Bipolar Solitary Wave Interactions within the Schamel Equation

Abstract

Pair soliton interactions play a significant role in the dynamics of soliton turbulence. The interaction of solitons with different polarities is particularly crucial in the context of abnormally large wave formation, often referred to as freak or rogue waves, as these interactions result in an increase in the maximum wave field. In this article, we investigate the features and properties of bipolar solitary wave interactions within the framework of the non-integrable Schamel equation, contrasting them with the integrable modified Korteweg-de Vries (mKdV) equation. We show that in bipolar solitary wave interactions involving two solitary waves with significantly different amplitudes in magnitude, the behavior closely resembles what is observed in the mKdV equation. However, when solitary waves have similar amplitudes in modulus, the maximum value of their interaction remains less than the sum of their initial amplitudes. This distinguishes these interactions from integrable models, where the resulting impulse amplitude equals the sum of the soliton amplitudes before interaction. Furthermore, in the Schamel equation, smaller solitary waves can transfer some energy to larger ones, leading to an increase in the larger soliton amplitude and a decrease in the smaller one amplitude. This effect is particularly prominent when the initial solitary waves have similar amplitudes. Consequently, large solitary waves can accumulate energy, which is crucial in scenarios involving soliton turbulence or soliton gas, where numerous solitons interact repeatedly. In this sense, non-integrability can be considered a factor that triggers the formation of rogue waves.