Elliptic-rogue waves and modulational instability in nonlinear soliton equations.

We present elliptic-rogue wave solutions for integrable nonlinear soliton equations in rational form by elliptic functions. Unlike solutions generated on the plane wave background, these solutions depict rogue waves emerging on elliptic function backgrounds. By refining the modified squared wave function method in tandem with the Darboux-Bäcklund transformation, we establish a quantitative correspondence between elliptic-rogue waves and the modulational instability. This connection reveals that the modulational instability of elliptic function solutions triggers rational-form solutions displaying elliptic-rogue waves, whereas the modulational stability of elliptic function solutions results in the rational-form solutions exhibiting the elliptic solitons or elliptic breathers. Moreover, this approach enables the derivation of higher-order elliptic-rogue waves, offering a versatile framework for constructing elliptic-rogue waves and exploring modulational stability in other integrable equations.