Wave structure and nonlinear balances in a family of evolutionary PDEs

SIAM Journal on Applied Dynamical Systems

Volumn 2, Issue 1, 2003, Pages 323-380

  Holm, Darryl D ^a,b   Staley, Martin F ^a  

^a CENTER FOR NONLINEAR STUDIES   (United States)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Hamilton's principle; Hamiltonian structures, symmetries, variational principles, conservation laws; Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction

Indexed keywords

EID: 4043101891     PISSN: 15360040     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1111111102410943     Document Type: Article

References (40)

1. M. J. ABLOWITZ AND P. A. CLARKSON, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991.
2. M. ALBER, R. CAMASSA, D. D. HOLM, AND J. E. MARSDEN, The geometry of peaked solitons and billiard solutions of a class of integrable PDEs, Lett. Math. Phys., 32 (1994), pp. 137-151.
3. M. ALBER, R. CAMASSA, D. D. HOLM, AND J. E. MARSDEN, On the link between umbilic geodesics and soliton solutions of nonlinear PDE's, Proc. Roy. Soc. London, 450 (1995), pp. 677-692.
4. M. ALBER, R. CAMASSA, V. N. FEDOROV, D. D. HOLM, AND J. E. MARSDEN, On billiard solutions of nonlinear PDE's, Phys. Lett. A, 264 (1999), pp. 171-178.
5. M. ALBER, R. CAMASSA, V. N. FEDOROV, D. D. HOLM, AND J. E. MARSDEN, The complex geometry of piecewise solutions of integrable nonlinear PDE's of shallow water and Dym type, Comm. Math. Phys., 221 (2001), pp. 197-227.
6. R. BEALS, D. H. SATTINGER, AND J. SZMIGIELSKI, Multipeakons and the classical moment problem, Adv. Math., 154 (2000), pp. 229-257.
7. R. CAMASSA AND D. D. HOLM, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), pp. 1661-1664; also available online from http://xxx.lanl.gov/abs/patt-sol/9305002.
8. R. CAMASSA, D. D. HOLM, AND J. M. HYMAN, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), pp. 1-33.
9. S. CHEN, C. FOIAS, D. D. HOLM, E. J. OLSON, E. S. TITI, AND S. WYNNE, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett., 81 (1998), pp. 5338-5341; also available online from http://xxx.lanl.gov/abs/chao-dyn/9804026.
10. A. CONSTANTIN, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London, 457 (2001), pp. 953-970.
11. A. CONSTANTIN AND J. ESCHER, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), pp. 303-328.
12. A. CONSTANTIN AND J. ESCHER, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), pp. 229-243.
13. A. CONSTANTIN AND W. A. STRAUSS, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), pp. 603-610.
14. H.-H. DAI AND Y. HUO, Solitary shock waves and other travelling waves in general compressible hyperelastic rods, Proc. Roy. Soc. London, 456 (2000), pp. 331-363.
15. A. DEGASPERIS, D. D. HOLM, AND A. N. W. HONE, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), pp. 1463-1474; also available online from http://xxx.lanl.gov/abs/nlin.SI/0205023.
16. A. DEGASPERIS AND M. PROCESI, Asymptotic integrability, in Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta, eds., World Scientific, River Edge, NJ, 1999, pp. 23-37.
17. H. DULLIN, G. GOTTWALD, AND D. D. HOLM, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), pp. 194501-194504.
18. H. R. DULLIN, D. D. HOLM, AND G. GOTTWALD, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynam. Res., 33 (2003), pp. 73-95.
19. A. S. FOKAS, On a class of physically important integrable equations, Phys. D, 87 (1995), pp. 145-150.
20. A. S. FOKAS AND B. FUCHSSTEINER, Backlund-transformations for hereditary symmetries, Nonlinear Anal., 5 (1981), pp. 423-432.
21. A. S. FOKAS AND Q. M. LIU, Asymptotic integrability of water waves, Phys. Rev. Lett., 77 (1996), pp. 2347-2351.
22. O. FRINGER AND D. D. HOLM, Integrable vs. nonintegrable geodesic soliton behavior, Phys. D, 150 (2001), pp. 237-263; also available online from http://xxx.lanl.gov/abs/solv-int/9903007.
23. B. FUCHSSTEINER AND A. S. FOKAS, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), pp. 47-66.
24. B. FUCHSSTEINER, The Lie algebra structure of nonlinear evolution equations admitting infinite-dimensional abelian symmetry groups, Progr. Theoret. Phys., 65 (1981), pp. 861-876.
25. B. FUCHSSTEINER, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), pp. 229-243.
26. D. D. HOLM, J. E. MARSDEN, AND T. S. RATIU, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), pp. 1-81.
27. D. D. HOLM, J. E. MARSDEN, AND T. S. RATIU, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (1998), pp. 4173-4177.
28. D. D. HOLM, R. B. LOWRIE, AND M. F. STALEY, Shock-Capturing Properties of the Burgers-αβ Equation, manuscript.
29. D. D. HOLM AND M. F. STALEY, Momentum Filament Dynamics in a Family of 2 + 1 Evolutionary PDEs, manuscript.
30. D. D. HOLM AND E. S. TITI, PDE Results for Peakon Dynamics, manuscript.
31. J. K. HUNTER AND R. H. SAXTON, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), pp. 1498-1521.
32. R. S. JOHNSON, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech., 455 (2002), pp. 63-82.
33. Y. KODAMA, On integrable systems with higher order corrections, Phys. Lett. A, 107 (1985), pp. 245-249.
34. Y. KODAMA, Normal forms for weakly dispersive wave equations, Phys. Lett. A, 112 (1985), pp. 193-196.
35. Y. KODAMA, On solitary wave interaction, Phys. Lett. A, 123 (1987), pp. 276-282.
36. H. P. KRUSE, J. SCHREULE, AND W. DU, A two-dimensional version of the CH equation, in Symmetry and Perturbation Theory: SPT 2001, D. Bambusi, G. Gaeta, and M. Cadoni, eds., World Scientific, New York, 2001, pp. 120-127.
37. Z. LI AND N. R. SIBGATULLIN, An improved theory of long waves on the water surface, J. Appl. Math. Mech., 61 (1997), pp. 177-182.
38. A. V. MIKHAILOV AND V. S. NOVIKOV, Perturbative symmetry approach, J. Phys. A, 35 (2002), pp. 4775-4790.
39. M. I. MILLER, A. TROUVÉ, AND L. YOUNES, On the metrics and Euler-Lagrange equations of computational anatomy, Ann. Rev. Biomed. Engrg., 4 (2002), pp. 375-405.
40. D. MUMFORD, Pattern theory and vision, in Questions Mathématiques En Traitement Du Signal et de L'Image, Institut Henri Poincaré, Paris, 1998, pp. 7-13.