Wave breaking and solitary wave solutions for a generalized Novikov equation

Abstract In the paper, we mainly study wave-breaking and solitary wave solutions for a generalized Novikov equation. We first obtain a new blow-up result for strong solutions of a generalized Novikov equation in H s ( R ) for any s > 5 2 . We then study the solitary wave solutions and find a single solitary wave solution which blows up in H 1 ( R ) . This implies that the weak solutions of the equation may not be unique.

[1]  Jing Ping Wang,et al.  Integrable peakon equations with cubic nonlinearity , 2008, 0805.4310.

[2]  Z. Yin,et al.  A note on the Cauchy problem of the Novikov equation , 2013 .

[3]  Yongsheng Li,et al.  The Cauchy problem for the integrable Novikov equation , 2012 .

[4]  Adrian Constantin,et al.  The trajectories of particles in Stokes waves , 2006 .

[5]  Xavier Raynaud,et al.  A convergent numerical scheme for the Camassa--Holm equation based on multipeakons , 2005 .

[6]  Joachim Escher,et al.  Particle trajectories in solitary water waves , 2007 .

[7]  Vladimir S. Novikov,et al.  Generalizations of the Camassa–Holm equation , 2009 .

[8]  Z. Yin,et al.  Global weak solutions for the Novikov equation , 2011 .

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

[10]  L. Molinet,et al.  Ill-posedness of the Camassa–Holm and related equations in the critical space , 2018, Journal of Differential Equations.

[11]  A. Constantin.  On the scattering problem for the Camassa-Holm equation , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[13]  Z. Yin,et al.  Well-posedness and global existence for the Novikov equation , 2012 .

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

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

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

[17]  A. Constantin.  Existence of permanent and breaking waves for a shallow water equation: a geometric approach , 2000 .

[18]  J. Escher,et al.  Analyticity of periodic traveling free surface water waves with vorticity , 2011 .

[19]  Ping Zhang,et al.  On the weak solutions to a shallow water equation , 2000 .

[20]  Global weak solutions for a generalized Novikov equation , 2019 .

[21]  Yongsheng Li,et al.  The Cauchy problem for the Novikov equation , 2012, Nonlinear Differential Equations and Applications NoDEA.

[22]  Z. Yin,et al.  The Cauchy problem for a generalized Novikov equation , 2017 .

[23]  A. Constantin.  Particle trajectories in extreme Stokes waves , 2012 .

[24]  Adrian Constantin,et al.  A shallow water equation on the circle , 1999 .

[25]  Z. Yin,et al.  Remarks on the well-posedness of Camassa–Holm type equations in Besov spaces , 2016 .

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

[27]  V. Gerdjikov,et al.  Inverse scattering transform for the Camassa–Holm equation , 2006, Inverse Problems.

[28]  Gerald Teschl,et al.  Long-time Asymptotics for the Camassa-Holm Equation , 2009, SIAM J. Math. Anal..

[29]  Tony Lyons.  Particle Trajectories in Extreme Stokes Waves Over Inifinte Depth , 2014 .

[30]  R. Danchin.  A few remarks on the Camassa-Holm equation , 2001, Differential and Integral Equations.

[31]  A. Constantin,et al.  Global Weak Solutions for a Shallow Water Equation , 2000 .

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

[33]  R. Danchin.  A note on well-posedness for Camassa-Holm equation , 2003 .

[34]  Guillermo Rodriguez-Blanco.  On the Cauchy problem for the Camassa-Holm equation , 2001 .

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