Korteweg-de Vries Research Articles

Derivation of Sawada-Kotera and Kaup-Kupershmidt Equations KdV Flow Equations from Modified Nonlinear Schrödinger Equation (MNLS)ns from Modfied Nonlinear Schrödinger Equation (MNLS)

Mathematical models of problems that arise in almost every branch of science are nonlinear evolution equations (NLEE). As a result, nonlinear evolution equations have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear evolution equations and solution methods. The main reason for this situation is that nonlinear evolution equations involve the problem of nonlinear wave propagation. In this study, (1 + 1) dimensional fifth-order nonlinear Korteweg-de Vries (fKdV) type equations were obtained by applying the multi-scale method known as the perturbation method for the modified nonlinear Schrödinger (MNLS) equation. Thus, we showed the relationship between KdV equations and MNLS type equations.

Solitonic Analysis of the Newly Introduced Three-Dimensional Nonlinear Dynamical Equations in Fluid Mediums

The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features.

An extended (2+1)-dimensional modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation: Lax pair and Darboux transformation

Abstract The aim of this paper is to study an extended modified Korteweg-de Vries Calogero-Bogoyavlenskii-Schiff (mKdV-CBS) equation and present its Lax pair with a spectral parameter. Meanwhile, a Miura transformation is explored, which reveals the relationship between solutions of the extended mKdV-CBS equation and the extended (2+1)-dimensional Korteweg-de Vries (KdV) equation. On the basis of the obtained Lax pair and the existing research results, Darboux transformation is derived, which plays a crucial role in presenting soliton solutions. Besides, soliton molecules are given by the velocity resonance mechanism.

Integrable flows on null curves in the Anti-de Sitter 3-space

Abstract We formulate integrable flows related to the Korteweg–De Vries (KdV) hierarchy on null curves in the anti-de Sitter 3-space ( AdS ). Exploiting the specific properties of the geometry of AdS , we analyze their interrelationships with Pinkall flows in centro-affine geometry. We show that closed stationary solutions of the lower order flow can be explicitly found in terms of periodic solutions of a Lamé equation. In addition, we study the evolution of non-stationary curves arising from a 3-parameter family of periodic solutions of the KdV equation.

KdV conformal symmetry breaking in nearly AdS2

We study the gauge theory formulation of Jackiw-Teitelboim gravity and propose Korteweg-de Vries asymptotic conditions that generalize the asymptotic dynamics of the theory. They permit to construct an enlarged set of boundary actions formed by higher order generalizations of the Schwarzian derivative that contain the Schwarzian as lower term in a tower of SL(2, ℝ) invariants. They are extracted from the KdV Hamiltonians and can be obtained recursively. As a result, the conformal symmetry breaking observed in nearly AdS2 is characterized by a much larger set of dynamical modes associated to the stationary KdV hierarchy. We study quantum perturbation theory for the generalized Schwarzian action including the symplectic measure and compute the one-loop correction to the partition function. We find that despite the non-linear nature of the higher-Schwarzian contribution, it acquires a manageable expression that renders a curious leading temperature dependence on the entropy S = #Ta for a an odd integer.

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems.

The spatial Whitham equation

The Whitham equation is a non-local, nonlinear partial differential equation that models the temporal evolution of spatial profiles of surface displacement of water waves. However, many laboratory and field measurements record time series at fixed spatial locations. To directly model data of this type, it is desirable to have equations that model the spatial evolution of time series. The spatial Whitham (sWhitham) equation, proposed as the spatial generalization of the Whitham equation, fills this need. In this paper, we study this equation and apply it to water-wave experiments on shallow and deep water. We compute periodic travelling-wave solutions to the sWhitham equation and examine their properties, including their stability. Results for small-amplitude solutions align with known results for the Whitham equation. This suggests that the systems are consistent in the weakly nonlinear regime. At larger amplitudes, there are some discrepancies. Notably, the sWhitham equation does not appear to admit cusped solutions of maximal wave height. In the second part, we compare predictions from the temporal and spatial Korteweg–de Vries and Whitham equations with measurements from laboratory experiments. We show that the sWhitham equation accurately models measurements of tsunami-like waves of depression, waves of elevation, and solitary waves on shallow water. Its predictions also compare favourably with experimental measurements of waves of depression and elevation on deep water. Accuracy is increased by adding a phenomenological damping term. Finally, we show that neither the sWhitham nor the temporal Whitham equation accurately models the evolution of wave packets on deep water.

Excitation of ion acoustic solitary waves and shock waves in decaying plasma between biased parallel-plate electrodes

In this study, we explore the excitation of ion acoustic solitary waves and shock waves in a decaying plasma confined between biased parallel-plate electrodes using one-dimensional particle-in-cell simulations. Our findings demonstrate that the oscillating electric field at the sheath edge generates a sequence of ion acoustic solitary waves, which exhibit characteristics consistent with the Korteweg–de Vries (KdV) equations. We observe that as the electron temperature decreases, the intervals between adjacent wave pulses shorten progressively, leading to the eventual formation of a coherent shock wave structure. These findings highlight the critical role of kinetic simulations in elucidating the dynamics of plasma decay processes.

Electron-acoustic solitary waves via suprathermal populations in Saturn's magnetosphere: Bifurcation, sensitivity, and stability analysis

Cassini and Voyager space missions observed non-thermal electron populations (with varying characteristics) in Saturn's magnetosphere, which can be correctly described using kappa distributions. Based on these observations, our objective is to inspect the evolution of electron-acoustic solitary waves (EASWs) within Saturn's magnetosphere. The propagation of weakly nonlinear (EASWs) in a collisional plasma system comprising a cold electron fluid, hot electrons following a kappa distribution, and stationary ions is investigated. By employing the reductive perturbation technique, the Korteweg–de Vries Burgers (KdV–B) equation is derived. An exact solution of the KdV–B equation, with a conformable time-derivative, is found using the unified method. It is observed that plasma current-induced collision between electrons and ions leads to remarkable dissipation, generating EASWs. Furthermore, when studying the sensitivity of the system, the appearance of a positive potential is depicted as external forces vanish, which may be due to stationary ions. Additionally, bifurcation, stability, and significant influence of plasma characteristics are considered.

Uniform solitary wave theory for viscous flow over topography

The flow of a density stratified fluid over obstacles has been intensively explored from a natural and scientific point of view. This flow has been successfully governed by using the forced Korteweg–de Vries-Burgers equation that generated solitons in a viscous flow. This is done by adding the viscous term beyond the Korteweg–de Vries approximation. It is based on the conservation laws of the Korteweg–de Vries-Burgers equation for mass and energy, and assumes that the upstream wavetrains are composed of solitary waves. Our results show that the influence of viscosity plays a key role in determining the upstream solitary wave amplitude of the bore. A good comparison is obtained between the numerical and analytical solutions.

Congested traffic patterns of mixed lattice hydrodynamic model combining the perceptual range differences with passing effect

With the aid of Vehicle-to-everything (V2X) technologies for network connectivity, connected autonomous vehicles (CAVs) can broaden the drivers' perceptual boundaries and receive a greater quantity of exogenous vehicle information, thereby governing the vehicle's acceleration information of the next moment. Nonetheless, constrained by contemporary communication networks and sophisticated vehicle control technology, the process of promoting CAVs is long-lasting, and throughout this stage of transition, both CAVs and human-driven vehicles (HDVs) will happen on the road. Moreover, passing behaviour is uncommon in the traffic flow model despite being a fundamental microscopic driving behaviour, particularly within mixed-traffic situations. To bridge that hole, we introduce the percentage ratios of CAVs into the lattice hydrodynamic model by integrating the perceptual range differences between two different types of vehicles with passing effects. Subsequently, the stability norm associated with the new model is ascertained by performing the linear stability analysis. When the above-mentioned stability condition is not achieved, we investigate the complex behaviour of the new model, and the associated existing conditions, as well as the modified Korteweg-de Vries (mKdV) equation, are determined simultaneously. When the passing ratio is inadequate, no jam and kink jam make up the whole phase region; while when the passing ratio surpasses the minimum, the initial unstable region can be segregated into two extra segments: the chaotic sub-region and the kink jam sub-region, the density wave progressively transitioned from being a kink-Bando traffic wave to a chaotic phase. Lastly, the findings of the numerical experiment identify with the theoretical derivation made above.

New Processing Technique of Jacobian Elliptic Equation and Its Application to the (3+1)-Dimensional Modified Korteweg de Vries–Zakharov–Kuznetsov Equation

This article introduces two kinds of processing techniques to solve Jacobian elliptic equations and obtain rich periodic wave solutions. Then, the equation was used as an auxiliary equation to solve the (3+1)-dimensional modified Korteweg de Vries–Zakharov–Kuznetsov (mKDV-ZK) equation. Combined with the mapping method, a large number of new types of exact periodic wave solutions were obtained, many of which were rarely found in previous research. Numerical simulations have demonstrated the evolution of various periodic waves in (3+1)-dimensional mKDV-ZK. The solutions and wave phenomena obtained in this article will help expand our understanding of the equation.

Nonlinear evolution of disturbances in higher time-derivative theories

We investigate the evolution of localized initial value profiles when propagated in integrable versions of higher time-derivative theories. In contrast to the standard cases in nonlinear integrable systems, where these profiles evolve into a specific number of N-soliton solutions as dictated by the conservation laws, in the higher time-derivative theories the theoretical prediction is that the initial profiles can settle into either two-soliton solutions or into any number of N-soliton solutions. In the latter case this implies that the solutions exhibit oscillations that spread in time but remain finite. We confirm these analytical predictions by explicitly solving the associated Cauchy problem numerically with multiple initial profiles for various higher time-derivative versions of integrable modified Korteweg-de Vries equations. In the case with the theoretical possibility of a decay into two-soliton solutions, the emergence of underlying singularities may prevent the profiles from fully developing or may be accompanied by oscillatory, chargeless standing waves at the origin.

Evolution of water wave packets by wind in shallow water

We use the Korteweg–de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV–Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

Domain decomposition for physics-data combined neural network based parametric reduced order modelling

This paper proposes a novel domain decomposition (DD) method for the parametric reduced-order model based on the Physics-Data Combined Neural Network (DD-PDCNN). The computational domain is partitioned into a number of subdomains, and a Physics-Data Combined Neural Network (PDCNN) is constructed for each subdomain, which not only represents the dynamics within its region but also considers interactions with neighbouring subdomains. The interface conditions between subdomains are considered by averaging solutions at the neighbourhoods, incorporating averaged solutions, predicted solutions at current and next time levels into reduced governing equations. The implicit coupling among the subdomains ensures global continuity and establishes boundary conditions for each subdomain.The performance of this new DD-PDCNN is compared with that of PDCNN without domain decomposition, data-driven model and traditional Reduced Basis(RB) model. The capability of this model is tested using a number of parametric nonlinear problems, such as the Korteweg-de Vries (KdV) equation, the two-dimensional Kovasznay flow and the two-dimensional incompressible Navier–Stokes equation. The results indicate that the proposed method offers an effective way to construct an accurate reduced order model for parameterised PDEs through machine learning. In particular, in specific parameter ranges where distinct physical phenomenon regions are prominent, the method outperforms the model without domain decomposition, demonstrating excellent performance on several challenging problems.

Boundary controllability for the Korteweg–de Vries–Burgers equation on a finite domain

Boundary controllability for the Korteweg–de Vries–Burgers equation on a finite domain

Soliton molecules, bifurcation solitons and interaction solutions of a generalized (2 + 1)-dimensional korteweg-de vries system for the shallow-water waves

The central purpose of this paper is exploring the soliton molecules, bifurcation solitons and interaction solutions of the Korteweg–de Vries system based on the Hirota bilinear method. The studied system acts as an extension of the classic KdV system for the shallow-water waves, and is very useful to contribute in nonlinear wave phenomena. Firstly, the soliton molecules are obtained by adding resonance parameters in N-soliton. Then the interaction solutions between soliton/breather and soliton molecules are studied, as well as the interaction between two soliton molecules by using N-soliton. Moreover, a class of novel bifurcation solitons are derived, including Y-type bifurcation solitons, X-type bifurcation solitons and multiple-bifurcation solitons. In the end, the dynamic properties of soliton molecules, bifurcation solitons as well as the interaction solutions are presented graphically. The developed solutions of this research are all new and can enable us apprehend the nonlinear dynamic behaviors of the generalized (2+1)-dimensional Korteweg–de Vries system better.

Polynomial dynamical systems associated with the KdV hierarchy

In 1974, S.P. Novikov introduced the stationary n-equations of the Korteweg–de Vries hierarchy, namely the n-Novikov equations. These are associated with integrable polynomial dynamical systems, with polynomial 2n integrals, in ℂ3n. In this paper, we construct an infinite-dimensional polynomial dynamical system that is universal for all dynamical systems corresponding to the n-Novikov equations. Thus, we solve the well-known problem of the relationship between the n-Novikov equations for different n.

Multiple soliton solutions and other travelling wave solutions to new structured (2+1)-dimensional integro-partial differential equation using efficient technique

The Ito equation belongs to the Korteweg–de Vries (KdV) family and is commonly employed to predict how ships roll in regular seas. Additionally, it characterizes the interaction between two internal long waves. In the 1980s, Ito extended the bilinear KdV equation, resulting in the well-known (1+1)-dimensional and (2+1)-dimensional Ito equations. In this study finds numerous classes of exact solutions for a new structured (2 + 1)-dimensional Ito integro-differential equation using the help of the Mathematica software. The Improved Modified Extended Tanh Function Scheme (IMETFS) is utilised to address the aforementioned equation analytically. Bright, dark, and singular soliton solutions are produced. Additionally, periodic, exponential, rational, singular periodic, and Weierstrass elliptic doubly periodic results are achieved. The method employed includes the nonlinear evolution equations that arise in a variety of real-world situations, and it is efficient, applicable, and simple to handle. For certain obtained solutions, specific options of free constants are presented in 3D, 2D, and contour graphical depictions.

A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations

In this paper, the space-time generalized finite difference scheme is applied to solve the nonlinear high-order Korteweg-de Vries equations in multiple dimensions. The proposed numerical scheme combines the space-time generalized finite difference method, the Levenberg-Marquardt algorithm, and a time-marching approach. The space-time generalized finite difference method treats the temporal axis as a spatial axis, enabling the proposed scheme to discretize all derivatives in the governing equation. This is accomplished through Taylor series expansion and the moving least squares method. Due to the expandability of the Taylor series to any order, the proposed numerical scheme excels in efficiently handling mixed and higher-order derivatives. These capabilities are distinct advantages of the proposed scheme. The resulting system of algebraic equations is sparse but overdetermined. Therefore, the Levenberg-Marquardt algorithm is directly applied to solve this nonlinear algebraic system. During the calculation process, the time-marching approach reduces computational effort and improves efficiency by dividing the space-time domain.