Soliton Molecule and State Transition Waves in the $$(n+1)$$-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani Equation

This paper investigates nonlinear localized waves and their transition dynamics for the $$(n+1)$$-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani (KdVSKR) equation. Focusing on the representative $$(3+1)$$-dimensional case, bound-state structure and soliton molecules are generated by means of the Hirota bilinear method and introducing an appropriate phase-velocity resonance condition. In addition, through the complex-conjugate parameter reductions, the first-order breather waves are derived. On this basis, the conversion mechanism is established through the geometric regulation of characteristic line. Enforcing parallelism of the characteristic line in selected planes, various state transition waves are produced from the first-order breather. Consequently, a variety of transition patterns are given, including quasi-periodic waves, multi-peak localized waves, oscillatory M-shaped waves, M-shaped waves, W-shaped waves, and quasi-anti-dark solitons, with distinct behaviors depending on the spatial plane in which the transition conditions are imposed. The presented results enrich the wave classification of the $$(n+1)$$-dimensional KdVSKR equation, providing a unified mechanism for constructing soliton molecules as well as higher-dimensional state-transition dynamics.