Wave Breaking and Measure of Momentum Support for an Integrable Camassa‐Holm System with Two Components

We discuss a new integrable two-component Camassa-Holm equation which describes the motion of fluid. This paper is concerned with the wave breaking mechanism for the Cauchy problem with periodic condition where two special classes of initial data are involved. Moreover, the estimate of momentum support is also shown.

[1]  Guilong Gui,et al.  On the global existence and wave-breaking criteria for the two-component Camassa-Holm system , 2010 .

[2]  Ying Fu,et al.  Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons , 2010 .

[3]  Wave breaking for a modified two-component Camassa–Holm system , 2012 .

[4]  Athanassios S. Fokas,et al.  Symplectic structures, their B?acklund transformation and hereditary symmetries , 1981 .

[5]  Yong Zhou,et al.  Wave breaking for a shallow water equation , 2004 .

[6]  Octavian G. Mustafa,et al.  On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system , 2009 .

[7]  W. Strauss,et al.  Stability of peakons , 2000 .

[8]  Yong Zhou,et al.  Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation , 2006 .

[9]  A. Bressan,et al.  GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION , 2007 .

[10]  Yong Zhou,et al.  Blow up and propagation speed of solutions to the DGH equation , 2009 .

[11]  Zhengguang Guo,et al.  On a two-component Degasperis–Procesi shallow water system , 2010 .

[12]  D. Henry Infinite propagation speed for a two component Camassa-Holm equation , 2009 .

[13]  The Cauchy problem for a two-component generalized θ-equations , 2010 .

[14]  Yong Zhou,et al.  A new asymptotic behavior of solutions to the Camassa-Holm equation , 2012 .

[15]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

[16]  A. Constantin,et al.  The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations , 2007, 0709.0905.

[17]  Shun-Guang Kang,et al.  The support of the momentum density of the Camassa-Holm equation , 2011, Appl. Math. Lett..

[18]  Joachim Escher,et al.  Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation , 2007 .

[19]  Youjin Zhang,et al.  A Two-component Generalization of the Camassa-Holm Equation and its Solutions , 2005, nlin/0501028.

[20]  Eigenvalues associated with the vortex patch in 2-D Euler equations , 2004 .

[21]  J. Escher,et al.  Global existence and blow-up for a shallow water equation , 1998 .

[22]  P. Olver,et al.  Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation , 2000 .

[23]  A. Constantin Finite propagation speed for the Camassa-Holm equation , 2005 .

[24]  Yue Liu,et al.  Stability of Solitary Waves and Wave-Breaking Phenomena for the Two-Component Camassa–Holm System , 2009 .

[25]  Marcus Wunsch,et al.  The Generalized Hunter-Saxton System , 2010, SIAM J. Math. Anal..

[26]  G. Misiołek Classical solutions of the periodic Camassa—Holm equation , 2002 .

[27]  Yong Zhou,et al.  Wave Breaking of the Camassa–Holm Equation , 2012, J. Nonlinear Sci..

[28]  Z. Yin,et al.  Global existence and blow-up phenomena for a periodic 2-component Camassa–Holm equation , 2011, Monatshefte für Mathematik.

[29]  R. Johnson,et al.  Camassa–Holm, Korteweg–de Vries and related models for water waves , 2002, Journal of Fluid Mechanics.

[30]  Yong Zhou,et al.  On Solutions to a Two‐Component Generalized Camassa‐Holm Equation , 2010 .

[31]  Yong Zhou Wave breaking for a periodic shallow water equation , 2004 .

[32]  G. Falqui,et al.  On a Camassa-Holm type equation with two dependent variables , 2005, nlin/0505059.

[33]  J. Escher,et al.  Wave breaking for nonlinear nonlocal shallow water equations , 1998 .

[34]  Rossen I. Ivanov,et al.  On an integrable two-component Camassa–Holm shallow water system , 2008, 0806.0868.

[35]  Zhengguang Guo,et al.  Persistence Properties and Unique Continuation of Solutions to a Two-component Camassa–Holm Equation , 2011 .

[36]  Zhengguang Guo Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa–Holm equation , 2008 .

[37]  H. McKean Breakdown of a shallow water equation , 1998 .

[38]  A. Bressan,et al.  Global Conservative Solutions of the Camassa–Holm Equation , 2007 .

[39]  A. Alexandrou Himonas,et al.  The Cauchy problem for an integrable shallow-water equation , 2001, Differential and Integral Equations.

[40]  J. Escher,et al.  Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation , 1998 .

[41]  Z. Popowicz A 2-component or N=2 supersymmetric Camassa–Holm equation , 2005, nlin/0509050.

[42]  Zhengguang Guo,et al.  Blow-up and global solutions to a new integrable model with two components , 2010 .

[43]  W. N. Everitt Spectral Theory and Differential Equations , 1975 .

[44]  P. Olver,et al.  Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.