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In classical shallow water wave (SWW) theory, there exist two integrable one-dimensional SWW equation [HirotaSatsuma (HS) type and Ablowitz-Kaup-Newell-Segur (AKNS) type] in the Boussinesq approximation. In this paper, we mainly focus on the integrable SWW equation of AKNS type. The nonlocal symmetry in form of square spectral function is derived starting from its Lax pair. Infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter. These nonlocal symmetries can be localized and the SWW equation is extended to enlarged system with auxiliary dependent variables. Then Darboux transformation for the prolonged system is found by solving the initial value problem. Similarity reductions related to the nonlocal symmetry and explicit group invariant solutions are obtained. It is shown that the soliton-cnoidal wave interaction solution, which represents soliton lying on a cnoidal periodic wave background, can be obtained analytically. Interesting characteristics of the interaction solution between soliton and cnoidal periodic wave are displayed graphically.

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This paper is focused on a nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle whose action is a quadratic function of the derivatives of the field. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. The approach follows very closely the method of energy-dependent coordinates proposed by Bressan, Zhang and Zheng [6,7]. By introducing a new set of variables, which resolve all singularities due to the possible concentration of energy, the equation can be rewritten as a semilinear system. We construct the global weak solution by expressing the solution of the semilinear system in terms of the original variables.