CTE Solvability, Nonlocal Symmetries and Exact Solutions of Dispersive Water Wave System

A consistent tanh expansion (CTE) method is developed for the dispersion water wave (DWW) system. For the CTE solvable DWW system, there are two branches related to tanh expansion, the main branch is consistent while the auxiliary branch is not consistent. From the consistent branch, we can obtain infinitely many exact significant solutions including the soliton-resonant solutions and soliton-periodic wave interactions. From the inconsistent branch, only one special solution can be found. The CTE related nonlocal symmetries are also proposed. The nonlocal symmetries can be localized to find finite Backlund transformations by prolonging the model to an enlarged one.