Shallow-water soliton dynamics beyond the Korteweg–de Vries equation

An alternative way for the derivation of the new Korteweg–de Vries (KdV)-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It is obtained in the second-order perturbative approach in the weakly nonlinear, dispersive, and long wavelength limit. Only treating all these terms in the second-order perturbation theory made the derivation of this KdV-type equation possible. The motion of a wave, which starts as a KdV soliton, is studied according to the new equation in several cases by numerical simulations. The quantitative changes of a soliton's velocity and amplitude appear to be directly related to bottom variations. Changes of the soliton's velocity appear to be almost linearly anticorrelated with changes of water depth, whereas correlation of variation of soliton's amplitude with changes of water depth looks less linear. When the bottom is flat, the new terms narrow down the family of exact solutions, but at least one single soliton survives. This is also checked by numerics.

Figure 1

Schematic view of the geometry of the shallow-water wave problem.

Figure 2

Time evolution of the initial KdV soliton according to Eq. (18) for bottom shape function $h_{-}\left(x\right)$ and $\alpha =\beta =0.1,\delta =0.2$. See detailed explanations in the text.

Figure 3

The same as described in the legend of Fig. 2 but for the bottom shape function $h_{+}\left(x\right)$.

Figure 4

Anticorrelations between the soliton's velocity and the water depth. Dots indicate the average velocities of the tops of solitons for given positions, lines with the same color the shape of the bottom function.

Figure 5

Correlations between the soliton's amplitude and the water depth. Case $\alpha =\beta =0.1,\delta =0.2$. Dots indicate amplitudes of solitons for given positions, lines with the same color the shape of the bottom function.

Figure 6

Distortions of solitary wave due to the motion over an extended obstacle for $\alpha =\beta =0.1,\delta =0.2$. See details in the text.

Figure 7

Distortions of a solitary wave due to motion over an extended well. Case $\alpha =\beta =0.1,\delta =0.2$.

Figure 8

Comparison of shapes of the main part of the waves at $t=440,520,600$ from Fig. 6 (left) and Fig. 7 (right) with the shape of the KdV soliton (sol) after shifts to the same position.

Figure 9

Time evolution of the exact soliton Eq. (34) according to the the second-order Eq. (20) obtained in numerical simulations.

Figure 10

The same as described in the legend of Fig. 6 but for initial condition given by the exact second-order soliton Eq. (34).

Figure 11

The same as described in the legend of Fig. 7 but for initial condition given by the exact second-order soliton Eq. (34).