On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation

Communications in Mathematical Physics

Volumn 257, Issue 3, 2005, Pages 667-701

  Tian, Lixin ^a   Gui, Guilong ^a   Liu, Yue ^b  

^a JIANGSU UNIVERSITY   (China)

^b UNIVERSITY OF TEXAS AT ARLINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21344447273     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-005-1356-z     Document Type: Article

References (36)

1. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11). 1661-1664 (1993)
2. Fuchssteiner, B., Fokus, A. S.: Symplrctic structures, their Bucklund transformation and hereditary symmetries. Physica D 4, 47-66 (1981)
3. Johnson, R. S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid. Mech. 457, 63-82 (2002)
4. Dullin, R., Gottwald, G., Holm, D.: An integrahle shallow water equation with linear and nonlinear dispersion. Phys.Rev. Lett. 87(9). 4501-4504 (2001)
5. Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1-33 (1994)
6. Fisher, M., Schiff, J.: The Camassa Holm equations: conserved quantities and the initial value problem. Phy. Lett. A. 259(3), 371-376 (1999)
7. Clarkson, P.A., Mansfield, H.L., Pricstley, T.J.: Symmetries of a class of nonlinear third-order partial differential equations. Math. Compul. Modelling 25 (8-9), 195-212 (1997)
8. Kraenkel, R.A., Senthilvelan, M., Zenchuk, A.I.: On the integrable perturbations of the Camassa-Holm equation. J. Math. Phys. 41(5). 3160-3169 (2000)
9. Cooper. F., Shepard, H.: Solitons in the Camassa - Holm shallow water equation. Phys. Lett. A 194(4). 246-250 (1994)
10. Tian, L., Xu. G., Liu. Z.: The concave or convex peaked and smooth soliton solutions of CamassaHolm equation. Appl. Math. Mech. 123(5), 557-567 (2002)
11. Tian, L., Song, X.: New peaked solitary wave solutions of the generalized Camassa- Holm equation. Chaos, Solitons and Fractals 19(3). 621-637 (2004)
12. Tian, L., Yin, J.: New compacton solutions and solitary solutions of fully nonlinear generalized Camassa-Holm equations. Chaos. Soliton and Fractals 20(4), 289-299 (2004)
13. Constantin, A., Eschcr J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Sup. Pisa 26, 303-328 (1998)
14. Constantin, A.: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann.Inst.Fourier (Grenoble) 50, 321-362 (2000)
15. Constantin, A.: The Hamitonian structure of the Camassa-Holm equation. Hxposition. Math. 15, 53-85 (1997)
16. Constantin, A., McKean, H. P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52(8). 949-982 (1999)
17. Constantin, A., Strauss, W.A.: Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 12, 415-422 (2002)
18. Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. R.Soc . London A 457, 953-970 (2001)
19. Lenells, J.: The Scattering approach for the Camassa-Holm equation. J. Nonliear Math.Phys. 9(4). 389-393 (2002)
20. Beals, R., Sattinger, D., Szmigielski, J.: Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math. 140, 190-206 (1998)
21. Constantin, A., Escher, J.: Wave Breaking for Nonlinear Nonlocal Shallow Water Equations. Acta Math. 181, 229-243 (1998)
22. Danchin, R.: A Few Remarks on the Camassa-Holm Equation. Differential and Integral Equations 14, 953-988 (2001)
23. Guo, BL., Liu, ZR.: Peaked wave solutions of CH - γ equation. Sci China (Ser. A) 33(4). 325-337 (2003)
24. Tang, M., Yang, C.: Hxtension on peaked wave solutions of CH - γ equation. Chaos. Solitons and Fractals 20, 815-825 (2004)
25. Bona, J., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans.Royal Soc. London Series A 278, 555-601 (1975)
26. Kato, T.: On the Cauchy problem for the (generalized) KdV equation. Studies in Applied Mathematics. Advances in Mathematics Supplementary. Vol.8. NewYork-London: Academic Press. 1983, pp. 93-128
27. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. J.Funct.Anal. 74, 160-197 (1987)
28. Gelfand, I.M., Dorfman, I.Ya.R.: Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl. 13, 248-262 (1979)
29. Li, Y., Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eq. 162, 27-63 (2000)
30. Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal. 46, 309-327 (2001)
31. Constantin, A., Hschcr, J.: Global weak solutions for a shallow water equation. Indiana Univ. Math. J 47(4), 1527-1545 (1998)
32. Dunford, N., Schwartz, J.T.: Linear operators. Vol.2. New York: Wiley. 1988
33. Kato, T., Ponce, G.: Commutator estimates and the Huler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891-907 (1988)
34. Ablowitz, M., Clarkson, P.: Soliton, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press. 1993
35. Deift, P., Trubowits, H.: Inverse scattering on the line. Comm Pure Appl. Math. 32, 121-251 (1979)
36. Li, Y.A., Olver, P.J.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eq. 162, 27-63 (2000)