December  2021, 14(12): 4659-4675. doi: 10.3934/dcdss.2021132

Qualitative analysis for a new generalized 2-component Camassa-Holm system

1. 

College of Mathematics Science and National Center for Applied Mathematics, Chongqing Normal University, Chongqing 401331, China

2. 

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

* Corresponding author: Shanshan Zheng

Received  August 2021 Revised  August 2021 Published  December 2021 Early access  October 2021

Fund Project: The first author Zhou is supported by National Science Foundation of China (Grant No. 11971082), Natural Science Foundation of Chongqing (Grant No. csts2020jcyj-jqX0022) and the Chongqing's youth talent support program, Science and Technology Research Program of Chongqing Municipal Educational Commission (Grant Nos.KJZD-M201900501 and KJQN202000518). The second author Zheng is supported by Graduate research innovation project of Chongqing (Grant No. CYS21277)

This paper considers the Cauchy problem for a 2-component Camassa-Holm system
$ \begin{equation*} m_t = ( u m)_x+ u _xm- v m, \ \ n_t = ( u n)_x+ u _xn+ v n, \end{equation*} $
where
$ n+m = \frac{1}{2}( u _{xx}-4 u ) $
,
$ n-m = v _x $
, this model was proposed in [2] from a novel method to the Euler-Bernoulli Beam on the basis of an inhomogeneous matrix string problem. The local well-posedness in Sobolev spaces
$ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $
with
$ s>\frac{5}{2} $
of this system was investigated through the Kato's theory, then the blow-up criterion for this system was described by the technique on energy methods. Finally, we established the analyticity in both time and space variables of the solutions for this system with a given analytic initial data.
Citation: Shouming Zhou, Shanshan Zheng. Qualitative analysis for a new generalized 2-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4659-4675. doi: 10.3934/dcdss.2021132
References:
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R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

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R. M. ChenL. FanH. Gao and Y. Liu, Breaking waves and solitary waves to the rotation-two-component Camassa-Holm system, SIAM J. Math. Anal., 49 (2017), 3573-3602.  doi: 10.1137/16M1073005.  Google Scholar

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R. M. ChenW. LianD. Wang and R. Xu, A rigidity property for the Novikov equation and the asymptotic stability of peakons, Arch. Ration. Mech. Anal., 241 (2021), 497-533.  doi: 10.1007/s00205-021-01658-z.  Google Scholar

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Y. ChenH. Gao and Y. Liu, On the Cauchy problem for the two-component Dullin-Gottwald-Holm system, Discrete Contin. Dyn. Syst., 33 (2013), 3407-3441.  doi: 10.3934/dcds.2013.33.3407.  Google Scholar

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J. Chu and J. Escher, Variational formulations of steady rotational equatorial waves, Adv. Nonlinear Anal., 10 (2021), 534-547.  doi: 10.1515/anona-2020-0146.  Google Scholar

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G. M. Coclite and L. d. Ruvo, Discontinuous solutions for the short-pulse master mode-locking equation, AIMS Math., 4 (2019), 437-462.  doi: 10.3934/math.2019.3.437.  Google Scholar

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A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

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A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[12]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar

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[15]

J. EscherD. HenryB. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271.  doi: 10.1007/s10231-014-0461-z.  Google Scholar

[16]

E. D. Farnum and J. N. Kutz, Dynamics of a low-dimensional model for short pulse mode locking, Photonics, 2 (2015), 865-882.  doi: 10.3390/photonics2030865.  Google Scholar

[17]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their B$\ddot{a}$klund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

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G. L. GuiY. Liu and T. X. Tian, Global existence and blow-up phenomena for the peakon b-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234.  doi: 10.1512/iumj.2008.57.3213.  Google Scholar

[19]

A. Himonas and C. Holliman, On well-posedness of the Degasperis-Procesi equation, Discrete Contin. Dyn. Syst., 31 (2011), 469-488.  doi: 10.3934/dcds.2011.31.469.  Google Scholar

[20]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.  doi: 10.1137/S1111111102410943.  Google Scholar

[21]

R. I. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267-2280.  doi: 10.1098/rsta.2007.2007.  Google Scholar

[22]

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.   Google Scholar

[23]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.  Google Scholar

[24]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[25]

M. Li and Z. Yin, Global existence and local well-posedness of the single-cycle pulse equation, J. Math. Phys., 58 (2017), 101515, 16pp. doi: 10.1063/1.5001381.  Google Scholar

[26]

W. Luo and Z. Yin, Gevrey regularity and analyticity for Camassa-Holm type systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1061-1079.   Google Scholar

[27]

Y. LiuD. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291-310.  doi: 10.4310/DPDE.2009.v6.n4.a1.  Google Scholar

[28]

Z. LuoZ. Qiao and Z. Yin, On the Cauchy problem for a modified Camassa-Holm equation, Monatsh. Math., 193 (2020), 857-877.  doi: 10.1007/s00605-020-01426-3.  Google Scholar

[29]

Y. MiY. LiuB. Guo and T. Luo, The Cauchy problem for a generalized Camassa-Holm equation, J. Differential Equations, 266 (2019), 6739-6770.  doi: 10.1016/j.jde.2018.11.019.  Google Scholar

[30]

A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.  doi: 10.1088/0305-4470/35/22/309.  Google Scholar

[31]

B. Moon, Single peaked traveling wave solutions to a generalized $\mu$-Novikov equation, Adv. Nonlinear Anal., 10 (2021), 66-75.  doi: 10.1515/anona-2020-0106.  Google Scholar

[32]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E(3), 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.  Google Scholar

[33]

D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Comm. Partial Differential Equations, 35 (2010), 613-629.  doi: 10.1080/03605300903509104.  Google Scholar

[34]

T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.  doi: 10.1016/j.physd.2004.04.007.  Google Scholar

[35]

S. WuJ. Escher and Z. Yin, Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 633-645.  doi: 10.3934/dcdsb.2009.12.633.  Google Scholar

[36]

R. Xu and Y. Yang, Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Discrete Contin. Dyn. Syst., 40 (2020), 6507-6527.  doi: 10.3934/dcds.2020288.  Google Scholar

[37]

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.  Google Scholar

[38]

M. YangY. Li and Z. Qiao, Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system, Discrete Contin. Dyn. Syst., 40 (2020), 2475-2493.  doi: 10.3934/dcds.2020122.  Google Scholar

[39]

Z. ZhaqilaoQ. Hu and Z. Qiao, Multi-soliton solutions and the Cauchy problem for a two-component short pulse system, Nonlinearity, 30 (2017), 3773-3798.  doi: 10.1088/1361-6544/aa7e9c.  Google Scholar

[40]

S. Zhou, Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs, J. Evol. Equ., 14 (2014), 727-747.  doi: 10.1007/s00028-014-0236-4.  Google Scholar

[41]

M. ZhuY. Liu and Y. Mi, Wave-breaking phenomena and persistence properties for the nonlocal rotation-Camassa-Holm equation, Ann. Mat. Pura Appl., 199 (2020), 355-377.  doi: 10.1007/s10231-019-00882-5.  Google Scholar

show all references

References:
[1]

A. AlsaediB. AhmadM. Kirane and B. T. Torebek, Blowing-up solutions of the time-fractional dispersive equations, Adv. Nonlinear Anal., 10 (2021), 952-971.  doi: 10.1515/anona-2020-0153.  Google Scholar

[2]

R. Beals and J. Szmigielski, A 2-Component Camassa-Holm Equation, Euler-Bernoulli Beam Problem and Non-Commutative Continued Fractions, arXiv: 2011.05964. Google Scholar

[3]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[4]

G. ChenR. M. Chen and Y. Liu, Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation, Indiana Univ. Math. J., 67 (2018), 2393-2433.  doi: 10.1512/iumj.2018.67.7510.  Google Scholar

[5]

R. M. ChenL. FanH. Gao and Y. Liu, Breaking waves and solitary waves to the rotation-two-component Camassa-Holm system, SIAM J. Math. Anal., 49 (2017), 3573-3602.  doi: 10.1137/16M1073005.  Google Scholar

[6]

R. M. ChenW. LianD. Wang and R. Xu, A rigidity property for the Novikov equation and the asymptotic stability of peakons, Arch. Ration. Mech. Anal., 241 (2021), 497-533.  doi: 10.1007/s00205-021-01658-z.  Google Scholar

[7]

Y. ChenH. Gao and Y. Liu, On the Cauchy problem for the two-component Dullin-Gottwald-Holm system, Discrete Contin. Dyn. Syst., 33 (2013), 3407-3441.  doi: 10.3934/dcds.2013.33.3407.  Google Scholar

[8]

J. Chu and J. Escher, Variational formulations of steady rotational equatorial waves, Adv. Nonlinear Anal., 10 (2021), 534-547.  doi: 10.1515/anona-2020-0146.  Google Scholar

[9]

G. M. Coclite and L. d. Ruvo, Discontinuous solutions for the short-pulse master mode-locking equation, AIMS Math., 4 (2019), 437-462.  doi: 10.3934/math.2019.3.437.  Google Scholar

[10]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[11]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[12]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar

[13]

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.  Google Scholar

[14]

J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, J. Reine Angew. Math., 624 (2008), 51-80.  doi: 10.1515/CRELLE.2008.080.  Google Scholar

[15]

J. EscherD. HenryB. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271.  doi: 10.1007/s10231-014-0461-z.  Google Scholar

[16]

E. D. Farnum and J. N. Kutz, Dynamics of a low-dimensional model for short pulse mode locking, Photonics, 2 (2015), 865-882.  doi: 10.3390/photonics2030865.  Google Scholar

[17]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their B$\ddot{a}$klund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[18]

G. L. GuiY. Liu and T. X. Tian, Global existence and blow-up phenomena for the peakon b-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234.  doi: 10.1512/iumj.2008.57.3213.  Google Scholar

[19]

A. Himonas and C. Holliman, On well-posedness of the Degasperis-Procesi equation, Discrete Contin. Dyn. Syst., 31 (2011), 469-488.  doi: 10.3934/dcds.2011.31.469.  Google Scholar

[20]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.  doi: 10.1137/S1111111102410943.  Google Scholar

[21]

R. I. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267-2280.  doi: 10.1098/rsta.2007.2007.  Google Scholar

[22]

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.   Google Scholar

[23]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.  Google Scholar

[24]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[25]

M. Li and Z. Yin, Global existence and local well-posedness of the single-cycle pulse equation, J. Math. Phys., 58 (2017), 101515, 16pp. doi: 10.1063/1.5001381.  Google Scholar

[26]

W. Luo and Z. Yin, Gevrey regularity and analyticity for Camassa-Holm type systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1061-1079.   Google Scholar

[27]

Y. LiuD. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291-310.  doi: 10.4310/DPDE.2009.v6.n4.a1.  Google Scholar

[28]

Z. LuoZ. Qiao and Z. Yin, On the Cauchy problem for a modified Camassa-Holm equation, Monatsh. Math., 193 (2020), 857-877.  doi: 10.1007/s00605-020-01426-3.  Google Scholar

[29]

Y. MiY. LiuB. Guo and T. Luo, The Cauchy problem for a generalized Camassa-Holm equation, J. Differential Equations, 266 (2019), 6739-6770.  doi: 10.1016/j.jde.2018.11.019.  Google Scholar

[30]

A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.  doi: 10.1088/0305-4470/35/22/309.  Google Scholar

[31]

B. Moon, Single peaked traveling wave solutions to a generalized $\mu$-Novikov equation, Adv. Nonlinear Anal., 10 (2021), 66-75.  doi: 10.1515/anona-2020-0106.  Google Scholar

[32]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E(3), 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.  Google Scholar

[33]

D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Comm. Partial Differential Equations, 35 (2010), 613-629.  doi: 10.1080/03605300903509104.  Google Scholar

[34]

T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.  doi: 10.1016/j.physd.2004.04.007.  Google Scholar

[35]

S. WuJ. Escher and Z. Yin, Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 633-645.  doi: 10.3934/dcdsb.2009.12.633.  Google Scholar

[36]

R. Xu and Y. Yang, Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Discrete Contin. Dyn. Syst., 40 (2020), 6507-6527.  doi: 10.3934/dcds.2020288.  Google Scholar

[37]

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.  Google Scholar

[38]

M. YangY. Li and Z. Qiao, Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system, Discrete Contin. Dyn. Syst., 40 (2020), 2475-2493.  doi: 10.3934/dcds.2020122.  Google Scholar

[39]

Z. ZhaqilaoQ. Hu and Z. Qiao, Multi-soliton solutions and the Cauchy problem for a two-component short pulse system, Nonlinearity, 30 (2017), 3773-3798.  doi: 10.1088/1361-6544/aa7e9c.  Google Scholar

[40]

S. Zhou, Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs, J. Evol. Equ., 14 (2014), 727-747.  doi: 10.1007/s00028-014-0236-4.  Google Scholar

[41]

M. ZhuY. Liu and Y. Mi, Wave-breaking phenomena and persistence properties for the nonlocal rotation-Camassa-Holm equation, Ann. Mat. Pura Appl., 199 (2020), 355-377.  doi: 10.1007/s10231-019-00882-5.  Google Scholar

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