Long waves in oceanic shallow water: Symbolic computation on the bilinear forms and Backlund transformations for the Whitham-Broer-Kaup system

For the Earth, water is at the core of sustainable development and at the heart of adaptation to climate change. For the Enceladus, the sixth-largest moon of the Saturn, Cassini spacecraft discovers a possible global ocean of liquid water beneath an icy crust. Oceanic water waves are one of the most common phenomena in nature. Hereby, on the Whitham-Broer-Kaup system for the dispersive long waves in the oceanic shallow water, with respect to the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, our binary Bell polynomials and symbolic computation lead to two sets of the bilinear forms, two sets of the N-soliton solutions and two sets of the auto-Backlund transformations with the sample solitons, where N is a positive integer. Our bilinear forms and auto-Backlund transformations are different from those reported in the existing literatures. All of our results are dependent on the coefficients in the system which represent different oceanic-water-wave dispersion/diffusion powers.