Number: Mech 2/90
Author(s): ENGELBRECHT, J.
Title: Solutions to the perturbed KdV equation. 32 p.
Language: English
ABSTRACT. The perturbation to the KdV equation is caused by an external
force, i.e. in mathematical terms the governing equation has a r.h.s. which in
this paper is taken in the form of a cubic polynomial with respect to the
dependent variable. Under such an excitation a soliton may be amplified in the
course of propagation.Two asymptotic approaches are used to describe the
amplitude changes. Beside the implicit solutions obtained, the numerical
solutions to the amplitude equations are given and analyzed. The analysis shows
the existence of stationary states for a perturbed soliton under the given
external force. The stationary solutions to the perturbed KdV equation are found
by the numerical integration. These solutions are characterized by slow envelope
oscillations which at weak perturbations die out for large times. The period of
these oscillations depends upon the value of the small parameter for the given
r.h.s. The phase portraits u-u' and u-u'' differ considerably from those in the
classical unperturbed case. Strong perturbation, however, leads to the loss of
stability.