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Homogenization of second order discrete model with local perturbation and application to traffic flow
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 |
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.
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. Bona, M. 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. Chen, H.-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. Gardner, J. M. Greene, M. 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. Gardner, J. M. Greene, M. 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] | |
[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. Bona, M. 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. Chen, H.-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. Gardner, J. M. Greene, M. 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. Gardner, J. M. Greene, M. 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] | |
[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. |






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