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March  2017, 37(3): 1489-1507. doi: 10.3934/dcds.2017061

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

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

Received  October 2015 Revised  October 2016 Published  December 2016

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

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.

Citation: Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
References:
[1]

A. Ali and H. Kalisch, A dispersive model for undular bores, Anal. Math. Phys., 2 (2012), 347-366.  doi: 10.1007/s13324-012-0040-7.

[2]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[3]

H. Borluk and H. Kalisch, Particle dynamics in the KdV approximation, Wave Motion, 49 (2012), 691-709.  doi: 10.1016/j.wavemoti.2012.04.007.

[4]

J. Boussinesq, Essai sur la théorie des eaux courantes, (French) [Essay on the theory of running water], Mémoires présentés par divers savant á l'Acad. des Sci. Inst. Nat. France, ⅩⅩⅢ, (1877), 1-680. 

[5]

Y.-Y. ChenH.-C. Hsu and H.-H. Hwung, Experimental study of the particle paths in solitary water waves, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 370 (2012), 1629-1637. 

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.

[7]

A. Constantin, Solitons from the Lagrangian perspective, Discrete Continuous Dynam. Systems -A, 19 (2007), 469-481.  doi: 10.3934/dcds.2007.19.469.

[8]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[9]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7.

[10]

A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73.  doi: 10.2991/jnmp.2008.15.s2.5.

[11]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299.

[12]

A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2008), 1-18.  doi: 10.1007/s00021-005-0214-2.

[13]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Letters, 19 (1967), 1095-1097.  doi: 10.1103/PhysRevLett.19.1095.

[14]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Korteweg-deVries equation and generalization. {VI}. {M}ethods for exact solution, Comm. Pure Appl. Math., 27 (1974), 97-133.  doi: 10.1002/cpa.3160270108.

[15]

D. Henry, Steady periodic flow induced by the Korteweg-de Vries equation, Wave Motion, 46 (2009), 403-411.  doi: 10.1016/j.wavemoti.2009.06.007.

[16]

R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Physical Review Letters, 27 (1971), 1192-1194.  doi: 10.1103/PhysRevLett.27.1192.

[17]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056.

[18]

H. Kalisch, Personal communications.

[19]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.

[20]

J. S. Russell, Report on Waves, Report of the fourteenth meeting of the British Association for the Advancement of Science 39 (1844).

[21]

G. B. Whitham, Linear and Nonlinear Waves Wiley-Interscience [John Wiley & Sons], New-York, 1974.

[22]

N. J. Zabusky and M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Physical Review Letters, 15 (1965), 240-243.  doi: 10.1103/PhysRevLett.15.240.

show all references

References:
[1]

A. Ali and H. Kalisch, A dispersive model for undular bores, Anal. Math. Phys., 2 (2012), 347-366.  doi: 10.1007/s13324-012-0040-7.

[2]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[3]

H. Borluk and H. Kalisch, Particle dynamics in the KdV approximation, Wave Motion, 49 (2012), 691-709.  doi: 10.1016/j.wavemoti.2012.04.007.

[4]

J. Boussinesq, Essai sur la théorie des eaux courantes, (French) [Essay on the theory of running water], Mémoires présentés par divers savant á l'Acad. des Sci. Inst. Nat. France, ⅩⅩⅢ, (1877), 1-680. 

[5]

Y.-Y. ChenH.-C. Hsu and H.-H. Hwung, Experimental study of the particle paths in solitary water waves, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 370 (2012), 1629-1637. 

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.

[7]

A. Constantin, Solitons from the Lagrangian perspective, Discrete Continuous Dynam. Systems -A, 19 (2007), 469-481.  doi: 10.3934/dcds.2007.19.469.

[8]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[9]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7.

[10]

A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73.  doi: 10.2991/jnmp.2008.15.s2.5.

[11]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299.

[12]

A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2008), 1-18.  doi: 10.1007/s00021-005-0214-2.

[13]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Letters, 19 (1967), 1095-1097.  doi: 10.1103/PhysRevLett.19.1095.

[14]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Korteweg-deVries equation and generalization. {VI}. {M}ethods for exact solution, Comm. Pure Appl. Math., 27 (1974), 97-133.  doi: 10.1002/cpa.3160270108.

[15]

D. Henry, Steady periodic flow induced by the Korteweg-de Vries equation, Wave Motion, 46 (2009), 403-411.  doi: 10.1016/j.wavemoti.2009.06.007.

[16]

R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Physical Review Letters, 27 (1971), 1192-1194.  doi: 10.1103/PhysRevLett.27.1192.

[17]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056.

[18]

H. Kalisch, Personal communications.

[19]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.

[20]

J. S. Russell, Report on Waves, Report of the fourteenth meeting of the British Association for the Advancement of Science 39 (1844).

[21]

G. B. Whitham, Linear and Nonlinear Waves Wiley-Interscience [John Wiley & Sons], New-York, 1974.

[22]

N. J. Zabusky and M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Physical Review Letters, 15 (1965), 240-243.  doi: 10.1103/PhysRevLett.15.240.

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