Rogue-wave solutions of the Zakharov equation

Using the bilinear transformation method, we derive general rogue-wave solutions of the Zakharov equation. We present these $N$th-order rogue-wave solutions explicitly in terms of $N$th-order determinants whose matrix elements have simple expressions. We show that the fundamental rogue wave is a line rogue wave with a line profile on the plane $(x,y)$ arising from a constant background at $t\ll0$ and then gradually tending to the constant background for $t\gg0$. Higher-order rogue waves arising from a constant background and later disappearing into it describe the interaction of several fundamental line rogue waves. We also consider different structures of higher-order rogue waves. We present differences between rogue waves of the Zakharov equation and of the first type of the Davey–Stewartson equation analytically and graphically.