Bäcklund transformation, conservation laws, and inverse scattering transform of a model integrodifferential equation for water waves

The Backlund transformation (BT), an infinite number of conservation laws, and the inverse scattering transform (IST) of a model integrodifferential equation for water waves in fluids of finite depth [Y. Matsuno, J. Math. Phys. 29, 49(1989)] are constructed by employing the bilinear transformation method. The model equation is also shown to pass the Painleve test. These facts prove the complete integrability of the equation. Both the deep‐ and shallow‐water limits of various results thus obtained are then investigated in detail. In addition, a new method to evaluate conserved quantities for pure N‐soliton is developed by utilizing actively the time part of the BT. It is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the Benjamin–Ono and the Korteweg–de Vries equations. The most important problem left open in this paper is to solve various IST equations.