Quantum mechanics

Rogue wave

Algebraic number

Korteweg–de Vries equation

Applied mathematics

Elliptic function

Periodic function

Eigenvalues and eigenvectors

Jacobian matrix and determinant

Physics

Nonlinear system

Mathematics

Eigenfunction

Mathematical analysis

Journal of Nonlinear Science

We address the most general periodic travelling wave of the modified Korteweg–de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there exist three symmetric pairs of simple eigenvalues away from the imaginary axis, and we give a new representation of the second non-periodic solution to the Lax equations for the same eigenvalues. We show that Darboux transformations with the non-periodic solutions to the Lax equations produce rogue waves on the periodic background, which are either brought from infinity by propagating algebraic solitons or formed in a finite region of the time-space plane.

Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background