A Numerical Solution for the Coexisting Field of Surface and Internal Solitary Waves

The numerical solutions for the coexisting fields of surface and internal solitary waves have been obtained, where the set of nonlinear equations based on the variational principle for steady waves are solved using the Newton- Raphson method. The relative phase velocity of surface-mode solitary waves is smaller in the coexisting fields of surface and internal solitary waves than in the cases without the coexistence of internal waves. The relative phase velocity of internal-mode solitary waves is also smaller in the coexisting fields of surface and internal solitary waves than in the cases without surface waves. The interfacial position of an internal mode internal solitary wave in a coexisting field of surface and internal waves can exceed the critical level determined in the corresponding case without a surface wave. The wave height ratio between internal-mode surface and internal solitary waves is smaller than the corresponding linear shallow water wave solution, and the difference increases, as the relative wave height of internal-mode internal solitary waves is increased.