Rogue waves and rational solutions of the Hirota equation

The Hirota equation is a modified nonlinear Schrödinger equation (NLSE) that takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. In describing wave propagation in the ocean and optical fibers, it can be viewed as an approximation which is more accurate than the NLSE. We have modified the Darboux transformation technique to show how to construct the hierarchy of rational solutions of the Hirota equation. We present explicit forms for the two lower-order solutions. Each one is a regular (nonsingular) rational solution with a single maximum that can describe a rogue wave in this model. Numerical simulations reveal the appearance of these solutions in a chaotic field generated from a perturbed continuous wave solution.

Figure 1

(Color online) Breather solution of the modified KdV Eq. (4) for $p=0.45$ and $q=0.5$.

Figure 2

(Color online) First-order Hirota breather [Eq. (5)] for $a_1=0.2$ and ${\alpha }_3=0.2$.

Figure 3

(Color online) First-order rational solution of Hirota equation [Eq. (9)] for ${\alpha }_3=0.2$. The arbitrary real parameter ${\alpha }_3$ is responsible for the skew angle of the “ridge” of this solution relative to the axes.

Figure 4

(Color online) Second-order Hirota rational soliton [Eq. (26)] for ${\alpha }_3=0.17$.

Figure 5

(Color online) Contour plot [(dimensionless) $t$ horizontal and $x$ vertical] of the second-order Hirota rational soliton [Eq. (26)] for ${\alpha }_3=0.17$.

Figure 6

Numerical simulations for the Hirota equation, showing the field maxima on evolution. Parameters of the simulation are presented inside of this figure.

Figure 7

(Color online) Numerical solution of the Hirota equation around a maximum of the field which approximately equals to 5, for ${\alpha }_3=0.17$ (absolute maximum in Fig. 6). Note its similarity to the rogue wave shown in Fig. 5.