Employing the dynamics of poles in the complex plane to describe properties of rogue waves: case studies using the Boussinesq and complex modified Korteweg–de Vries equations

The dynamics and properties of rogue waves of two classical evolution equations are studied in terms of trajectories of the poles of the exact solutions, by analytically continuing the spatial variable to be complex. The Boussinesq equation describes the motion of hydrodynamic waves in two opposite directions in the shallow water regime. The complex modified Korteweg–de Vries equation is relevant for wave packets in nonlinear media governed by a higher-order nonlinear Schrödinger equation, for the special case where second-order dispersion and cubic self-interaction are absent. On examining the movement of poles of the exact solutions for rogue waves, the real parts of the poles correlate well with the locations of maximum displacements in the physical space. This phenomenon holds for high precision numerically for the first-, second- and third-order rogue waves of the Boussinesq equation. A similar principle is also valid for the first- and second-order rogue waves of the complex modified Korteweg–de Vries equation. The imaginary parts of the poles can generate useful information too. For these two evolution models, a smaller imaginary part in the complex plane is associated with a larger amplitude of the rogue wave in physical space. An empirical formula is proposed which works well for the three lowest orders of rogue waves of the Boussinesq equation.