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    KdV-like soliton gas: similarity and difference in integrable and non-integrable models
    (Elsevier BV, 2025-11-01)
    A comparison of the statistical characteristics of a rarefied soliton gas is carried out within the framework of integrable and non-integrable equations from the Korteweg-de Vries (KdV) hierarchy. As examples, multi-soliton solutions of the modified KdV equation, and the modular Schamel equation are considered. A common property of the dynamics of bipolar solitons is the formation of rogue waves, which do not occur in unipolar gases. The fourth moment of the wave field (kurtosis) increases compared to the initial value in the case of a bipolar gas, and decreases for a unipolar gas. In the case of integrable KdV equations, the characteristics of the soliton gas reach a stationary level, while in non-integrable equations they remain functions of time. The inelastic transfer of energy from small solitons to large ones occur, and large waves become “more extreme” against the background of small solitons. The tendency of the occurrence of an anomalously large wave (soliton - champion) in non-integrable systems are discussed.
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    Dynamics of Irregular Wave Fields in the Schamel Equation Framework
    (Pleiades Publishing, 2025-02-01)
    The present paper is devoted to the study of the dynamics of narrowband wave fields within the nonintegrable Schamel equation, which plays an important role in plasma physics, wave dynamics in metamaterials, and electrical circuits. A Monte Carlo approach is used to obtain a large number of random independent realizations of the wave fields, allowing for an investigation of the evolution of the following statistical characteristics: spectra, moments, and distribution functions. The simulations are conducted for different values of the Ursell number (the ratio of nonlinearity to dispersion) to study the impact of nonlinearity and dispersion on the processes under consideration. The features of freak waves appeared in the random wave fields are discussed.
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