Vries Equation Research Articles

Interactions of localized wave and dynamics analysis in the new generalized stochastic fractional potential-KdV equation.

In this paper, we investigate the new generalized stochastic fractional potential-Korteweg-de Vries equation, which describes nonlinear optical solitons and photon propagation in circuits and multicomponent plasmas. Inspired by Kolmogorov-Arnold network and our earlier work, we enhance the improved bilinear neural network method by using a large number of activation functions instead of neurons. This method incorporates the concept of simulating more complicated activation functions with fewer parameters, with more diverse activation functions to generate more complex and rare analytical solutions. On this basis, constraints are introduced into the method, reducing a significant amount of computational workload. We also construct neural network architectures, such as "2-3-1," "2-2-3-1," "2-3-3-1," and "2-3-2-1" using this method. Maple software is employed to obtain many exact analytical solutions by selecting appropriate parameters, such as the superposition of double-period lump solutions, lump-rogue wave solutions, and three interaction solutions. The results show that these solutions exhibit more complex waveforms than those obtained by conventional methods, which is of great significance for the electrical systems and multicomponent fluids to which the equation is applied. This novel method shows significant advantages when applied to fractional-order equations and is expected to be increasingly widely used in the study of nonlinear partial differential equations.

Inverse scattering transform for continuous and discrete space‐time‐shifted integrable equations

AbstractNonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space‐time‐shifted nonlinear Schrödinger (NLS) and space‐time‐shifted modified Korteweg‐de Vries equations are studied. Additionally, the semidiscrete IST is developed for the time, space, and space‐time‐shifted variants of the Ablowitz–Ladik integrable discretization of the NLS. One‐soliton solutions are constructed for all continuous and discrete cases.

Exploring multi-soliton patterns, bifurcation analysis, and chaos in (2+1) dimensions: A study on nonlinear dynamics

This study explores the intrinsic characteristics of the (2+1)-dimensional Schwarz-Korteweg-de Vries equation used to describe shallow water waves. The multiple solitons are successfully constructed using a technique involving multiple exponential functions. Graphical representations of the results are provided in 3D, 2D, and density plots to assess the compatibility of the solutions. Also, the dynamic nature of studied equation is examined based on bifurcation and chaos theory for nonlinear systems. Bifurcation signifies how our dynamical system is affected by physical parameters in planar dynamical system. After that, we apply the external force on planar dynamical system to show the chaotic like behavior of the studied model. Such behavior is confirmed by utilizing different chaos detecting tools. The obtained results serve to explain the effectiveness and applicability of the utilized methodologies in comprehend the exact solution and qualitative behavior of nonlinear physical models.

A new analytic approach and its application to new generalized Korteweg‐de Vries and modified Korteweg‐de Vries equations

The current study is important from two perspectives. Firstly, in this article, we suggest a novel analytical technique for creating the exact solutions to nonlinear partial differential equations (NLPDEs). In order to study the dynamical behaviors of various wave phenomena, we can construct the several exact solutions in the form of Jacobi elliptic solutions, hyperbolic solutions, trigonometric solutions, and exponential solutions by using this method. Secondly, we consider a more generalized form of the (2+1)‐dimensional Korteweg–de Vries (KdV) and modified Korteweg‐de Vries (mKdV) equations that plays an important role in describing the shallow water. We successfully extract several soliton solutions for the examined equation. By choosing the appropriate parameters, some graphs of the discovered solutions have been represented in the figures. Every obtained solution has been demonstrated to satisfy the corresponding equation. The findings demonstrate that the method can be applied to solve a number of nonlinear evolution equations. The new solutions and the paper's findings could improve our understanding of how the waves move through shallow water and open up new research avenues.

Solutions and continuum limits to nonlocal discrete modified Korteweg‐de Vries equations

In this paper, we take advantage of the bilinearization reduction method to consider the local and nonlocal reduction of a discrete Ablowitz‐Kaup‐Newell‐Segur equation. Exact solutions in double Casoratian form to the reduced nonlocal discrete modified Korteweg‐de Vries equations are constructed. The dynamics of soliton solutions are analyzed and illustrated by asymptotic analysis. Moreover, both semi‐continuous limit and full continuous limit are applied to obtain the local and nonlocal semi‐discrete modified Korteweg‐de Vries equations, as well as the local and nonlocal continuous modified Korteweg‐de Vries equations.

Multidomain spectral approach to rational‐order fractional derivatives

AbstractWe propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multidomain approach; after transformations in accordance with the underlying curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw–Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg‐de Vries equations and compare these to results obtained with a discrete Fourier transform.

Hermite‐Shannon‐Cosine spectral method for fractional partial differential equations

AbstractShannon‐Cosine wavelet function possesses almost all excellent characteristics such as interpolation, compact support, and smoothness. As an interpolation wavelet function, it could be applied in fractional partial differential equations effectively. However, when solving engineering problems in a finite interval, the treatment of the boundary is still not smooth enough. So, the Hermite‐Shannon‐Cosine interval wavelet is constructed using the Hermite interpolation function to achieve smoother transitions at the boundary of the interval, thereby reducing boundary effects. Based on this, a method for solving Fractional PDEs is proposed, the method's performance and its processing capability at the interval boundary are verified by taking the Fractional Fokker‐Planck equation and the Time‐Fractional Korteweg‐de Vries equation as examples. Compared with the multi‐scale Faber‐Schauder wavelet collocation method, Point‐Symmetric interval wavelet spectral method, Dynamic interval wavelet spectral method, and so forth, the experimental results show that the method performs better in terms of numerical accuracy and effectiveness.

Traveling wave solutions of a coupled Schrödinger-Korteweg-de Vries equation by the generalized coupled trial equation method

The coupled Schrödinger-Korteweg-de Vries equation is a critical system of in nonlinear evolution equations. It describes various processes in dusty plasma, such as Langmuir waves, dust-acoustic waves, and electromagnetic waves. This paper uses the generalized coupled trial equation method to solve the equation. By the complete discrimination system for polynomial, a series of exact traveling wave solutions are obtained, including discontinuous periodic solutions, solitary wave solutions, and Jacobian elliptical function solutions. In addition, to determine the existence of the solutions and understand their properties, we draw three-dimensional images of the modules of the solutions with Mathematica. We obtain more comprehensive and accurate solutions than previous studies, and the results give the system more profound physical significance.

Effect of the ion drag force on head‐on collisions between two nonlinear dust acoustic waves in a strongly coupled dusty plasma

AbstractThe effect of dust grain pressure on the head‐on collision of two dust‐acoustic solitary waves in non‐magnetized, collisionless, and strongly coupled dust plasma has been investigated [J. Plasma Phys. (2020), 86, 905,860,111]. In this paper, the model is augmented by incorporating the ion drag force on dust grains. By way of the extended perturbation method of Poincaré‐Lighthill‐Kuo, Korteweg‐de Vries equations are derived. The wave characteristics along with the phase shifts after the head‐on collision are analytically established and numerically assessed. It revealed a non‐negligible decrease in the wave amplitude and the phase shift.

Large‐time asymptotics to the focusing nonlocal modified Kortweg‐de Vries equation with step‐like boundary conditions

AbstractWe investigate the large‐time asymptotics of solution for the Cauchy problem of the nonlinear focusing nonlocal modified Kortweg‐de Vries (MKdV) equation with step‐like initial data, that is, as , as , where A is an arbitrary positive real number. We first develop the direct scattering theory to establish the basic Riemann‐Hilbert (RH) problem associated with step‐like initial data. Thanks to the symmetries , of nonlocal MKdV equation, we investigate the asymptotics as and , respectively. Our main technique is to use the steepest descent analysis to deform the original matrix‐valued RH problem to corresponded regular RH problem, which could be explicitly solved. Finally, we obtain the different large‐time asymptotic behaviors of the solution of the Cauchy problem for focusing nonlocal MKdV equation in different space‐time sectors , , , and on the whole ‐plane.

A new numerical approach to Gardner Kawahara equation in magneto‐acoustic waves in plasma physics

AbstractThe basic idea of this article is to investigate the numerical solutions of Gardner Kawahara equation, a particular case of extended Korteweg‐de Vries equation, by means of finite element method. For this purpose, a collocation finite element method based on trigonometric quintic B‐spline basis functions is presented. The standard finite difference method is used to discretize time derivative and Crank–Nicolson approach is used to obtain more accurate numerical results. Then, von Neumann stability analysis is performed for the numerical scheme obtained using collocation finite element method. Several numerical examples are presented and discussed to exhibit the feasibility and capability of the finite element method and trigonometric B‐spline basis functions. More specifically, the error norms and are reported for numerous time and space discretization values in tables. Graphical representations of the solutions describing motion of wave are presented.

Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations

In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear partial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s≥0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s≥sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s<sc, the formal solutions are generically sc-Gevrey. In the latter case, we give an explicit example in which the solution is s'-Gevrey for no s'<sc. As a practical illustration, we apply our results to the generalized Burgers-Korteweg-de Vries equation.

On the singular limit of a boundary delayed Kuramoto-Sivashinsky-Korteweg-de Vries equation: Well-posedness and stability results

On the singular limit of a boundary delayed Kuramoto-Sivashinsky-Korteweg-de Vries equation: Well-posedness and stability results

Boundary behavior of the solution to the linear Korteweg‐De Vries equation on the half line

AbstractIn this paper, we consider the solution to the linear Korteweg‐De Vries (KdV) equation, both homogeneous and forced, on the quadrant via the unified transform method of Fokas and we provide a complete rigorous study of the integrals of the formula provided by the method, especially focusing on the explicit verification of the considered initial‐boundary‐value problems (IBVPs), with generic data, as well as on the uniform convergence of all its derivatives, as approaches the boundary of the quadrant, and their rapid decay as .

Linear and Nonlinear Spherically Symmetric Perturbation in Astrophysical Compact Objects

A degenerate quantum plasma system (DQPS) consisting of arbitrary number of degenerate light (electron, proton, neutron, etc.) and heavy (light nuclei/element, heavy nuclei/element, etc.) particle species is considered to investigate the spherically symmetric perturbation (both linear and nonlinear) of self-gravitational field. The fundamental characteristics (amplitude, width, etc.) of small amplitude self-gravitational potential structures, which are found to be formed in the considered plasma system, are examined by the reductive perturbation technique. The derivation and numerical analysis of the modified Korteweg-de Vries (mKdV) equation are obtained. The analytical results are applied in astrophysical compact objects like white dwarfs and neutron stars, which are the examples of DQPS. Journal of Engineering Science 13(1), 2022, 97-104

Ion-acoustic multisolitons in pair-ion plasmas containing degenerate electrons: overtaking collisions at critical compositions

Overtaking collisions of ion-acoustic (IA) solitons are examined in an unmagnetized pair-ion plasma containing degenerate electrons. A Korteweg-de Vries (KdV) equation governing nonlinear IA solitons in such a degenerate plasma model is obtained. At critical compositions, the nonlinear coefficient of the KdV equation vanishes. Therefore, a modified KdV (mKdV) equation describing nonlinear IA solitons in the degenerate plasma model has been derived by extending the analytic analysis to this critical case. Hirota’s bilinear method has been applied to obtain the multisolitons solution and their corresponding phase shifts in these two situations. For an in-depth analysis, we investigated the dynamics of two solitons of the same polarity and opposite polarities resulting from the overtaking collision for discrete times. In particular, the phase shifts are obtained for the first time at critical components in degenerate plasmas. In these two cases, it is found that the magnitude of the phase shifts due to the overtaking interaction of two solitons increases with increasing the negative ion-to-positive ion number density ratio. The present study may help explain the properties of nonlinear waves in white dwarfs that support the propagation of solitons.

Circular undular hydraulic jumps in turbulent free-surface flows

Near-critical turbulent axisymmetric free-surface flow over a horizontal bottom is investigated. Assuming that undular hydraulic jumps occur at large radii, an asymptotic analysis of the governing equations is performed in the double limit of very large Reynolds numbers and Froude numbers close to the critical value 1. The results are kept free of turbulence modelling due to a specific coupling of the two limiting processes. The final result of the asymptotic analysis is a new steady-state version of an extended Korteweg–de Vries (KdV) equation for the free-surface elevation. The extended KdV equation is derived as a uniformly valid differential equation, describing the flow near the origin of the undular jump as well as far downstream. Numerical solutions of the extended KdV equation show that circular undular jumps can develop if the reference state is located in the region where the effect due to axisymmetry prevails over the effect of friction. In this case, the solution oscillates over a very long distance until the accumulating friction effects force a breakdown. The comparison between the theories of undular jumps in turbulent and inviscid flows shows that friction is of minor importance near the development of the undular jump. However, friction has to be taken into account to describe the flow further downstream. Remarkably, the extended KdV equation is valid and yields undular solutions for both turbulent source and sink flow.

Density waves in an improved car-following model under intelligent transportation system environment

The intelligent transportation system (ITS) can effectively utilize the existing transportation facilities and improve traffic safety and efficiency. To study and evaluate the dynamic characteristics of traffic flow, a novel car-following model is proposed based on the two-velocity difference model (TVDM) with connected consideration of headway memory and backward looking effect under ITS environment. The linear stability theory is used to acquire the stability condition of the improved model, which shows that the proposed model can effectively strengthen the traffic flow stability. According to the nonlinear theory, the time-dependent Ginzburg–Landau (TDGL) equation and modified Korteweg–de Vries (mKdV) equation are obtained to exhibit the evolution characteristics of traffic flow density wave. At last, numerical simulation is performed to further validate that the improved model can enhance the traffic flow stability and eliminate traffic congestion.

Nonlinear and supernonlinear ion-acoustic wave phenomena in an electron-positron-pair-ion quantum plasma

A four-component quantum plasma consisting of electrons, positrons, negative heavy ions and positive light ions is proposed in this work. The dynamics of nonlinear and supernonlinear ion-acoustic waves are studied in the framework of the Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations which are derived employing the reductive perturbation technique. Using Galilean transformation these two evolution equations are transformed into planar dynamical systems. All possible phase portraits and corresponding small-amplitude Sagdeev's pseudopotential of these dynamical systems are presented graphically. The unique topology of phase portrait and a maxima in between two minimas in pseudopotential curves clearly establish quantum ion-acoustic superperiodic waves. Solitary, periodic and superperiodic wave solutions corresponding respectively to homoclinic, periodic and superperiodic orbits in phase portraits are obtained numerically and the influence of different parameters on these waves is observed. Further, various kinds of analytical wave solutions for the two evolution equations are discussed.

Influence of ions nonextensivity on the dynamics of dust acoustic double layers in a magnetized self-gravitating dusty plasma

The existence and polarity of dust acoustic double layers (DADLs) in a magnetized self-gravitating opposite polarity dusty plasma (OPDP) have been investigated. The constituents of the OPDP model are two species of positively and negatively charged dust grains, Maxwellian electrons, and nonextensive distributed ions. Employing the reduction perturbation method to solve basic fluid equations for dust grains, the extended Korteweg-de Vries (E-KdV) equation was derived, and its solution was extracted and analyzed. Results show that DADLs behavior is influenced by several new relevant configurational parameters, including the nonextensivity of ions, strength, and direction of the static magnetic field, as well as the self-gravitational field. In the proposed OPDP model positive and negative double-layer polarities may be formed. The polarity of OPDP is severely affected by the nonextensivity of plasma ions and the mass ratio of positive dusts to negatives. Results of higher-order nonlinear equation for solitary structure deny the presence of solitons in the physical condition of modeled plasma. In this work, simultaneous effects of an external magnetic field, self-gravitational force, and plasma ions’ nonextensivity have been studied extensively, which can be applied in various material processing plasmas, fusion plasmas, to space plasmas such as cometary tails, and Earth’s mesosphere.