Order Korteweg-de Vries Equation Research Articles

The generalized Korteweg-de Vries equation for interacting phonons in a coherent state basis

The higher order Korteweg-de Vries (KdV) equation of interacting phonons is obtained for a one-dimensional anharmonic lattice where fifth order anharmonicity in the potential is taken and justified from the classical continuum approximation method. The possible solitary wave solutions are discussed.

On a periodic boundary value problem with a finite number of gaps in its spectrum

(--co, &>, (4 9 n;>, (A, 3 A>, t&T WY (3) are called instability intervals. All but the first interval in (3) are finite and may shrink to a point under special conditions. Erdelyi [3] discovered situations where all but a finite number of finite instability intervals vanish when q(t) is a suitable elliptic function. Lax [9] showed that a function q(t) which satisfies the nth order Korteweg-de Vries equation requires (1) to have no more than n non-vanishing finite instability intervals. Conversely, if all finite instability intervals vanish, then q(t) in (1) is necessarily a constant. This fact was proved by Borg [ 11, Hochstadt [ 71 and Ungar [ 121. When all but n finite instability intervals vanish q(t) satisfies a differential equation of the form

N-Soliton Solution of the Higher Order Wave Equation for a Fluid of Finite Depth

Exact N -soliton solution of the higher order wave equation for a stratified fluid layer of finite depth is obtained by employing Hirota's bilinear transformation method. The solution reduces to the N -soliton solution of the higher order Korteweg-de Vries equation in the shallow water limit and to that of the higher order Benjamin-One equation in the deep water limit. The asymptotic form of the solution for large time is derived.

The prolongation structure of a higher order Korteweg-de Vries equation

We have used Wahlquist & Estabrook’s prolongation method to construct a Lie algebra for the higher order Korteweg de Vries (K.deV.) equation q 4 x + α qq 2 x + β q 2 x + γ q 3 ) x + q t = 0 with ( q nx =

Dynamical Processes of the Dressed Ion Acoustic Solitons

Dynamical processes of the dressed ion acoustic solitons have been numerically analyzed by solving the coupled set of the first order Korteweg-de Vries equation and the second order equation, which has been derived on a basis of the reductive perturbation theory. A single soliton dressed by clouds of thc second order perturbation potential exhausts ion acoustic wave behind when it travels down free space. On the other hand, in collision processes of two dressed solitons, the clouds around the Korteweg-da Vries soliton core redistribute themselves in such a way to equalize the heights of colliding pairs. Structure of the clouds suggests that effective attractive interaction acts between the colliding pairs of the dressed solitons.

A Bäcklund Transformation for a Higher Order Korteweg-De Vries Equation

A Backlund transformation in bilinear forms is presented for a higher order Korteweg-de Vries equation, which is a natural extension of the Korteweg-de Vries equation written in a bilinear form. The Backlund transformation in ordinary forms and the inverse scattering scheme are given for this higher order equation. Also, the first eleven of the infinity of conservation quantities for this equation are derived from the Backlung transformation.

Soliton solutions and the higher order Korteweg–de Vries equations

A prolongation structure is determined for a single equation from which the Korteweg–de Vries and all of its higher order forms can be derived. As result, inverse scattering problems for all of the equations are determined simultaneously.