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

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  • Flamarion, Marcelo V.
  • Pelinovsky, Efim
  • Melnikov, Ioann

Abstract

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.

Suggested Citation

  • Flamarion, Marcelo V. & Pelinovsky, Efim & Melnikov, Ioann, 2025. "Spring–mass behavior of solitons under the influence of an external force field within the modified Korteweg–de Vries equation," Chaos, Solitons & Fractals, Elsevier, vol. 196(C).
    • Handle: RePEc:eee:chsofr:v:196:y:2025:i:c:s0960077925004357
    DOI: 10.1016/j.chaos.2025.116422
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    References listed on IDEAS

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    1. Marcelo V. Flamarion & Efim Pelinovsky, 2022. "Solitary Wave Interactions with an External Periodic Force: The Extended Korteweg-de Vries Framework," Mathematics, MDPI, vol. 10(23), pages 1-11, December.
    2. Kochurin, E.A., 2025. "Anisotropy of free-surface wave turbulence induced by a horizontal magnetic (electric) field," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    3. Flamarion, Marcelo V. & Pelinovsky, Efim, 2022. "Soliton interactions with an external forcing: The modified Korteweg–de Vries framework," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    4. Flamarion, Marcelo V. & Pelinovsky, Efim, 2024. "Auto-resonance process under the interaction of solitons with external force and dissipation," Chaos, Solitons & Fractals, Elsevier, vol. 189(P2).
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