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In classical shallow water wave (SWW) theory, there exist two integrableone-dimensional SWW equation [Hirota-Satsuma (HS) type andAblowitz-Kaup-Newell-Segur (AKNS) type] in the Boussinesq approximation. Inthis paper, we mainly focus on the integrable SWW equation of AKNS type. Thenonlocal symmetry in form of square spectral function is derived starting fromits Lax pair. Infinitely many nonlocal symmetries are presented by introducingthe arbitrary spectrum parameter. These nonlocal symmetries can be localizedand the SWW equation is extended to enlarged system with auxiliary dependentvariables. Then Darboux transformation for the prolonged system is found bysolving the initial value problem. Similarity reductions related to thenonlocal symmetry and explicit group invariant solutions are obtained. It isshown that the soliton-cnoidal wave interaction solution, which representssoliton lying on a cnoidal periodic wave background, can be obtainedanalytically. Interesting characteristics of the interaction solution betweensoliton and cnoidal periodic wave are displayed graphically.

Nonlocal symmetry, Darboux transformation and soliton-cnoidal wave interaction solution for the shallow water wave equation