KdV Equation Research Articles

General boundary value problems of a class of fifth order KdV equations on a bounded interval

General boundary value problems of a class of fifth order KdV equations on a bounded interval

Weak compactons of nonlinearly dispersive KdV and KP equations

AbstractA weak formulation is devised for the equation, which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.

Comment on “Solitary Wave Solutions in (2+1) Dimensions: The KdV Equation Derived from Ideal Fluid Models”, IJTP (2024) 63:105

Comment on “Solitary Wave Solutions in (2+1) Dimensions: The KdV Equation Derived from Ideal Fluid Models”, IJTP (2024) 63:105

Flow patterns induced by a moving disturbance in rotational flows within the forced Korteweg–de Vries equation

Flow patterns induced by a moving disturbance in rotational flows within the forced Korteweg–de Vries equation

The exact analytical solutions of the (2+1)-dimensional extended Korteweg–de Vries equation using bilinear neural network method and bilinear residual network method

The [Formula: see text]-dimensional extended Korteweg–de Vries (KdV) equation is predominantly utilized to elucidate the propagation of waves that exhibit both dispersive and nonlinear characteristics within the domain of nonlinear physics. This paper employs the bilinear neural network method (BNNM) to derive the exact analytical solutions of the equation. By constructing various bilinear neural network models, we obtain lump solution, breather solution and periodic interaction solution of the equation. The bilinear residual network method (BRNM) is an extension of BNNM. We apply BRNM under specific constraints to obtain breather solution of the equation, thereby offering a broader conceptual framework. Subsequently, various 3D plots, contour plots, density plots and x-curves are used to illustrate the physical properties and dynamic behaviors of these waves.

Fredholm Determinant and Wronskian Representations of the Solutions to the Schrödinger Equation with a KdV-Potential

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials.

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.

Novel nonlinear wave transitions and interactions for (2+1)-dimensional generalized fifth-order KdV equation

The (2 + 1)-dimensional generalized fifth-order KdV (2GKdV) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKdV equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped (XS/YS), asymmetric M-Shaped (MS), W-Shaped (WS), Space-Curved (SC) and Oscillation M-Shaped (OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.

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.

Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G′/G-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.

Data-driven solutions and parameter estimations of a family of higher-order KdV equations based on physics informed neural networks

Physics informed neural network (PINN) demonstrates powerful capabilities in solving forward and inverse problems of nonlinear partial differential equations (NLPDEs) through combining data-driven and physical constraints. In this paper, two PINN methods that adopt tanh and sine as activation functions, respectively, are used to study data-driven solutions and parameter estimations of a family of high order KdV equations. Compared to the standard PINN with the tanh activation function, the PINN framework using the sine activation function can effectively learn the single soliton solution, double soliton solution, periodic traveling wave solution, and kink solution of the proposed equations with higher precision. The PINN framework using the sine activation function shows better performance in parameter estimation. In addition, the experiments show that the complexity of the equation influences the accuracy and efficiency of the PINN method. The outcomes of this study are poised to enhance the application of deep learning techniques in solving solutions and modeling of higher-order NLPDEs.

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.

A (2+1) modified KdV equation with time-dependent coefficients: exploring soliton solution via Darboux transformation and artificial neural network approach

A (2+1) modified KdV equation with time-dependent coefficients: exploring soliton solution via Darboux transformation and artificial neural network approach

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.

Analytical Solutions of the Fractional Hirota–Satsuma Coupled KdV Equation along with Analysis of Bifurcation, Sensitivity and Chaotic Behaviors

Analytical Solutions of the Fractional Hirota–Satsuma Coupled KdV Equation along with Analysis of Bifurcation, Sensitivity and Chaotic Behaviors

A fast compact difference scheme on graded meshes for solving the time-fractional nonlinear KdV equation

The fractional KdV equation has significant physical implications, the study of its efficient method has great scientific significance. A fast compact difference (FCD) scheme on graded meshes is used to address the time-fractional nonlinear KdV equation with an initial singularity. The temporal discretization employs the fast L1 algorithm, while a fourth-order compact scheme is utilized for spatial discretization. The Newton's method is used to approximate the nonlinear term. The fast L1 algorithm can greatly shorten the CPU calculation time without losing the precision of numerical solution. We verify the existence and uniqueness of numerical solutions, while also provide unconditional stability and convergence analysis. Theoretical analysis is validated by numerical results, indicating the FCD scheme on graded meshes has high-accuracy and time-saving performance.

Wave breaking, dispersive shock wave, and phase shift for the defocusing complex modified KdV equation.

We study the problem of wave breaking for a simple wave propagating to a quiescent medium in the framework of the defocusing complex modified KdV (cmKdV) equation. It is assumed that a cubic root singularity is formed at the wave-breaking point. The dispersive regularization of wave breaking leads to the generation of a dispersive shock wave (DSW). We describe the DSW as a modulated periodic wave in the framework of the Gurevich-Pitaevskii approach based on the Whitham modulation theory. The generalized hodograph method is used to solve the Whitham equations, and the boundaries of the DSW are found. Most importantly, we determine the correct phase shift for the DSW from the generalized phase relationships and the modified Gurevich-Pitaevskii matching conditions, so that a complete description of the DSW is obtained rather than just its envelope. All of our analytical predictions agree well with the numerical simulations.

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.

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.