Patterns of rogue waves in the sharp-line Maxwell–Bloch system

The Maxwell–Bloch system describes light-matter interactions in a semi-infinitely long one dimensional two-level optical medium. Rogue wave patterns in the Maxwell–Bloch system under sharp-line limit are analytically studied. 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 and shear. It is also shown that when multiple internal parameters in the rogue wave solutions get large, new rogue wave patterns would arise, including heart-shaped structures, fan-shaped structures, and many others. Analytically, these patterns are determined by the root structure of the Adler–Moser polynomials through a linear transformation. Comparison between analytical predictions of these rogue patterns and true solutions shows excellent agreement.