Extreme water waves are very large waves that appear unexpectedly even under relatively calm conditions in the open ocean, and dispersive wave equations can describe the extreme water wave phenomena. In this paper, collision dynamics of breather and soliton molecule in the (2+1)-dimensional Sawada-Kotera equation have been investigated. Firstly, soliton molecules and breathers are analytically constructed under velocity resonance conditions of the \(N\) -soliton solutions. The collisions exhibit elastic behavior, forming X-shaped collisions with transient linear “stems” that split into dual V-shaped wavefronts; Secondly, asymptotic analysis reveals stationary localized stem structures: slopes and lengths remain time-invariant, and lengths of the stem structures are governed by coefficients \(A_{ij}\) . For breather-breather collisions, these invariants can be confirmed by trajectory equations; Thirdly, soliton molecule-breather collisions are reduced hierarchically: fourth-order collisions decompose into third-order soliton-breather collisions, preserving stem invariants and establishing sequential slope relationships; Finally, the methodology extends to higher-order collisions via iterative asymptotic reduction, consistently yielding elementary X-shaped collisions. This work unifies geometric invariants across solitonic and breather collisions in integrable systems.
