Kortweg-de Vries Research Articles

Excitation of ion-acoustic rarefactive solitons in a two-electron-temperature plasma

Rarefactive ion-acoustic solitons have been observed in a two-electron-temperature plasma. Some of the characteristics can be interpreted by the solution of the Kortweg–de Vries (K–dV) equation. The Mach number of the solitons is a function of the temperature ratio of hot and cold components.

The quantum inverse method and the monodromy matrix in integrable field theories

Under investigation in this paper is a variable-coefficient modified Kortweg-de Vries (vc-mKdV) model describing certain situations from the fluid mechanics, ocean dynamics and plasma physics. N-fold Darboux transformation (DT) of a variable-coefficient Ablowitz–Kaup–Newell–Segur spectral problem is constructed via a gauge transformation. Multi-solitonic solutions in terms of the double Wronskian for the vc-mKdV model are derived by the reduction of the N-fold DT. Three types of the solitonic interactions are discussed through figures: (1) Overtaking collision; (2) Head-on collision; (3) Parallel solitons. Nonlinear, dispersive and dissipative terms have the effects on the velocities of the solitonic waves while the amplitudes of the waves depend on the perturbation term.

The Bäcklund trasnformation for the integrable many-body systems

Under investigation in this paper is a variable-coefficient modified Kortweg-de Vries (vc-mKdV) model describing certain situations from the fluid mechanics, ocean dynamics and plasma physics. N-fold Darboux transformation (DT) of a variable-coefficient Ablowitz–Kaup–Newell–Segur spectral problem is constructed via a gauge transformation. Multi-solitonic solutions in terms of the double Wronskian for the vc-mKdV model are derived by the reduction of the N-fold DT. Three types of the solitonic interactions are discussed through figures: (1) Overtaking collision; (2) Head-on collision; (3) Parallel solitons. Nonlinear, dispersive and dissipative terms have the effects on the velocities of the solitonic waves while the amplitudes of the waves depend on the perturbation term.

Complete integrability of the Kortweg-de Vries equation under perturbation around its solution: Lie-Backlund symmetry approach

It is shown that when the Korteweg-de Vries equation is perturbed about a particular solution the resulting evolution equations to all order of perturbations admit infinitely many Lie-Backlund symmetries. The corresponding commuting constants of motion are derived and thereby complete integrability can be established.

Simple multisoliton solutions

Malfliet has proposed a simple solution method for the Kortweg–de Vries equation. We show that a slight modification in this method makes it effective in obtaining multisoliton solutions for a number of other nonlinear dispersive wave equations.

Connection between the infinite sequence of Lie–Bäcklund symmetries of the Korteweg–de Vries and sine-Gordon equations

From the observation that the infinite sequence of Lie–Bäcklund symmetries of the potential modified Kortweg–deVries (PMK–dV) and the sine–Gordon (s–G) equations are identical, it is shown that there exists a simple connection between the Lie–Bäcklund symmetries (written in the form of evolution equations) of the Korteweg–deVries (K–dV) and s–G equations. Further, this connection is similar to the one obtained by Chodos for the conserved quantities of K–dV and s–G equations. We also point out that the result of Chodos can be realized from the equality of conserved densities of PMK–dV and s–G systems.

On the pseudo-parabolic regularization of the generalized Kortweg-de Vries equation

On the pseudo-parabolic regularization of the generalized Kortweg-de Vries equation

Solitary waves in a finite depth fluid

It is shown that U( xi )=u0(cosh2 a xi +(ab)-2 sinh2 a xi )-1 is the exact stationary wave solution to the Whitham equation in a two-layer fluid of finite depth and represents the natural connection between the Benjamin-Ono deep water (a to 0) and Kortweg-de Vries shallow water (b to infinity ) theories.

Asymptotic development of finite amplitude cylindrical ion acoustic waves

A finite amplitude cylindrical ion acoustic wave develops at large distances from the exciter an oscillatory structure which resembles an Airy function but its amplitude oscillations are asymmetric. This phenomenon can be described by solutions of the linearized Kortweg-de Vries equation in self-similar variables. This equation has been solved for integral values of a parameter gamma . New solutions which can be represented as the Weyl transform of the Airy function are obtained for fractional values of gamma . The measured rate of spatial damping of the oscillations and the variation of their period with respect to distance and Debye length agree well with those predicted by these solutions.

Propagation of Drift Waves of Small but Finite Amplitude

It is shown that the drift wave of small but finite amplitude propagating in a plasma of hot electrons and cold ions is governed by a modified Kortweg-de Vries equation in two-dimensional space comprising the external magnetic field and the direction normal to a density gradient. The solitary wave is obtained explicitly.

Bäcklund Transformation for Solutions of the Korteweg-de Vries Equation

We present a B\"acklund transformation operating on solutions of the Kortweg-de Vries equation. It provides a unifying approach to the theory of solitary waves (solutions). Recursive application of the transformation to any solution of the Korteweg-de Vries equation generates a hierarchy of solutions for which an algebraic recursion relation is obtained. Every such hierarchy (one of which is the known family of pure multisoliton solutions) thus admits a nonlinear superposition principle.

Reductive Perturbation Method in Nonlinear Wave Propagation II. Application to Hydromagnetic Waves in Cold Plasma

The Reductive Perturbation Method established in a previous paper is applied to the analysis of hydromagnetic wave of small but finite amplitude propagating in a cold plasma. It is shown that the basic system of equations belongs to the exceptional case defined in that paper and in the lowest order of perturbation it can be reduced to the Kortweg-de Vries equation for all angles of propagation.