Solution Of Korteweg-de Vries Equation Research Articles

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

We provide a general solution for a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was established that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this article supports once again an empirical rule saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the method of inverse scattering problem.

Head-on collisional effects of ion acoustic waves in pair-ion plasma: topological and non-topological solitons

The phenomena concerning head-on collision between the two topological and non-topological solitons are investigated in unmagnetized and collisionless plasma system consisting of weakly relativistic cold positive ( $$H^{+}$$ , $${Xe}^{+}$$ and $$K^{+}$$ ) and negative ( $$H^{-}$$ , $${SF}_{6}^{-}$$ , $$O_{2}^{-}$$ and $$C_{7}F_{14}^{-}$$ ) ion species with superthermal electrons. To do this, the two-sided Korteweg–de Vries (KdV) equations are derived using extended Poincare–Lighthill–Kue method. The solutions of KdV equations are obtained by solitary ansatz technique. Productions as well as collisional effects of non-topological and topological solitons are investigated taking the experimentally obtained plasma parameters into accounts. The study reveals that the nonlinearity of non-topological solitons become more pronounced for $$K^{+}-C_{7}F_{14}^{-}$$ plasma as compared to the other ionic species plasmas. The masses and densities of negative ion species play a vital role on the potential profiles of topological solitons. A new type large amplitude wave is produced in the colliding region due to collision depending on plasma parameters.

Dust-acoustic cnoidal waves in a magnetized quantum dusty plasma

ABSTRACT An investigation of nonlinear dust-acoustic cnoidal waves in a magnetized quantum dusty plasma consisting of inertial dust, inertialess electrons and ions is presented. The Korteweg–de Vries (KdV) equation is obtained by employing reductive perturbation technique and analytical solution of KdV equation is derived by employing the Sagdeev pseudopotential approach. Numerical analysis of the solution is performed to study the characteristics of the dust-acoustic cnoidal waves. In this study, positive (negative) potential dust-acoustic cnoidal waves are observed for the case of positive (negative) dust grains. The influence of various parameters such as quantum diffraction parameter of electrons and ions, magnetic field strength, obliqueness, concentration of ions and Fermi temperature (electrons and ions) has been studied. In the limiting case, dust-acoustic cnoidal wave solution is found to reduce to solitary wave solution.

Metastability of solitary waves in diatomic FPUT lattices$^\dagger$

It is known that long waves in spatially periodic polymer Fermi-Pasta-Ulam-Tsingou lattices are well-approximated for long, but not infinite, times by suitably scaled solutions of Korteweg-de Vries equations. It is also known that dimer FPUT lattices possess nanopteron solutions, i.e., traveling wave solutions which are the superposition of a KdV-like solitary wave and a very small amplitude ripple. Such solutions have infinite mechanical energy. In this article we investigate numerically what happens over very long time scales (longer than the time of validity for the KdV approximation) to solutions of diatomic FPUT which are initially suitably scaled (finite energy) KdV solitary waves. That is we omit the ripple. What we find is that the solitary wave continuously leaves behind a very small amplitude “oscillatory wake.” This periodic tail saps energy from the solitary wave at a very slow (numerically sub-exponential) rate. We take this as evidence that the diatomic FPUT “solitary wave” is in fact quasi-stationary or metastable.

Quasi-soliton solution of Korteweg-de Vries equation and its application in ion acoustic waves

Investigation of interaction between solitons and their background small amplitude waves has been an interesting topic in numerical study for more than three decades. A classical soliton accompanied with oscillatory tails to infinite extent in space, is an interesting quasi-soliton, which has been revealed in experimental study and really observed. However, analytical solution of such a special quasi-soliton structure is rarely considered. In this paper, two branches of soliton-cnoidal wave solution as well as the two-soliton solution of the Korteweg-de Vries (KdV) equation are obtained by the generalized tanh expansion method. The exact relation between the soliton-cnoidal wave solution and the classical soliton solution of the KdV equation is established. By choosing suitable wave parameters, the quasi-soliton behavior of the soliton-cnoidal wave solution is revealed. It is found that with modulus of the Jacobi elliptic function approaching to zero asymptotically, the oscillating tails can be minimized and the soliton core of the soliton-cnoidal wave turns closer to the classical soliton solution. In addition, the quasi-soliton structure is revealed in a plasma physics system. By the reductive perturbation approach, the KdV equation modeling ion acoustic waves in an ideal homogeneous magnetized plasma is derived. It is confirmed that the waveform of the quasi-soliton is significantly influenced by the electron distribution, temperature ratio of ion to electron, magnetic field strength, and magnetic field direction. Interestingly, the amplitude of the quasi-soliton keeps constant due to the -independence of nonlinear coefficient A. The width of the soliton core and the wavelength of the surrounded periodic wave become constant with the further increase of . The explicit soliton-cnoidal wave solution with quasi-soliton behavior obtained here is applicable to many physical scenarios. For instance, the quasi-soliton structure can be viewed as a classical soliton with perturbations, and can correct the classical soliton in both theoretical and experimental study.

Similarity Projective Excitations and Similaritons for Generalized Korteweg–de Vries System with Variable Coefficients

A general similarity projective solution to the generalized Korteweg–de Vries (KdV) system with varying coefficients is derived with the aid of direct similarity projective approach. All the allowed exact solutions of the standard KdV equation can be converted into the corresponding exact solutions of the generalized KdV system. In terms of the known exact solution of the KdV equation, some typical localized excitations with self-similar behaviors and similaritons are revealed by prescribing appropriate system parameters.

A New Homotopy Analysis Method for Approximating the Analytic Solution of KdV Equation

In this study a new technique of the Homotopy Analysis Method (nHAM) is applied to obtain an approximate analytic solution of the well-known Korteweg-de Vries (KdV) equation. This method removes the extra terms and decreases the time taken in the original HAM by converting the KdV equation to a system of first order differential equations. The resulted nHAM solution at third order approximation is then compared with that of the exact soliton solution of the KdV equation and found to be in excellent agreement.

Multi-Resolution Analysis of Wavelet Like Soliton Solutions of KdV Equations

Many physical phenomena are described by non- linear partial differential equations. These equations have soliton solutions which exhibit wavelet features called wavelet like solitons. Such wavelet like solitons have expansions in Gaussian family wavelets. In this work, using the fact that the wavelet like soliton has Gaussian representation, multi- resolution analysis which is based on wavelets is carried out to obtain better approximation with the application of wavelet- Galerkin and wavelet-Petrokov-Galerkin methods for soliton solution of Korteweg-de Vries equation which appears in the study of waves in shallow water in the fluid dynamics. In the end, experimental data processing employing Gaussian representation of soliton solution is discussed.

Approximate Method for Studying the Waves Propagating along the Interface between Air‐Water

This paper is devoted to consider the approximate solutions of the nonlinear water wave problem for a fluid layer of finite depth in the presence of gravity. The method of multiple‐scale expansion is employed to obtain the Korteweg‐de Vries (KdV) equations for solitons, which describes the behavior of the system for free surface between air and water in a nonlinear approach. The solutions of the water wave problem split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as the solutions of KdV equations. The solution of KdV equations is obtained analytically by using a reliable modification of Laplace decomposition method (LDM), namely, the modified Laplace decomposition method (MLDM) is presented. This procedure is a powerful tool for solving large amount of nonlinear problems. The proposed method provides the solution as a series which may converge to the exact solution of the problem. Also, the convergence analysis of the proposed method is given. Finally, we observe that the elevation of the water waves is in form of traveling solitary waves. The horizontal and vertical of the velocity components have nonlinear characters.

Numerical algorithms for solutions of Korteweg–de Vries equation

AbstractThe nonlinear Korteweg–de Vries (KdVE) equation is solved numerically using both Lagrange polynomials based differential quadrature and cosine expansion‐based differential quadrature methods. The first test example is travelling single solitary wave solution of KdVE and the second test example is interaction of two solitary waves, whereas the other three examples are wave production from solitary waves. Maximum error norm and root mean square error norm are computed, and numerical comparison with some earlier works is done for the first two examples, the lowest four conserved quantities are computed for all test examples. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Numerical Solution of Korteweg-de Vries Equation by the Fourier Pseudospectral Method

In this paper, we present a numerical solution of one-dimensional Korteweg-de Vries equation with variant boundary conditions by the Fourier pseudospectral method. Four test problem with known exact solutions were studied to demonstrate the accuracy of the present method. An artificial viscosity was proposed to improve the accuracy of the numerical scheme. The obtained results were compared with the exact solution of each problem and found to be in good agreement with each other.

Traveling Wave Solution of Korteweg-de Vries Equation using He's Homotopy Perturbation Method

Article Traveling Wave Solution of Korteweg-de Vries Equation using He's Homotopy Perturbation Method was published on June 1, 2007 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 8, issue 2).

Solution of KdV equation by computer algebra

In this paper we use the real exponential approach by draw up Mathematica program to calculate the KdV (Korteweg–de Vries) equation. We get the same soliton solutions as conventional analytical methods of nonlinear equation.

On a higher dimensional miuratransform

We investigate a spatial modified Miura transform. To describe this transform we have to solve a non-linear first-order system of partial differential equations. This investigation will be done by the help of quaternionic analysis. The main goal is to find a hypercomplex factorization of the Schrodinger equation. In one dimension Miura'stransformation is needed to map solutions of the modified Korteweg-de Vries equation into solutions of Korteweg-de Vries equation.

Asymptotic Behaviour of Collisionless Shock in Nonlinear LC Circuit

Asymptotic property of collisionless shock has been examined experimentally using a nonlinear LC circuit which simulates the Toda lattice. The shock front velocity depends on the magnitude of a discontinuity as a collisional shock, but the collisionless shock is not stationary. That is, soliton like peaks are generated behind the shock front as an initial step voltage propagates in the circuit. The time interval between these peaks increases logarithmically in time but decreases in proportion to square root of an initial amplitude. The results are qualitatively explained by a solution of Korteweg-de Vries equation.