Bright and dark rogue internal waves: The Gardner equation approach

We have found “bright and dark” solutions of the Gardner equation which can model internal rogue waves in three-layer fluids. We provide the first four “bright” and “dark” exact rational solutions to the Gardner equation. These are the lowest-order solutions of the corresponding hierarchies of rogue-wave solutions of this equation. They have been obtained from the rogue-wave solutions of a modified Korteweg-de Vries equation by using a Lorentz-type transformation. The maximal (and minimal) amplitudes and the background levels of these solutions for arbitrary order are deduced, based on the lowest-order examples. These solutions can be useful for explanations of extremely large amplitude internal waves in the ocean, as well as for abnormally large-amplitude waves in other areas of nonlinear physics, such as optics and dusty plasmas.

Figure 1

The first-order bright rational solution of the Gardner equation is given by Eq. (8). Here the background is $-1$. The maximum, at $t=0$, is 1, thus giving an amplitude of 2.

Figure 2

The first-order dark rational solution of the Gardner equation from Eq. (9). Here the background level is 0. Minimum at $t=0$ is $-2$. In order to isolate clearly dark rogue-wave solutions from bright ones, a distinct color scheme (yellow-green) is used for the dark rogue waves here and in all figures below.

Figure 3

The second-order rational solution of the GE given by Eq. (11). The “bright” rogue wave in the middle is clearly visible. Here the background is 0. Maximum at the origin is 2.

Figure 4

The “dark” second-order rogue-wave solution given by Eq. (14). Here the background is $-1$. The central minimum is $-3$.

Figure 5

Third-order rogue-wave solution of Gardner equation given by Eq. (15). Here the background level is $-1$ while the maximal amplitude at the origin is 3.

Figure 6

“Dark” third-order rational solution to the GE given by Eq. (16). This solution is inverted relative to Fig. 5 and shifted down. Here the background is 0. The central minimum is $-4$.

Figure 7

Fourth-order “bright” rogue wave of the Gardner equation, given by Eq. (A1). Here the background level is 0. Maximal amplitude is 4.

Figure 8

“Dark” fourth-order rational solution of the GE, given by Eq. (17) and Eq. (A2). This solution is inverted relative to Fig. 7 and shifted down. Here the background is $-1$. The central minimum is $-5$.