The qualitative properties of the particle trajectories of the $N$-solitons solution of the KdV equation are recovered from the first order velocity field by the introduction of the stream function. Numerical simulations show an accurate depth dependance of the particles trajectories for solitary waves. Failure of the free surface kinematic boundary condition for the first order type velocity field is highlighted.
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Figure 3.
Interaction between two solitons. The cross (resp. circle) represents the position of the maximum of the faster (resp. slower) soliton if no interaction would have occured. Figure a) is the state of the
Figure 4. Comparison of the numerical approximation of the particle trajectories for the first order velocity field (top left) and the higher order velocity field (top right). Zoom on the end of the particle trajectories for the first order velocity field (bottom left) and the higher order velocity field (bottom right). The depth of the fluid is 30 cm and the height of the solitary wave is 5.46 cm. The dashed line represents the undisturbed water surface.
Figure 5.
Total displacement (
Figure 6. Numerical approximation of the particle trajectories for the 2-solitons solution. The particles trajectories are in black, the initial position of the 2-solitons is in dashed black and the final position is in gray. The height of the soliton in front is 0.4cm and the soliton behind is 0.3cm. The depth of the water is 1cm.
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