Painlevé analysis and higher-order rogue waves of a generalized (3+1)-dimensional shallow water wave equation

Considering the importance of ever-increasing interest in exploring localized waves, we investigate a generalized (3+1)-dimensional Hirota-Satsuma-Ito equation describing unidirectional propagation shallow-water waves and perform Painlev\'e analysis to understand its integrability nature. We construct explicit form higher-order rogue wave solutions by adopting Hirota's bilinearization polynomial functions. Further, explore their dynamics detail, depicting different pattern formation that reveal potential advantages with available arbitrary constants manipulation mechanism. Particularly, demonstrate existence singly-localized line-rogue doubly-localized multiple (single, triple, sextuple) structures generating triangular pentagon type geometrical patterns controllable orientations can be altered appropriately tuning parameters. The presented will an essential inclusion context higher-dimensional systems.