# American Institute of Mathematical Sciences

November 2018, 38(11): 5415-5442. doi: 10.3934/dcds.2018239

## On a new two-component $b$-family peakon system with cubic nonlinearity

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 2 Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 3 School of Mathematical & Statistical Sciences, University of Texas Rio Grande Valley, 1201 W. University, Dr. Edinburg, Texas 78539, USA 4 College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

Received  August 2017 Revised  November 2017 Published  August 2018

Fund Project: Authors to whom correspondence should be addressed through the following three E-mails: kaiyan@hust.edu.cn, zhijun.qiao@utrgv.edu, zhangmath@126.com

In this paper, we propose a two-component $b$-family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component $b$-family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan provided the initial data decay exponentially and algebraically.

Citation: Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239
##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der MathematischenWissenschaften, Vol. 343, Berlin-Heidelberg-NewYork: Springer, 2011. doi: 10.1007/978-3-642-16830-7. [2] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [3] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [4] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. [5] G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91. doi: 10.1016/j.jfa.2005.07.008. [6] A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757. [7] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328. [8] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. [10] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math.(2), 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [11] A. Constantin, R. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012. [12] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comm. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6. [13] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [14] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. [15] A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [16] H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. [17] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. [18] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions, Theo. Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. [19] A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perturbation Theory, Rome, 1998, World Sci. Publishing, River Edge, NJ, 1999, 23–37. [20] H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Letters, 87 (2001), 4501-4504. [21] J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153. doi: 10.1007/s00209-010-0778-2. [22] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010. [23] A. Fokas, On a class of physically important integrable equations, Physica D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O. [24] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [25] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm eqaution, Physica D, 95 (1996), 229-243. doi: 10.1016/0167-2789(96)00048-6. [26] X. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity, 22 (2009), 1847-1856. doi: 10.1088/0951-7715/22/8/004. [27] G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008. [28] A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479. doi: 10.1088/0951-7715/25/2/449. [29] A. A. Himonas and J. Holmes, Hölder continuity of the solution map for the Novikov equation, J. Math. Phys., 54 (2013), 061501, 11pp. doi: 10.1063/1.4807729. [30] Y. Hou, P. Zhao, E. Fan and Z. Qiao, Algebro-geometric solutions for the Degasperis-Procesi hierarchy, SIAM J. Math. Anal., 45 (2013), 1216-1266. doi: 10.1137/12089689X. [31] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. [32] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Lecture Notes in Math., Springer Verlag, Berlin, 448 (1975), 25–70. [33] Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820. doi: 10.1007/s00220-006-0082-5. [34] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3. [35] H. P. McKean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874. doi: 10.4310/AJM.1998.v2.n4.a10. [36] V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002. [37] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. [38] Z. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys., 239 (2003), 309-341. doi: 10.1007/s00220-003-0880-y. [39] Z. Qiao, Integrable hierarchy (the DP hierarchy), 3 by 3 constrained systems, and parametric and stationary solutions, Acta Applicandae Mathematicae, 83 (2004), 199-220. doi: 10.1023/B:ACAP.0000038872.88367.dd. [40] Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9pp. doi: 10.1063/1.2365758. [41] Z. Qiao, New integrable hierarchy, its parametric solutions, cuspons, one-peak solutions, and M/W-shape peak solitons, J. Math. Phys., 48 (2007), 082701, 20pp. doi: 10.1063/1.2759830. [42] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. [43] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. [44] G. B. Whitham, Linear and Nonlinear Waves, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9781118032954. [45] X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 11 (2012), 707-727. [46] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [47] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269 (2011), 1113-1127. doi: 10.1007/s00209-010-0775-5. [48] K. Yan, Z. Qiao and Z. Yin, Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Comm. Math. Phys., 336 (2015), 581-617. doi: 10.1007/s00220-014-2236-1. [49] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. [50] Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. doi: 10.1016/j.jfa.2003.07.010.

show all references

##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der MathematischenWissenschaften, Vol. 343, Berlin-Heidelberg-NewYork: Springer, 2011. doi: 10.1007/978-3-642-16830-7. [2] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [3] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [4] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. [5] G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91. doi: 10.1016/j.jfa.2005.07.008. [6] A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757. [7] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328. [8] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. [10] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math.(2), 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [11] A. Constantin, R. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012. [12] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comm. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6. [13] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [14] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. [15] A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [16] H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. [17] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. [18] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions, Theo. Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. [19] A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perturbation Theory, Rome, 1998, World Sci. Publishing, River Edge, NJ, 1999, 23–37. [20] H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Letters, 87 (2001), 4501-4504. [21] J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153. doi: 10.1007/s00209-010-0778-2. [22] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010. [23] A. Fokas, On a class of physically important integrable equations, Physica D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O. [24] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [25] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm eqaution, Physica D, 95 (1996), 229-243. doi: 10.1016/0167-2789(96)00048-6. [26] X. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity, 22 (2009), 1847-1856. doi: 10.1088/0951-7715/22/8/004. [27] G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008. [28] A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479. doi: 10.1088/0951-7715/25/2/449. [29] A. A. Himonas and J. Holmes, Hölder continuity of the solution map for the Novikov equation, J. Math. Phys., 54 (2013), 061501, 11pp. doi: 10.1063/1.4807729. [30] Y. Hou, P. Zhao, E. Fan and Z. Qiao, Algebro-geometric solutions for the Degasperis-Procesi hierarchy, SIAM J. Math. Anal., 45 (2013), 1216-1266. doi: 10.1137/12089689X. [31] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. [32] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Lecture Notes in Math., Springer Verlag, Berlin, 448 (1975), 25–70. [33] Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820. doi: 10.1007/s00220-006-0082-5. [34] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3. [35] H. P. McKean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874. doi: 10.4310/AJM.1998.v2.n4.a10. [36] V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002. [37] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. [38] Z. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys., 239 (2003), 309-341. doi: 10.1007/s00220-003-0880-y. [39] Z. Qiao, Integrable hierarchy (the DP hierarchy), 3 by 3 constrained systems, and parametric and stationary solutions, Acta Applicandae Mathematicae, 83 (2004), 199-220. doi: 10.1023/B:ACAP.0000038872.88367.dd. [40] Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9pp. doi: 10.1063/1.2365758. [41] Z. Qiao, New integrable hierarchy, its parametric solutions, cuspons, one-peak solutions, and M/W-shape peak solitons, J. Math. Phys., 48 (2007), 082701, 20pp. doi: 10.1063/1.2759830. [42] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. [43] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. [44] G. B. Whitham, Linear and Nonlinear Waves, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9781118032954. [45] X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 11 (2012), 707-727. [46] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [47] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269 (2011), 1113-1127. doi: 10.1007/s00209-010-0775-5. [48] K. Yan, Z. Qiao and Z. Yin, Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Comm. Math. Phys., 336 (2015), 581-617. doi: 10.1007/s00220-014-2236-1. [49] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. [50] Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. doi: 10.1016/j.jfa.2003.07.010.
two-peakon solutions $u$ and $v$ given by (2.10) with $A_1 = 1$, $A_2 = 2$, $A_3 = 3$ and $c = 1$
Two-peakon solutions $u$ and $v$ given by (2.11) with $A_1 = 0$, $A_2 = 1$, $A_3 = 3$ and $c = 0$
Two-peakon solutions $u$ and $v$ given by (2.12) with $A_1 = 0$, $A_2 = -1$, $A_3 = -3$ and $c = \frac{1}{10}$
Conservation laws
 CH equation DP equation Novikov equation Is $\int_{\mathbb{R}}\, (u m)(t, x) d x$ conserved? yes no yes Is $\int_{\mathbb{R}}\, m(t, x) d x$ conserved? yes yes no
 CH equation DP equation Novikov equation Is $\int_{\mathbb{R}}\, (u m)(t, x) d x$ conserved? yes no yes Is $\int_{\mathbb{R}}\, m(t, x) d x$ conserved? yes yes no
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