  Shallow water waves phenomena in nature attract the attention of scholars and
play an important role in fields such as tsunamis, tidal waves, solitary waves,
and hydraulic engineering. Hereby,
fortheshallowwaterwavesphenomenainvariousnaturalenvironments,
westudytheKdV-Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation. Based on the
binary Bell polynomial theory, a new general bilinear B\"acklund
transformation, Lax pair and infinite conservation laws of the KdV-CBS equation
are derived, and it is proved that it is completely integrable in Lax pair
sense. Various types of mixed solutions are constructed by using a combination
of Homoclinic test method and symbolic computations. These findings have
important significance for the discipline, offering vital insights into the
intricate dynamics of the KdV-CBS equation. We hope that our research results
could help the researchers understand the nonlinear complex phenomena of the
shallow water waves in oceans, rivers and coastal areas. Furthermore, the
present work can be directly applied to other nonlinear equations.
