A new two-component system modelling shallow-water waves
Author:
Delia Ionescu-Kruse
Journal:
Quart. Appl. Math. 73 (2015), 331-346
MSC (2010):
Primary 35Q35, 76B15, 76M30, 37K05, 76B25
DOI:
https://doi.org/10.1090/S0033-569X-2015-01369-1
Published electronically:
March 30, 2015
Original version:
Previous version posted March 16, 2015
Corrected version:
Current version corrects publisher's errors in both Equations (1.1) and (1.2).
MathSciNet review:
3357497
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: For propagation of surface shallow-water waves on irrotational flows, we derive a new two-component system. The system is obtained by a variational approach in the Lagrangian formalism. The system has a noncanonical Hamiltonian formulation. We also find its exact solitary-wave solutions.
References
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- V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295
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- Adrian Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 321–362 (English, with English and French summaries). MR 1775353
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- Adrian Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. MR 2867413
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- A. Constantin, T. Kappeler, B. Kolev, and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom. 31 (2007), no. 2, 155–180. MR 2326419, DOI https://doi.org/10.1007/s10455-006-9042-8
- Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), no. 1, 165–186. MR 2481064, DOI https://doi.org/10.1007/s00205-008-0128-2
- Joachim Escher, Non-metric two-component Euler equations on the circle, Monatsh. Math. 167 (2012), no. 3-4, 449–459. MR 2961293, DOI https://doi.org/10.1007/s00605-011-0323-3
- Joachim Escher and Boris Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z. 269 (2011), no. 3-4, 1137–1153. MR 2860280, DOI https://doi.org/10.1007/s00209-010-0778-2
- A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech. 78 (1976), 237–246.
- Guilong Gui and Yue Liu, On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math. 69 (2011), no. 3, 445–464. MR 2850740, DOI https://doi.org/10.1090/S0033-569X-2011-01216-5
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- Darryl D. Holm, Hamiltonian structure for two-dimensional hydrodynamics with nonlinear dispersion, Phys. Fluids 31 (1988), no. 8, 2371–2373. MR 951328, DOI https://doi.org/10.1063/1.866587
- Delia Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys. 14 (2007), no. 3, 303–312. MR 2350091, DOI https://doi.org/10.2991/jnmp.2007.14.3.1
- Delia Ionescu-Kruse, Variational derivation of the Green-Naghdi shallow-water equations, J. Nonlinear Math. Phys. 19 (2012), no. suppl. 1, 1240001, 12. MR 2999395, DOI https://doi.org/10.1142/S1402925112400013
- Delia Ionescu-Kruse, Variational derivation of two-component Camassa-Holm shallow water system, Appl. Anal. 92 (2013), no. 6, 1241–1253. MR 3197932, DOI https://doi.org/10.1080/00036811.2012.667082
- R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555
- Yi A. Li, Linear stability of solitary waves of the Green-Naghdi equations, Comm. Pure Appl. Math. 54 (2001), no. 5, 501–536. MR 1811125, DOI https://doi.org/10.1002/cpa.1.abs
- Yi A. Li, Hamiltonian structure and linear stability of solitary waves of the Green-Naghdi equations, J. Nonlinear Math. Phys. 9 (2002), no. suppl. 1, 99–105. Recent advances in integrable systems (Kowloon, 2000). MR 1900188, DOI https://doi.org/10.2991/jnmp.2002.9.s1.9
- Yi A. Li, A shallow-water approximation to the full water wave problem, Comm. Pure Appl. Math. 59 (2006), no. 9, 1225–1285. MR 2237287, DOI https://doi.org/10.1002/cpa.20148
- Ju. I. Manin, Algebraic aspects of nonlinear differential equations, Current problems in mathematics, Vol. 11 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1978, pp. 5–152. (errata insert) (Russian). MR 0501136
- Michael Tabor and Yvain M. Trève (eds.), Mathematical methods in hydrodynamics and integrability in dynamical systems, AIP Conference Proceedings, vol. 88, American Institute of Physics, New York, 1982. MR 694249
- Y. Nutku, On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure, J. Math. Phys. 28 (1987), no. 11, 2579–2585. MR 913410, DOI https://doi.org/10.1063/1.527749
- Peter J. Olver and Yavuz Nutku, Hamiltonian structures for systems of hyperbolic conservation laws, J. Math. Phys. 29 (1988), no. 7, 1610–1619. MR 946335, DOI https://doi.org/10.1063/1.527909
- F. Serre, Contribution à l’étude des écoulements permanents et variables dans les canaux, La Houille Blanche 3 (1953), 374–388, and 6 (1953), 830–872.
- J. J. Stoker, Water waves, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1992. The mathematical theory with applications; Reprint of the 1957 original; A Wiley-Interscience Publication. MR 1153414
- C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Mathematical Phys. 10 (1969), 536–539. MR 271526, DOI https://doi.org/10.1063/1.1664873
References
- Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, second edition, revised and enlarged; with the assistance of Tudor Raţiu and Richard Cushman, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978. MR 515141 (81e:58025)
- V. I. Arnol′d, Mathematical methods of classical mechanics, translated from the Russian by K. Vogtmann and A. Weinstein, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. MR 997295 (90c:58046)
- Borys Alvarez-Samaniego and David Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J. 57 (2008), no. 1, 97–131. MR 2400253 (2010a:35016), DOI https://doi.org/10.1512/iumj.2008.57.3200
- Borys Alvarez-Samaniego and David Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math. 171 (2008), no. 3, 485–541. MR 2372806 (2009b:35324), DOI https://doi.org/10.1007/s00222-007-0088-4
- D. J. Benney, Some properties of long non-linear waves, Studies Appl. Math. 52 (1973) 45–50.
- J. Cavalcante and H. P. McKean, The classical shallow water equations: symplectic geometry, Phys. D 4 (1981/82), no. 2, 253–260. MR 653778 (84h:76010), DOI https://doi.org/10.1016/0167-2789%2882%2990066-5
- Adrian Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math. 15 (1997), no. 1, 53–85. MR 1438436 (98c:76011)
- Adrian Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, English, with English and French summaries, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 321–362. MR 1775353 (2002d:37125)
- Adrian Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys. 46 (2005), no. 2, 023506, 4. MR 2121730 (2005h:35303), DOI https://doi.org/10.1063/1.1845603
- Adrian Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. MR 2867413
- Adrian Constantin and Joachim Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z. 233 (2000), no. 1, 75–91. MR 1738352 (2001b:35258), DOI https://doi.org/10.1007/PL00004793
- A. Constantin, T. Kappeler, B. Kolev, and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom. 31 (2007), no. 2, 155–180. MR 2326419 (2008d:58005), DOI https://doi.org/10.1007/s10455-006-9042-8
- Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), no. 1, 165–186. MR 2481064 (2010f:35334), DOI https://doi.org/10.1007/s00205-008-0128-2
- Joachim Escher, Non-metric two-component Euler equations on the circle, Monatsh. Math. 167 (2012), no. 3-4, 449–459. MR 2961293, DOI https://doi.org/10.1007/s00605-011-0323-3
- Joachim Escher and Boris Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z. 269 (2011), no. 3-4, 1137–1153. MR 2860280 (2012m:37124), DOI https://doi.org/10.1007/s00209-010-0778-2
- A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech. 78 (1976), 237–246.
- Guilong Gui and Yue Liu, On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math. 69 (2011), no. 3, 445–464. MR 2850740 (2012i:35337)
- David Henry, Infinite propagation speed for the Degasperis-Procesi equation, J. Math. Anal. Appl. 311 (2005), no. 2, 755–759. MR 2168432 (2006d:35225), DOI https://doi.org/10.1016/j.jmaa.2005.03.001
- Darryl D. Holm, Hamiltonian structure for two-dimensional hydrodynamics with nonlinear dispersion, Phys. Fluids 31 (1988), no. 8, 2371–2373. MR 951328 (89f:76023), DOI https://doi.org/10.1063/1.866587
- Delia Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys. 14 (2007), no. 3, 303–312. MR 2350091 (2008j:76015), DOI https://doi.org/10.2991/jnmp.2007.14.3.1
- Delia Ionescu-Kruse, Variational derivation of the Green-Naghdi shallow-water equations, J. Nonlinear Math. Phys. 19 (2012), no. suppl. 1, 1240001, 12. MR 2999395, DOI https://doi.org/10.1142/S1402925112400013
- D. Ionescu-Kruse, Variational derivation of two-component Camassa-Holm shallow water system, Appl. Anal. 92 (2013), no. 6, 1241–1253. DOI:10.1080/00036811.2012.667082. MR 3197932
- R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555 (99m:76017)
- Yi A. Li, Linear stability of solitary waves of the Green-Naghdi equations, Comm. Pure Appl. Math. 54 (2001), no. 5, 501–536. MR 1811125 (2003b:35186), DOI https://doi.org/10.1002/cpa.1.abs
- Yi A. Li, Hamiltonian structure and linear stability of solitary waves of the Green-Naghdi equations, Recent advances in integrable systems (Kowloon, 2000), J. Nonlinear Math. Phys. 9 (2002), no. suppl. 1, 99–105. MR 1900188 (2003d:37126), DOI https://doi.org/10.2991/jnmp.2002.9.s1.9
- Yi A. Li, A shallow-water approximation to the full water wave problem, Comm. Pure Appl. Math. 59 (2006), no. 9, 1225–1285. MR 2237287 (2007d:76024), DOI https://doi.org/10.1002/cpa.20148
- Ju. I. Manin, Algebraic aspects of nonlinear differential equations (Russian), Current problems in mathematics, Vol. 11 (Russian), pp. 5–152 (errata insert), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1978. MR 0501136 (58 \#18567)
- Mathematical methods in hydrodynamics and integrability in dynamical systems, AIP Conference Proceedings, edited by Michael Tabor and Yvain M. Trève, vol. 88, American Institute of Physics, New York, 1982. MR 694249 (84c:76002)
- Y. Nutku, On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure, J. Math. Phys. 28 (1987), no. 11, 2579–2585. MR 913410 (88k:58058), DOI https://doi.org/10.1063/1.527749
- Peter J. Olver and Yavuz Nutku, Hamiltonian structures for systems of hyperbolic conservation laws, J. Math. Phys. 29 (1988), no. 7, 1610–1619. MR 946335 (90b:58085), DOI https://doi.org/10.1063/1.527909
- F. Serre, Contribution à l’étude des écoulements permanents et variables dans les canaux, La Houille Blanche 3 (1953), 374–388, and 6 (1953), 830–872.
- J. J. Stoker, Water waves, The mathematical theory with applications, reprint of the 1957 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1992. MR 1153414 (92m:76029)
- C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Mathematical Phys. 10 (1969), 536–539. MR 0271526 (42 \#6409)
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Additional Information
Delia Ionescu-Kruse
Affiliation:
Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 6, P.O. Box 1-764, 014700 Bucharest, Romania
Email:
Delia.Ionescu@imar.ro
Keywords:
Shallow-water waves,
variational methods,
Hamiltonian structures,
solitary waves
Received by editor(s):
April 26, 2013
Received by editor(s) in revised form:
May 16, 2013
Published electronically:
March 30, 2015
Article copyright:
© Copyright 2015
Brown University