KdV-like soliton gas: similarity and difference in integrable and non-integrable models

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.