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. CamassaD. 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. ConstantinR. 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. DegasperisD. 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. DullinG. 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. HouP. ZhaoE. 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. YanZ. 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. CamassaD. 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. ConstantinR. 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. DegasperisD. 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. DullinG. 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. HouP. ZhaoE. 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. YanZ. 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.

Figure 1.  two-peakon solutions $u$ and $v$ given by (2.10) with $A_1 = 1$, $A_2 = 2$, $A_3 = 3$ and $c = 1$
Figure 2.  Two-peakon solutions $u$ and $v$ given by (2.11) with $A_1 = 0$, $A_2 = 1$, $A_3 = 3$ and $c = 0$
Figure 3.  Two-peakon solutions $u$ and $v$ given by (2.12) with $A_1 = 0$, $A_2 = -1$, $A_3 = -3$ and $c = \frac{1}{10}$
Table 1.  Conservation laws
CH equationDP equationNovikov equation
Is $\int_{\mathbb{R}}\, (u m)(t, x) d x$ conserved?yesnoyes
Is $\int_{\mathbb{R}}\, m(t, x) d x$ conserved?yesyesno
CH equationDP equationNovikov equation
Is $\int_{\mathbb{R}}\, (u m)(t, x) d x$ conserved?yesnoyes
Is $\int_{\mathbb{R}}\, m(t, x) d x$ conserved?yesyesno
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