General high-order rogue wave to NLS-Boussinesq equation with the dynamical analysis

Abstract: General high-order rogue waves of the NLS-Boussinesq equation are obtained by the KP-hierarchy reduction theory. These rogue waves are expressed with the determinants, whose entries are all algebraic forms. It is found that the fundamental rogue waves can be classified three patterns: four-petals state, dark state, bright state by choosing different parameter $\alpha$ values. As the evolution of the parameter $\alpha$. An interesting phenomena is discovered: the rogue wave changes from four-petals state to dark state, whereafter to the bright state, which are consistent with the change of the corresponding critical points to the function of two variables. Furthermore, the dynamics of second-order and third-order rogue waves are presented in detail, which can be regarded as the nonlinear superposition of the fundamental rogue waves.