  In classical shallow water wave (SWW) theory, there exist two integrable
one-dimensional SWW equation [Hirota-Satsuma (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.
