Higher-order (2+4) Korteweg-de Vries - like equation for interfacial waves in a symmetric three-layer fluid

Physics of Fluids. 2011. Vol. 23. No. 11. P. 116602-1-13–116602-13.

O.E. Kurkina, A.A. Kurkin, T. Soomere, E.N. Pelinovsky, Рувинская Е. А.

We address a specific but possible situation in natural water bodies when the three-layer stratification has a symmetric nature, with equal depths of the uppermost and the lowermost layers. In such case, the coefficients at the leading nonlinear terms of the modified Korteweg-de Vries (mKdV) equation vanish simultaneously. It is shown that in such cases there exists a specific balance between the leading nonlinear and dispersive terms. An extension to the mKdV equation is derived by means of combination of a sequence of asymptotic methods. The resulting equation contains a cubic and a quintic nonlinearity of the same magnitude and possesses solitary wave solutions of different polarity. The properties of smaller solutions resemble those for the solutions of the mKdV equation whereas the height of the taller solutions is limited and they become table-like. It is demonstrated numerically that the collisions of solitary wave solutions to the resulting equation are weakly inelastic: the basic properties of the counterparts experience very limited changes but the interactions are certainly accompanied by a certain level of radiation of small-amplitude waves.

Ключевые слова: Korteweg-de Vries equation, solitons, stratified flow, surface waves (fluid), ветровые, длинные неприливные и внутренние волны, моделирование физических процессов, математические модели волновых движений тяжелой жидкости, идеальная несжимаемая жидкость, теория волн и колебаний жидкости