KdV Equation Research Articles

The Nonlinear Schrödinger Equation Derived from the Third Order Korteweg-de Vries Equation Using Multiple Scales Method

Nonlinear equations of evolution (NLEE) are mathematical models used in various branches of science. As a result, nonlinear equations of evolution 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 equations of evolution and solution methods. Although the origin of nonlinear equations of evolution dates back to ancient times, there have been significant developments regarding these equations from the past to the present. The main reason for this situation is that nonlinear equations of evolution involve the problem of nonlinear wave propagation. In recent years, equations of formation have become increasingly important in applied mathematics. This work focuses on the perturbation approach, often known as many scales, for nonlinear evolution equations. The article focuses on the analysis of the (1+1) dimensional third-order nonlinear Korteweg-de Vries (KdV) equation using the multiple scales method, which resulted in obtaining nonlinear Schrödinger (NLS) type equations.

Wave propagation and soliton behavior in biomechanical tissues: A mathematical approach

This study presents a mathematical model for understanding wave propagation and soliton behavior in biomechanical tissues, explicitly focusing on the Achilles tendon. Utilizing the Korteweg-de Vries (KdV) equation, the research incorporates the Achilles tendons’ nonlinear elastic and viscoelastic properties to explore how mechanical waves propagate through this complex tissue. The tendon’s nonlinear elasticity leads to wave steepening, while its viscoelasticity introduces dispersive effects that counteract this steepening, resulting in the formation of solitons—stable, localized waves that maintain their shape as they propagate. Key findings from this study reveal that the formation and propagation of solitons are strongly influenced by the tendon’s mechanical properties. Numerical simulations show that stiffer tendons, characterized by a higher elasticity modulus, support faster soliton propagation, with wave speeds ranging from 18.9 m/s in damaged tendons to 28.6 m/s in stiffened tendons. Additionally, soliton amplitude increases with tissue stiffness, with the highest amplitude observed in stiffened tendons (5.1 mm) and the lowest in damaged tendons (3.2 mm). The study also demonstrates that energy dissipation due to the tendon’s viscoelasticity plays a critical role in soliton behavior. Damaged tendons exhibit the highest energy loss (18.6%), leading to shorter soliton propagation distances, while stiffer tendons retain more energy (96.1%) and allow solitons to travel further distances (up to 180 mm). Moreover, the balance between nonlinearity and dispersion is crucial for maintaining soliton stability. Excessive nonlinearity leads to unstable solitons, while higher levels of dispersion contribute to more stable waveforms.

Dynamics of the Collision of Two Nearly Equal Solitary Waves for the Zakharov–Kuznetsov Equation

AbstractWe study the dynamics of the collision of two solitary waves for the 2 and 3-dimensional Zakharov–Kuznetsov equation, a high-dimensional non-integrable version of the Korteweg-de Vries equation that appears as an asymptotic model in plasma physics. We describe the evolution of the solution behaving as a sum of 2-solitary waves of nearly equal speeds at time $$t=-\infty $$ t = - ∞ up to time $$t=+\infty $$ t = + ∞ . We show that this solution behaves as the sum of two modulated solitary waves and an error term which is small in $$H^1$$ H 1 for all time $$t \in {\mathbb {R}}$$ t ∈ R . Finally, we also prove the stability of this solution for large times around the collision. The proofs are a non-trivial extension of the ones of Martel and Merle for the quartic generalized Korteweg-de Vries equation to higher dimensions. First, despite the non-explicit nature of the solitary wave, we construct an approximate solution in an intrinsic way by canceling the error to the equation only in the natural directions of scaling and translation. Then, to control the difference between a solution and the approximate solution, we use a modified energy functional and a refined modulation estimate in the transverse variable. Moreover, we rely on the Hamiltonian structure of the ODE governing the distance between the waves, which cannot be approximated by explicit solutions, to close the bootstrap estimates on the parameters. We hope that the techniques introduced here are robust and will prove useful in studying the collision phenomena for other focusing non-linear dispersive equations with non-explicit solitary waves.

Impact of relativistic positron beam on ion-acoustic solitary, periodic and breather waves in Earths’ ionospheric region through the framework of KdV and modified KdV equation

Abstract This study explores the propagation characteristics of ion-acoustic periodic, soliton, and breather waves in electron-positron-ion (EPI) plasma with a relativistic positron beam. The Korteweg–de Vries (KdV) equation is obtained by applying the traditional reductive perturbation method (RPM) to the fundamental set of fluid equations. When the KdV model is unable to accurately represent the nonlinear system’s evolution, a modified Korteweg–de Vries (mKdV) equation is constructed. In both models, Jacobi elliptic functions are used to derive periodic solutions, and a connection between periodic waves and soliton solutions is established. Hirota’s bilinear method is used to generate breathers directly from the KdV type framework without utilizing the modified Schrödinger framework inferred from the KdV type framework, which is a prevalent method in studies of nonlinear waves. Numerical knowledge of various physical factors in the ionospheric region is incorporated into the model to elucidate wave propagation in the Earth’s upper atmosphere.

Accelerating solutions of the Korteweg-de Vries equation

Abstract The Korteweg-de Vries equation is a fundamental nonlinear equation that describes solitons with constant velocity. On the contrary, here we show that this equation also presents accelerated wavepacket solutions. This behavior is achieved by putting the Korteweg-de Vries equation in terms of the Painlevé I equation. The accelerated waveform solutions are explored numerically showing their accelerated behavior explicitly.

Experimental and theoretical studies of sea ice effects on internal solitary waves

Internal solitary waves in polar regions have attracted much interest recently. It is important to understand how sea ice affects them as this may have a profound influence on human activities and the environment. In this study, experiments on internal solitary waves with and without two types of sea ice (ice sheet and ice keel) are presented, as well as corresponding simulations using the Korteweg-de Vries (KdV) equation, the Benjamin-Ono (BO) equation, and the variable-coefficient Korteweg-de Vries (vKdV) equation, which is a derivation of the KdV equation. Comparison between experiments without sea ice and simulations using the KdV and BO equations proves the suitability of the former over the latter for this study. The experiments with sea ice and theoretical simulations using the vKdV equation provide evidence for wave deformation, oscillation occurring in the rear of the wave, and a decrease in amplitude. The latter suggests possibilities of energy dissipation or the emission of small amplitude linear waves. The sharp vertices of the ice result in occasional inconsistency with the vKdV predictions. Nonetheless, the vKdV equation is still suitable for modeling internal solitary waves under sea ice, giving generally accurate results that can assist further studies. This is the first time the vKdV equation has been applied to investigate the impacts of sea ice on internal solitary waves.

Review of Some Modified Generalized Korteweg–de Vries–Kuramoto–Sivashinsky Equations (Part II)

In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form N(ux), where ux denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto–Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in Rn. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case.

Review of Some Modified Generalized Korteweg–De Vries–Kuramoto–Sivashinsky (mgKdV-KS) Equations

This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg–De Vries equation, the Kuramoto–Sivashinsky equation, the generalized Korteweg–De Vries–Kuramoto–Sivashinsky equation and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane.

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Exact solitary wave solutions for a coupled gKdV–Schrödinger system by a new ODE reduction method

AbstractA new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with ‐power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with . No solutions for were known previously in the literature. For , four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized Hénon–Heiles form.

Mixed neural operator learning on the solitary wave propagation over slope topography and inverse problem

This study proposes the mixed neural operator (MNO) learning framework, which further combines with the particle swarm optimization (PSO) to address challenges of solitary wave propagation over topography. The forward problem is defined as the evolution prediction of the solitary wave propagating over topography, while the inverse problem is defined as an optimization to identify the topography parameter based on the solitary wave elevation. Both the forward and inverse problems can be considered within a single framework and the dataset are provided by the classical Korteweg–de Vries (KdV) equation. The MNO framework is shown to simulate the evolution of solitary waves over topography, accurately capturing the wave elevation under different topographical conditions. By comparing with different neural operators, it is found that the U-shape neural operator is the most suitable for the KdV equation simulation. The coefficient of determination for the inverse problem based on the combination of MNO and PSO can reach 0.992, showing great potential of the approach in topography recognition. Finally, the proposed learning framework is preliminary applied to the prediction of the tsunami runup onto a complex beach, and a good agreement is also achieved between the direct simulation and the learning framework prediction.

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

New generalized Jacobi–Galerkin operational matrices of derivatives: an algorithm for solving the time-fractional coupled KdV equations

The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances.

Obtaining new soliton solutions of the fractional generalized perturbed KdV equation

Abstract In this study, the fractional generalized perturbed KdV equation (gpKdV) with beta derivative is considered. The generalized exponential rational function method (GERFM) is applied to this equation for the first time in this study. Thus, dark soliton, bright soliton, singular soliton, mixed soliton, trigonometric function, rational trigonometric function, hyperbolic function and rational exponential function solutions of this fractional equation are obtained for the first time in this study. The 2D, 3D, and density plots, which effectively illustrate the behavior of these solitons, are shown for various values and specific ranges of the solutions.

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.