Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation

Ludovick Gagnon

doi:10.3934/dcds.2017061

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2017, Volume 37, Issue 3: 1489-1507. Doi: 10.3934/dcds.2017061

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Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation

Ludovick Gagnon

Sorbonnes Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis, Lions F-75005, Paris, France

Received: October 2015

Revised: October 2016

Published: May 2017

LG is supported by FQRNT and by ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7).

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Abstract

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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Mathematics Subject Classification: Primary:35C08, 76B15.

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Figure 1. A soliton solution of (1) represented in the Cartesian coordinates.

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Figure 2. The orbits of water particles obtained from the experimental measurements of the polystyrene beads motions at different water levels $b$ in the four experimental wave cases (a) $h_0$=20cm, $a$=7.07cm; (b) $h_0$=20cm, $a$=8.56cm; (c) $h_0$=30cm, $a$=5.46cm; (d) $h_0$=30cm, $a$=7.56cm.

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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 $2$-solitons solution before the interaction and b) is the state after the interaction. The frame is fixed at the speed of the slower soliton.

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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.

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Figure 5. Total displacement ($X$) in the x variable and maximal displacement ($Y$) in the y variable with respect to the initial vertical position above the flat bottom of the particle $b$ for the first order velocity field ($1^{st}$), the higher velocity field (Hi.) and the experimental results (Exp.).

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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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