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Water-wave symbolic computation for the Earth, Enceladus and Titan: The higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations
In the Solar System, water and water waves are commonly seen: For the Earth, water is “at the core of sustainable development” and “at the heart of adaptation to climate change”; For the Enceladus, Cassini spacecraft discovers a possible global ocean of liquid water beneath an icy crust; For the Titan, Cassini spacecraft suggests an icy shell floating atop a global ocean. Shallow water waves near the ocean beaches or in the lakes can be described by the Boussinesq-Burgers-type equations. In this Letter, on the higher-order Boussinesq-Burgers system, symbolic computation helps us to go from the two-dimensional Bell polynomials to construct two non-auto-Bäcklund transformations and to proceed from the Painlevé-Bäcklund format to obtain four auto-Bäcklund transformations with some soliton solutions. All of our results are shown to be dependent on the constant coefficient in the system.