Spring–mass behavior of solitons under the influence of an external force field within the modified Korteweg–de Vries equation

Chaos, Solitons and Fractals. 2025. Vol. 196. Article 116422.

Flamarion M., Pelinovsky E., Melnikov I.

We investigate the interaction of solitons with an external periodic field within the framework of the modified Korteweg–de Vries (mKdV) equation. In the case of small perturbation a simple dynamical system is used to
describe the soliton behavior. Equilibrium points of this dynamical system are computed when the external
force travels at a constant speed. Assuming that the external force moves with sinusoidal speed, we demonstrate
that the soliton behavior is qualitatively similar to the constant-speed case. Besides, a resonant frequency is
derived from the asymptotic theory without using the classical broad force approximation. The results obtained
from the dynamical system are compared with fully nonlinear direct numerical simulations, which reveal that
the soliton solution exhibits spiral-like behavior in the soliton amplitude versus soliton phase space. Moreover,
when the external force oscillates at the resonant frequency, the trajectories in the soliton phase versus soliton
amplitude exhibit chaotic behavior.

Keywords: asymptotics, нелинейные волны, солитоны, solitons, nonlinear waves, асимптотики, trapped waves, захваченные волны

Publication based on the results of: Качественная теория систем обыкновенных уравнений и уравнений в частных производных (2026)