KdV Equation Research Articles

Study of multi solitons, breather soliton structures in the earth's magnetotail region

The investigation of wave packets with varying magnitudes, which predominantly carry energy across different physical mediums, proves to be highly intriguing and unveils numerous nonlinear characteristics within Earth's magnetotail region and magnetosphere, as confirmed by extensive space observations. Nonlinear features in wave dynamics are elucidated through diverse nonlinear structures, among which breather structures hold prominence and manifest across various wave fields. We have considered an unmagnetized collisionless homogeneous hydrodynamical governing equations set in an electron-ion plasma comprising hot and cold ions. In this study, we have explored 1-soliton, 2-soliton, and breather structures in the framework of the Gardner equation (GE). We have derived the GE from the normalized set of governing equations employing the reductive perturbation technique (RPT). The GE, an extension of the Korteweg-de Vries (KdV) equation, encompasses the combined effects of quadratic and cubic nonlinearities. By employing the Hirota bilinear method (HBM), we have obtained multi-soliton solutions and breather soliton solutions. We have noticed the variations in electron temperature ratios and density concentration ratios substantially impact and reshape breather soliton structures. Such alterations in breather structures, influenced by different electron temperatures and concentration ratios, hold potential significance in understanding energy transport within Earth's magnetotail region. The alteration of shapes and soliton structures may also be investigated in the dynamics of wave fields upon interaction such as hydrodynamics, optics, optic fibers, signal processing, etc.

On the analytical soliton approximations to fractional forced Korteweg–de Vries equation arising in fluids and plasmas using two novel techniques

The current investigation examines the fractional forced Korteweg–de Vries (FF-KdV) equation, a critically significant evolution equation in various nonlinear branches of science. The equation in question and other associated equations are widely acknowledged for their broad applicability and potential for simulating a wide range of nonlinear phenomena in fluid physics, plasma physics, and various scientific domains. Consequently, the main goal of this study is to use the Yang homotopy perturbation method and the Yang transform decomposition method, along with the Caputo operator for analyzing the FF-KdV equation. The derived approximations are numerically examined and discussed. Our study will show that the two suggested methods are helpful, easy to use, and essential for looking at different nonlinear models that affect complex processes.

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A class of complex breather and soliton solutions to both KdV and mKdV equations are identified with a Pöschl-Teller type PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document}-symmetric potential. However, these solutions represent only the unbroken-PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document} phase owing to their isospectrality to an infinite potential well in the complex plane having real spectra. To obtain the broken-PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document} phase, an extension of the potential satisfying the sl2,R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$sl\\left( 2,\\mathbb {R}\\right)$$\\end{document} potential algebra is mandatory that additionally supports non-trivial zero-width resonances.

Wronskian Representation of the Solutions to the KdV Equation

A method to construct solutions to the Korteweg-de-Vries (KdV) equation in terms of wronskians is given. For this, a particular type of polynomials is considered and we obtain for each positive integer n, rational solutions in terms of determinants of order n. Explicit solutions can be easily constructed and rational solutions from order 1 until order 10 are given.

Interfacial conditioning in physics informed neural networks

Physics informed neural networks (PINNs) have effectively demonstrated the ability to approximate the solutions of a system of partial differential equations (PDEs) by embedding the governing equations and auxiliary conditions directly into the loss function using automatic differentiation. Despite demonstrating potential across diverse applications, PINNs have encountered challenges in accurately predicting solutions for time-dependent problems. In response, this study presents a novel methodology aimed at enhancing the predictive capability of PINNs for time-dependent scenarios. Our approach involves dividing the temporal domain into multiple subdomains and employing an adaptive weighting strategy at the initial condition and at the interfaces between these subdomains. By employing such interfacial conditioning in physics informed neural networks (IcPINN), we have solved several unsteady PDEs (e.g., Allen–Cahn equation, advection equation, Korteweg–De Vries equation, Cahn–Hilliard equation, and Navier–Stokes equations) and conducted a comparative analysis with numerical results. The results have demonstrated that IcPINN was successful in obtaining highly accurate results in each case without the need for using any labeled data.

The Cauchy Problem for the Nonlinear Complex Modified Korteweg-de Vries Equation with Additional Terms in the Class of Periodic Infinite-Gap Functions

The Cauchy Problem for the Nonlinear Complex Modified Korteweg-de Vries Equation with Additional Terms in the Class of Periodic Infinite-Gap Functions

Dynamical Behaviors and Abundant New Soliton Solutions of Two Nonlinear PDEs via an Efficient Expansion Method in Industrial Engineering

In this study, we discuss the dynamical behaviors and extract new interesting wave soliton solutions of the two significant well-known nonlinear partial differential equations (NPDEs), namely, the Korteweg–de Vries equation (KdVE) and the Jaulent–Miodek hierarchy equation (JMHE). This investigation has applications in pattern recognition, fluid dynamics, neural networks, mechanical systems, ecological systems, control theory, economic systems, bifurcation analysis, and chaotic phenomena. In addition, bifurcation analysis and the chaotic behavior of the KdVE and JMHE are the main issues of the present research. As a result, in this study, we obtain very effective advanced exact traveling wave solutions with the aid of the proposed mathematical method, and the solutions involve rational functions, hyperbolic functions, and trigonometric functions that play a vital role in illustrating and developing the models involving the KdVE and the JMHE. These new exact wave solutions lead to utilizing real problems and give an advanced explanation of our mentioned mathematical models that we did not yet have. Some of the attained solutions of the two equations are graphically displayed with 3D, 2D, and contour panels of different shapes, like periodic, singular periodic, kink, anti-kink, bell, anti-bell, soliton, and singular soliton wave solutions. The solutions obtained in this study of our considered equations can lead to the acceptance of our proposed method, effectively utilized to investigate the solutions for the mathematical models of various important complex problems in natural science and engineering.

Finite-gap solutions of the real modified Korteweg-de Vries equation

Рассмотрены методы построения конечнозонных решений вещественного классического модифицированного уравнения Кортевега-де Фриза и эллиптических конечнозонных потенциалов оператора Дирака. Оба метода используют преобразование Миуры, связывающее решения уравнения Кортевега-де Фриза и модифицированного уравнения Кортевега-де Фриза. Приведены примеры.

Stabilization of linear KdV equation with boundary time-delay feedback and internal saturation

This paper investigates the question of stabilization of linear Korteweg-de Vries equation with time-delay on the boundary feedback in presence of saturated source term. Under some conditions the well-posedness is proved. The exponential stability result is demonstrated, using an appropriate Lyapunov functional.

Determination of an unknown coefficient in the Korteweg–de Vries equation

Abstract In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in L 2 {L^{2}} -norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.

New derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons

The computation of the amplitude, α, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg–de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, ε2≪1, a new derivation of the “beyond all orders in ε” amplitude, α, is presented. It is shown by asymptotic matching techniques, extended to higher orders in ε, that the value of α can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [], not only sheds new light on the computation of α but also greatly facilitates its numerical determination to a remarkable precision for so small values of ε, which are beyond the capabilities of standard numerical methods. Published by the American Physical Society 2024

Dynamical properties of nonlinear dust ion-acoustic waves on the basis of the Schamel–KdV equation

Dynamical properties of nonlinear dust ion-acoustic waves on the basis of the Schamel–KdV equation

Uniqueness of lump solution to the KP‐I equation

AbstractThe KP‐I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution, which is a traveling wave, and the KP‐I equation in this case reduces to the Boussinesq equation. In this paper we classify all the ‘lump‐type’ solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman–Miller for the Schrödinger equation, we show that the lump‐type solutions are rational and their functions have to be polynomials of degree for some integer . In particular, this implies that the lump solution is the unique ground state of the KP‐I equation (as conjectured by Klein–Saut). The problem studied in this paper was mentioned in Airault–McKean–Moser, our result can be regarded as a two‐dimensional analogy of their theorem on the classification of rational solutions for the KdV equation.

Another look at the solution of the equations; the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation.

Another look at the solution of the equations; the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation.

The (2+1)‐dimensional Schwarzian Korteweg–de Vries equation and its generalizations with discrete Lax matrices

By using the discrete Lax matrices corresponding to and in the Adler–Bobenko–Suris list of quadrilateral lattice equations, we establish solutions of the (2+1)‐dimensional Schwarzian Korteweg–de Vries (SKdV) equation and its generalizations. According to the structure of integrable Hamiltonian systems provided by one discrete Lax matrix for the lattice, we construct a novel Lax representation for the (2+1)‐dimensional SKdV equation. On the basis of the Riemann surface and elliptic variables, the Hamiltonian phase flows related to the equation are linearized by the Abel–Jacobi coordinates. Consequently, we obtain finite genus solutions to the (2+1)‐dimensional SKdV equation. From the discrete Lax matrix of the lattice, we study the generalized (2+1)‐dimensional SKdV equation with a parameter , which can be reduced to a simple case of the Krichever–Novikov equation. Finally, we show that the case of can also be solved through a modified version of the other discrete Lax matrix for the equation.

Bridging trails in reflectionless potential deformation: Two paths and one horizon

In contrast to the conventional one-parameter class of isospectral deformation using translation, we calculate the two- and three-parameter classes of isospectral deformation of the well-known reflectionless potential by utilizing a far different approach of scaling methodology. Subsequently, using these results, we find that this more general class of deformations is not unique but instead subsume in the same class of conventional one-parameter translational deformation. We also provide a theoretical foundation for how these two incredibly different approaches converge at the same destination. Finally, we show that the most generic class of potentials, obtained by scaling deformation, are solutions of the nonlinear KdV equation.

Nonlinear Reduced Order Modelling for Korteweg-de Vries Equation

Efficient computational techniques that maintain the accuracy and invariant preservation property of the Korteweg-de Vries (KdV) equations have been studied by a wide range of researchers. In this paper, we introduce a reduced order model technique utilizing kernel principle component analysis (KPCA), a nonlinear version of the classical principle component analysis, in a non-intrusive way. The KPCA is applied to the data matrix, which is formed by the discrete solution vectors of KdV equation. In order to obtain the discrete solutions, the finite differences are used for spatial discretization, and linearly implicit Kahan's method for the temporal one. The back-mapping from the reduced dimensional space, is handled by a non-iterative formula based on the idea of multidimensional scaling (MDS) method. Through KPCA, we illustrate that the reduced order approximations conserve the invariants, i.e., Hamiltonian, momentum and mass structure of the KdV equation. The accuracy of reduced solutions, conservation of invariants, and computational speed enhancements facilitated by classical (linear) PCA and KPCA are exemplified through one-dimensional KdV equation.

Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

On decay of solutions to some perturbations of the Korteweg-de Vries equation

On decay of solutions to some perturbations of the Korteweg-de Vries equation

Higher order contribution to ion-acoustic Korteweg–de Vries (KdV) soliton in unmagnetized quantum plasmas with arbitrary degeneracy of electrons

The propagation of nonlinear ion-acoustic Korteweg–de Vries (KdV) solitons and dressed solitons (with higher order contribution term) in unmagnetized quantum plasmas with arbitrary degeneracy of electrons is investigated using quantum hydrodynamic model. The reductive perturbation method is used to derive the first order nonlinear ion-acoustic KdV equation and also higher order contribution of inhomogeneous differential equation is obtained for the dressed ion-acoustic wave soliton in arbitrary degenerate electron plasmas. Moreover, the higher order inhomogeneous differential equation is solved (nonsecular solution is obtained) using the renormalization method. The conditions for the validity of the higher order correction are also discussed in detail for ion-acoustic solitons in unmagnetized arbitrary degenerate electrons plasma case. The effects of chosen values of electron temperature, equilibrium fugacity and corresponding electron density of a physical quantum plasma system and quantum diffraction parameter H (for two conditions i.e., H < 2 or H > 2) on the amplitude and width of the KdV and dressed ion-acoustic wave solitons for both electrostatic potential hump (compressive) or dip (rarefactive) structures are investigated. The numerical plots are also presented for illustration with numerical values of a physical partially degenerate electrons plasma system exist in the astrophysical or laboratory conditions.