| Abstract: |
| We present a comparative study of the statistical properties of rarefied soliton gases within both integrable and non-integrable models from the Korteweg-de Vries (KdV) hierarchy. Focusing on multi-soliton solutions of the modified KdV equation and the modular Schamel equation, we show that bipolar soliton gases exhibit the formation of rogue waves, a feature absent in unipolar gases. This distinction is reflected in the evolution of higher-order statistics, with kurtosis increasing in the bipolar case and decreasing in the unipolar one. In integrable settings, the statistical characteristics relax to stationary states, whereas in non-integrable systems they remain time-dependent. We also observe inelastic energy transfer from smaller to larger solitons, leading to increasingly extreme wave events. The emergence of anomalously large structures, referred to as champion solitons, is discussed in the context of non-integrable dynamics. |
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