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On the concentration of semiclassical states for nonlinear Dirac equations
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 |
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.
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. |



![]() | CH equation | DP equation | Novikov equation |
Is | yes | no | yes |
Is | yes | yes | no |
![]() | CH equation | DP equation | Novikov equation |
Is | yes | no | yes |
Is | yes | yes | no |
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