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Author: search.filters.author.Marcelo V. FlamarionSubject: search.filters.subject.Asymptotic theory

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    Auto-resonance process under the interaction of solitons with external force and dissipation
    (Elsevier Ltd, 2024-12-01)
    Algebraic soliton interactions with an external force in the presence of Reynolds viscosity is investigated. In the absence of an external force, the soliton amplitude decays over time. However, when an external force is introduced, it acts as a restoring force, and in some cases, the soliton's amplitude is preserved. A dynamical system that governs the soliton amplitude and its crest position is obtained assuming a weak force and weak viscosity. For an external force with a Gaussian shape, the dynamical system has two equilibrium points, namely, a saddle and a stable spiral. Asymptotic results are compared with direct numerical simulations, and a strong qualitative agreement is observed. The stable spiral predicted by the asymptotic theory is stable in the sense that soliton solutions with a chosen amplitude and crest position near the spiral point are attracted to it, preserving their amplitude and location.
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    Solitons in dissipative systems subjected to random force within the Benjamin–Ono type equation
    (Elsevier Ltd, 2024-10-01)
    Solitary wave dynamics is investigated under the assumption of small dissipation and an external random force. Through a change of variables, the problem becomes homogeneous, allowing for the derivation of asymptotic algebraic soliton solutions. This change of variables makes the randomness manifest primarily on the soliton phases. Consequently, the averaged soliton field and the statistical moments can be computed analytically, assuming that the phase follows a uniform distribution. In the absence of Reynolds dissipation, we show that the soliton-averaged field tends to spread and dampen as the dispersion increases. In addition, in the presence of Reynolds dissipation, we demonstrate that algebraic solitons can transition between thick and thin soliton states. Moreover, when there is viscosity in the upper moving layer, the averaged soliton field exhibits a dynamic evolution from soliton to thick soliton to soliton, contingent upon the parameter settings.
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    Soliton dynamics in random fields: The Benjamin-Ono equation framework
    (Elsevier BV, 2025-08-01)
    Algebraic soliton interactions with a periodic or quasi-periodic random force are investigated via the Benjamin-Ono equation, which models internal waves in a two-layer fluid. The random force is modeled as a Fourier series with a finite number of modes and random phases uniformly distributed, while its frequency spectrum has a Gaussian shape centered at a peak frequency. The expected value of the averaged soliton wave field is computed asymptotically and compared with numerical results, with a strong agreement shown. We identify parameter regimes where the averaged soliton field splits into two steady pulses and a regime where the soliton field splits into two solitons traveling in opposite directions. In the latter case, the averaged soliton speeds are variable. In both scenarios, the soliton field is damped by the external force. Additionally, we identify a regime where the averaged soliton exhibits the following behavior: it splits into two distinct solitons and then recombines to form a single soliton. This motion is periodic over time.
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