General high-order rogue waves to nonlinear Schrödinger–Boussinesq equation with the dynamical analysis

General high-order rogue waves of the nonlinear Schrödinger–Boussinesq equation are obtained by the KP-hierarchy reduction theory, and the N-order rogue waves are expressed with the determinants, whose entries are all algebraic forms, which is shown in the theorem. It is found that the fundamental first-order rogue waves can be classified into three patterns: four-petal state, dark state, bright state by choosing different values of parameter α. An interesting phenomenon is discovered as the evolution of the parameter α: the rogue wave changes from four-petal state to dark state, whereafter bright state, which are consistent with the change in the corresponding critical points to the function of two variables. Furthermore, the dynamical property of second-order and third-order rogue waves is plotted, which can be regarded as the nonlinear superposition of the fundamental first-order rogue waves.