Pattern dynamics of higher-order rogue waves in the nonlinear Schrödinger–Boussinesq equation

Rogue wave patterns in the nonlinear Schrödinger–Boussinesq equation are analytically studied. Using bilinear Kadomtsev–Petviashvili (KP) reduction method, two types of higher-order rogue wave solutions in the nonlinear Schrödinger–Boussinesq equation are constructed. For the first type of rogue wave solutions where a certain quartic equation has one non-imaginary simple root, it is shown that when single internal parameter in bilinear expressions of rogue waves gets large, these waves would exhibit clear geometric patterns, which comprise fundamental (Peregrine) rogue waves arranged in shapes such as triangle, pentagon, heptagon and nonagon structures, with a possible lower-order rogue wave at the center. These rogue wave patterns are analytically determined from the root structure of the Yablonskii–Vorob’ev polynomial hierarchy through dilation, rotation, stretch, shear and translation. Comparison between analytical predictions of these rogue patterns and true solutions shows excellent agreement.