On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey

References

• Abdelouhab , L , Bona , J , Felland , M and Saut , J C . 1989 . Nonlocal models for nonlinear, dispersive waves . Physica D , 40 : 360 – 392 .
   Web of Science ®Google Scholar
• Beals , R , Sattinger , D and Szmigielski , J . 1999 . Multipeakons and a Theorem of Stieltjes . Inverse Problems , 15 : 1 – 4 .
   Web of Science ®Google Scholar
• Camassa , R and Holm , D.D. 1993 . An integrable shallow water equation with peaked solitons . Phys. Rev. Lett. , 71 : 1661 – 1664 .
   PubMed Web of Science ®Google Scholar
• Camassa , R , Holm , D.D. and Hyman , J . 1994 . An new integrable shallow water equation . Adv. Appl. Mech. , 31 : 1 – 34 .
   Google Scholar
• Constantin , A . 2000 . Existence of permanent and breaking waves for a shallow water equation: a geometrical approach . Annales de l’Institut Fourier , 50 : 321 – 362 .
   Web of Science ®Google Scholar
• Constantin , A . 2001 . On the scattering problem for the Camassa-Holm equation . Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. , 457 : 953 – 970 .
   Web of Science ®Google Scholar
• Constantin , A and Escher , J . 1998 . Global existence and blow-up for a shallow water equation . Annali Scuola Norm. Sup. Pisa , 26 : 303 – 328 .
   Google Scholar
• Constantin , A and Escher , J . 1998 . Wave breaking for nonlinear nonlocal shallow water equations . Acta Mathematica , 181 : 229 – 243 .
   Web of Science ®Google Scholar
• Constantin , A and Escher , J . 1998 . Global weak solutions for a shallow water equation . Indiana Univ. Math. J. , 47 : 1527 – 1545 .
   Web of Science ®Google Scholar
• Constantin , A and Kolev , B . 2002 . On the geometric approach to the motion of inertial mechanical systems . J. Phys. A , 35 : R51 – R79 .
   Google Scholar
• Constantin , A and Kolev , B . 2003 . Geodesic flow on the diffeomorphism group of the circle . Comment. Math. Helv. , 78 : 787 – 804 .
   Web of Science ®Google Scholar
• Constantin , A and Strauss , W . 2000 . Stability of peakons . Comm. Pure Appl. Math. , 53 : 603 – 610 .
   Web of Science ®Google Scholar
• Constantin , A and Molinet , L . 2000 . Global weak solutions for a shallow water equation . Comm. Math. Phys. , 211 : 45 – 61 .
   Web of Science ®Google Scholar
• Constantin , A and Molinet , L . 2001 . Orbital stability of solitary waves for a shallow water equation . Physica D , 157 : 75 – 89 .
   Web of Science ®Google Scholar
• Dai H H Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech. 127(1998), 193–207
   Google Scholar
• Danchin , R . 2001 . A few remarks on the Camassa-Holm equation . Diff. Int. Eq. , 14 : 953 – 980 .
   Google Scholar
• Danchin , R . 2003 . A note on well-posedness for Camassa-Holm equation . J. Differential Equations , 192 : 429 – 444 .
   Web of Science ®Google Scholar
• Fokas , A S and Fuchssteiner , B . 1981 . Symplectic structures, their Bäcklund transformation and hereditary symmetries . Physica D , 4 : 47 – 66 .
   Web of Science ®Google Scholar
• Himonas , A and Misiolek , G . 2001 . The Cauchy problem for an integrable shallow water equation . Diff. Int. Eq. , 14 : 821 – 831 .
   Google Scholar
• Johnson , R S and Camassa-Holm . 2002 . Korteweg-de Vries and related models for water waves . J. Fluid Mech. , 455 : 63 – 82 .
   Web of Science ®Google Scholar
• Kenig , C E , Ponce , G and Vega , L . 1993 . Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle . Comm. Pure Appl. Math. , 46 : 527 – 620 .
   Web of Science ®Google Scholar
• Koch , H and Tzvetkov , N . 2002 . Nonlinear wave interactions for the Benjamin-Ono equation . Preprint ,
   Google Scholar
• Lenells , J . 2002 . The scattering approach for the Camassa-Holm equation . J. Nonlinear Math. Phys. , 9 : 389 – 393 .
   Web of Science ®Google Scholar
• Li , Y and Olver , P . 2000 . Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation . J. Differential Equations , 162 : 27 – 63 .
   Web of Science ®Google Scholar
• Lions , J L . 1969 . Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires , Paris : Dunod .
   Google Scholar
• Misiolek , G . 1998 . A shallow water equation as a geodesic flow on the Bott-Virasoro group . J. Geom. Phys. , 24 : 203 – 208 .
   Web of Science ®Google Scholar
• Rodriguez-Blanco , G . 2001 . On the Cauchy problem for the Camassa-Holm equation . Nonlinear Anal. , 14 : 309 – 327 .
   Google Scholar
• Xin , Z and Zhang , P . 2000 . On the weak solutions to a shallow water equation . Comm. Pure Appl. Math. , 53 : 1411 – 1433 .
   Web of Science ®Google Scholar
• Xin , Z and Zhang , P . 2002 . On the uniqueness and large time behavior of the weak solutions to a shallow water equation . Comm. Partial Diff. Eq. , 27 : 1815 – 1844 .
   Web of Science ®Google Scholar