The Journal of Geometric Mechanics (JGM)

Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach
Pages: 281 - 293, Issue 3, September 2015

doi:10.3934/jgm.2015.7.281      Abstract        References        Full text (400.4K)           Related Articles

Joachim Escher - Institute for Applied Mathematics, University of Hanover, D-30167 Hanover, Germany (email)
Tony Lyons - School of Mathematical Sciences, University College Cork, Cork, Ireland (email)

1 V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.       
2 R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.       
3 R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.
4 A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.       
5 A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, Volume 81 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.       
6 A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.       
7 A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.       
8 A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.       
9 A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79.       
10 A.Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.       
11 A. Constantin and B. Kolev, $H^k$ metrics on the diffeomorphism group of the circle, J. Nonlin. Math. Phys., 10 (2003), 424-430.       
12 A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (Rome, 1998), World Sci. Publ., River Edge, NJ, (1999), 23-37.       
13 D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), 102-163.       
14 J. Escher, Non-metric two-component Euler equations on the circle, Monatsh. Math., 167 (2012), 449-459.       
15 J. Escher, D. Henry, B. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pur. App., (2014). To appear.
16 J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153.       
17 J. Escher and B. Kolev, Geodesic completeness for Sobolev $H^s$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 6 (2014), 335-372. To appear. Available at http://arxiv.org/abs/1308.3570.       
18 J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 14 (2014), 949-968. To appear. Available at http://arxiv.org/abs/1202.5122.       
19 J. Escer and Z. Yin, Well-posedness, blow-up phenomena and global solutions for the $b$-equation, J. Reine Angew. Math., 624 (2008), 51-80.       
20 D. Henry, Compactly supported solutions of a family of nonlinear differential equations, Dyn. Contin. Discrete Impuls Syst. Ser. A Math. Anal., 15 (2008), 145-150.       
21 D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Dyn. Syst. Ser. B, 12 (2009), 597-606.       
22 D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlin. Anal., 70 (2009), 1565-1573.       
23 R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-396.       
24 R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.       
25 R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111.       
26 B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation, Wave Motion, 46 (2009), 412-419.       
27 G. Misiołek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104.       

Go to top