SOLUTIONS TO THE PERTURBED KDV EQUATION

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 equations 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.