust plasma beta and the angle of propagation of different species, on the salient properties of DIAW solitons, rogons, and breather structures has been studied numerically.

An anatomization of pulse solitons of nerve impulse model via phase portraits, chaos and sensitivity analysis

Across the neuronal membrane, the Korteweg–de Vries equation regulates the spatial and temporal development of the density difference among extracellular and intracellular liquids. In this study, model of the nerve impulse is taken into consideration by means of bifurcation and chaotic theory. The considered model is subjected to an extended direct algebraic method, that revealed abundant analytical results containing hyperbolic, trigonometric and rational functions. Solitons are obtained in terms of pulse stimulation from density difference equation. These results are made accessible together with their fundamental requirements, which ensure the continuance of pulse solitons such as dark, singular and kink that utilize suitable physical parameters in 2D plots. Subsequently, the nonlinear density difference equation is transformed into a planar dynamical system via a Galilean transformation. Bifurcation theory is applied to the dynamical structure of the considered setup to investigate the possible phase portraits of the travelling wave results. Furthermore, we applied the numerical approaches to determine the periodic and quasiperiodic behaviours of the considered equation and the findings are exhibited graphically. Sensitivity and multistability analysis are additionally used to study the periodic and quasiperiodic behaviour for several initial constraints. At the end, the concluding remarks are presented.