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Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential
Conserved quantities, global existence and blow-up for a generalized CH equation
1. | Department of Mathematics, Hangzhou Dianzi University, Zhejiang 310018, China |
2. | Department of Mathematics, The University of Texas Rio Grande Valley, Edinburg, TX 78541, USA |
3. | College of Mathematics and statistics, Chongqing University, Chongqing 401331, China |
In this paper, we study conserved quantities, blow-up criterions, and global existence of solutions for a generalized CH equation. We investigate the classification of self-adjointness, conserved quantities for this equation from the viewpoint of Lie symmetry analysis. Then, based on these conserved quantities, blow-up criterions and global existence of solutions are presented.
References:
[1] |
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. |
[2] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl.(Singap.), 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[3] |
V. Busuioc,
On second grade fluids with vanishing viscosity, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1241-1246.
doi: 10.1016/S0764-4442(99)80447-9. |
[4] |
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. |
[5] |
A. Constantin,
On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.
doi: 10.1006/jfan.1997.3231. |
[6] |
A. Constantin,
Existence of permanent 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,
On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[8] |
A. Constantin and J. Escher,
Well-posedness, global existence, and blow-up 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,
Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Super. Pisa Cl. Sci., 26 (1998), 303-328.
|
[10] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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 D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[13] |
A. Constantin and H. 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. |
[14] |
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. |
[15] |
A. Degasperis, D. Holm and A. Hone,
A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474.
doi: 10.1023/A:1021186408422. |
[16] |
A. Degasperis, D. Holm and A. Hone,
Integral and non-integrable equations with peakons, in Nonlinear Physics: Theory and Experiment, Ⅱ (Gallipoli, 2002), World Sci. Publ., River Edge, NJ, (2003), 37-43.
doi: 10.1142/9789812704467_0005. |
[17] |
A. Degasperis and M. Procesi,
Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Scientific Publ., River Edge, NJ, (1999), 23-37.
|
[18] |
H. Dullin, G. Gottwald and D. Holm,
On asymptotically equivalent shallow water wave equations, Phys. D., 190 (2004), 1-14.
doi: 10.1016/j.physd.2003.11.004. |
[19] |
H. Dullin, G. Gottwald and D. Holm, An integrable shallow water equation with linear and nonlinear dispersion Phys. Rev. Lett. 87 (2001), 194501.
doi: 10.1103/PhysRevLett.87.194501. |
[20] |
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. |
[21] |
B. Fuchssteiner and A. Fokas,
Symplectic structures, their backlund transformation and hereditary symmetries, Phys. D, 4 (1981), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[22] |
M. Gandarias, Weak self-adjoint differential equations,
J. Phys. A: Math. Theor. 44 (2011), 262001. |
[23] |
M. Gandarias, M. Redondo and M. Bruzon,
Some weak self-adjoint Hamilton Jacobi Bellman equations arising in financial mathematics, Nonlinear Anal.: RWA, 13 (2012), 340-347.
doi: 10.1016/j.nonrwa.2011.07.041. |
[24] |
K. Grayshan and A. Himonas,
Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232.
|
[25] |
G. Gui, Y. Liu and T. 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. |
[26] |
A. Himonas and C. Holliman,
The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differ. Equations, 19 (2014), 161-200.
|
[27] |
A. Himonas and C. Thompson, Persistence properties and unique continuation for a generalized Camassa-Holm equation J. Math. Phys. 55 (2014), 091503, 12pp.
doi: 10.1063/1.4895572. |
[28] |
H. Holden and X. Raynaud,
Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.
doi: 10.3934/dcds.2009.24.1047. |
[29] |
D. Holm and M. 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. |
[30] |
J. Holmes,
Continuity properties of the data-to-solution map for the generalized Camassa-Holm equation, J. Math. Anal. Appl., 417 (2014), 635-642.
doi: 10.1016/j.jmaa.2014.03.033. |
[31] |
A. Hone, H. Lundmark and J. Szmigielski,
Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation, Dyn. Partial Differential Eqns., 6 (2009), 253-289.
doi: 10.4310/DPDE.2009.v6.n3.a3. |
[32] |
A. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity J. Phys. A 41 (2008), 372002, 10pp.
doi: 10.1088/1751-8113/41/37/372002. |
[33] |
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. |
[34] |
N. Ibragimov,
A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.
doi: 10.1016/j.jmaa.2006.10.078. |
[35] |
N. Ibragimov,
Quasi-self-adjoint differential equations, Archives of ALGA., 4 (2007), 55-60.
|
[36] |
N. Ibragimov, Nonlinear self-adjointness and conservation laws J. Phys. A: Math. Theor. 44 (2011), 432002, 8pp.
doi: 10.1088/1751-8113/44/43/432002. |
[37] |
N. Ibragimov, M. Torrisi and R. Traciná, Self-adjointness and conservation laws of a generalized Burgers equation J. Phys. A: Math. Theor. 44 (2011), 145201, 5pp.
doi: 10.1088/1751-8113/44/14/145201. |
[38] |
D. Ionescu-Kruse,
Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312.
doi: 10.2991/jnmp.2007.14.3.1. |
[39] |
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. |
[40] |
R. S. Johnson,
The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111.
doi: 10.1016/S0169-5983(03)00036-4. |
[41] |
T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations, (Proc. Sympos. , Dundee, 1974; dedicated to Konrad Jorgens), pp. 25-70. Lecture Notes in Math. , Vol. 448, Springer, Berlin, 1975. |
[42] |
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. |
[43] |
H. McKean,
Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.
doi: 10.4310/AJM.1998.v2.n4.a10. |
[44] |
A. Mikhailov and V. Novikov,
Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.
doi: 10.1088/0305-4470/35/22/309. |
[45] |
L. Ni and Y. Zhou,
Well-posedness and persistence properties for the Novikov equation, J. Differential Equations, 250 (2011), 3002-3021.
doi: 10.1016/j.jde.2011.01.030. |
[46] |
W. Niu and S. Zhang,
Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation, J. Math. Anal. Appl., 374 (2011), 166-177.
doi: 10.1016/j.jmaa.2010.08.002. |
[47] |
V. Novikov, Generalizations of the Camassa-Holm equation J. Phys. A 42 (2009), 342002, 14pp.
doi: 10.1088/1751-8113/42/34/342002. |
[48] |
P. Olver,
Applications of Lie Groups to Differential Equations New York: Springer-Verlag, 1993.
doi: 10.1007/978-1-4612-4350-2. |
[49] |
Z. Qiao,
The Camassa-Holm hierarchy, $N$-dimensional integrable systems, and algebrogeometric solution on a symplectic submanifold, Commu. Math. Phys., 239 (2003), 309-341.
doi: 10.1007/s00220-003-0880-y. |
[50] |
Z. Qiao,
Integrable hierarchy, $3× 3$ constrained systems, and parametric and stationary solutions, Acta Appl. Math., 83 (2004), 199-220.
doi: 10.1023/B:ACAP.0000038872.88367.dd. |
[51] |
L. Wei,
Conservation laws for a modified lubrication equation, Nonlinear Analysis: RWA, 26 (2015), 44-55.
doi: 10.1016/j.nonrwa.2015.04.005. |
[52] |
L. Wei and J. Zhang,
Self-adjointness and conservation laws for Kadomtsev-Petviashvili-Burgers equation, Nonlinear Analysis: RWA, 23 (2015), 123-128.
doi: 10.1016/j.nonrwa.2014.11.008. |
[53] |
X. Wu and Z. Yin,
Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Sup. Pisa CI. Sci., 11 (2012), 707-727.
|
[54] |
Z. Xin and P. Zhang,
On the weak solution 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. |
[55] |
W. Yan, Y. Li and Y. Zhang,
The Cauchy problem for the integrable Novikov equation, J. Differential Equations, 253 (2012), 298-318.
doi: 10.1016/j.jde.2012.03.015. |
[56] |
Z. Yin,
On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.
|
[57] |
S. Zhou and C. Mu,
The properties of solutions for a generalized $b$-family equation with peakons, J. Nonlinear Sci., 23 (2013), 863-889.
doi: 10.1007/s00332-013-9171-8. |
[58] |
S. Zhou,
The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted spaces, Discrete Contin. Dyn. Syst., 34 (2014), 4967-4986.
doi: 10.3934/dcds.2014.34.4967. |
[59] |
S. Zhou, C. Mu and L. Wang,
Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation, Discrete Contin. Dyn. Syst., 34 (2014), 843-867.
doi: 10.3934/dcds.2014.34.843. |
show all references
References:
[1] |
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. |
[2] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl.(Singap.), 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[3] |
V. Busuioc,
On second grade fluids with vanishing viscosity, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1241-1246.
doi: 10.1016/S0764-4442(99)80447-9. |
[4] |
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. |
[5] |
A. Constantin,
On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.
doi: 10.1006/jfan.1997.3231. |
[6] |
A. Constantin,
Existence of permanent 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,
On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[8] |
A. Constantin and J. Escher,
Well-posedness, global existence, and blow-up 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,
Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Super. Pisa Cl. Sci., 26 (1998), 303-328.
|
[10] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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 D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[13] |
A. Constantin and H. 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. |
[14] |
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. |
[15] |
A. Degasperis, D. Holm and A. Hone,
A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474.
doi: 10.1023/A:1021186408422. |
[16] |
A. Degasperis, D. Holm and A. Hone,
Integral and non-integrable equations with peakons, in Nonlinear Physics: Theory and Experiment, Ⅱ (Gallipoli, 2002), World Sci. Publ., River Edge, NJ, (2003), 37-43.
doi: 10.1142/9789812704467_0005. |
[17] |
A. Degasperis and M. Procesi,
Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Scientific Publ., River Edge, NJ, (1999), 23-37.
|
[18] |
H. Dullin, G. Gottwald and D. Holm,
On asymptotically equivalent shallow water wave equations, Phys. D., 190 (2004), 1-14.
doi: 10.1016/j.physd.2003.11.004. |
[19] |
H. Dullin, G. Gottwald and D. Holm, An integrable shallow water equation with linear and nonlinear dispersion Phys. Rev. Lett. 87 (2001), 194501.
doi: 10.1103/PhysRevLett.87.194501. |
[20] |
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. |
[21] |
B. Fuchssteiner and A. Fokas,
Symplectic structures, their backlund transformation and hereditary symmetries, Phys. D, 4 (1981), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[22] |
M. Gandarias, Weak self-adjoint differential equations,
J. Phys. A: Math. Theor. 44 (2011), 262001. |
[23] |
M. Gandarias, M. Redondo and M. Bruzon,
Some weak self-adjoint Hamilton Jacobi Bellman equations arising in financial mathematics, Nonlinear Anal.: RWA, 13 (2012), 340-347.
doi: 10.1016/j.nonrwa.2011.07.041. |
[24] |
K. Grayshan and A. Himonas,
Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232.
|
[25] |
G. Gui, Y. Liu and T. 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. |
[26] |
A. Himonas and C. Holliman,
The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differ. Equations, 19 (2014), 161-200.
|
[27] |
A. Himonas and C. Thompson, Persistence properties and unique continuation for a generalized Camassa-Holm equation J. Math. Phys. 55 (2014), 091503, 12pp.
doi: 10.1063/1.4895572. |
[28] |
H. Holden and X. Raynaud,
Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.
doi: 10.3934/dcds.2009.24.1047. |
[29] |
D. Holm and M. 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. |
[30] |
J. Holmes,
Continuity properties of the data-to-solution map for the generalized Camassa-Holm equation, J. Math. Anal. Appl., 417 (2014), 635-642.
doi: 10.1016/j.jmaa.2014.03.033. |
[31] |
A. Hone, H. Lundmark and J. Szmigielski,
Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation, Dyn. Partial Differential Eqns., 6 (2009), 253-289.
doi: 10.4310/DPDE.2009.v6.n3.a3. |
[32] |
A. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity J. Phys. A 41 (2008), 372002, 10pp.
doi: 10.1088/1751-8113/41/37/372002. |
[33] |
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. |
[34] |
N. Ibragimov,
A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.
doi: 10.1016/j.jmaa.2006.10.078. |
[35] |
N. Ibragimov,
Quasi-self-adjoint differential equations, Archives of ALGA., 4 (2007), 55-60.
|
[36] |
N. Ibragimov, Nonlinear self-adjointness and conservation laws J. Phys. A: Math. Theor. 44 (2011), 432002, 8pp.
doi: 10.1088/1751-8113/44/43/432002. |
[37] |
N. Ibragimov, M. Torrisi and R. Traciná, Self-adjointness and conservation laws of a generalized Burgers equation J. Phys. A: Math. Theor. 44 (2011), 145201, 5pp.
doi: 10.1088/1751-8113/44/14/145201. |
[38] |
D. Ionescu-Kruse,
Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312.
doi: 10.2991/jnmp.2007.14.3.1. |
[39] |
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. |
[40] |
R. S. Johnson,
The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111.
doi: 10.1016/S0169-5983(03)00036-4. |
[41] |
T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations, (Proc. Sympos. , Dundee, 1974; dedicated to Konrad Jorgens), pp. 25-70. Lecture Notes in Math. , Vol. 448, Springer, Berlin, 1975. |
[42] |
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. |
[43] |
H. McKean,
Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.
doi: 10.4310/AJM.1998.v2.n4.a10. |
[44] |
A. Mikhailov and V. Novikov,
Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.
doi: 10.1088/0305-4470/35/22/309. |
[45] |
L. Ni and Y. Zhou,
Well-posedness and persistence properties for the Novikov equation, J. Differential Equations, 250 (2011), 3002-3021.
doi: 10.1016/j.jde.2011.01.030. |
[46] |
W. Niu and S. Zhang,
Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation, J. Math. Anal. Appl., 374 (2011), 166-177.
doi: 10.1016/j.jmaa.2010.08.002. |
[47] |
V. Novikov, Generalizations of the Camassa-Holm equation J. Phys. A 42 (2009), 342002, 14pp.
doi: 10.1088/1751-8113/42/34/342002. |
[48] |
P. Olver,
Applications of Lie Groups to Differential Equations New York: Springer-Verlag, 1993.
doi: 10.1007/978-1-4612-4350-2. |
[49] |
Z. Qiao,
The Camassa-Holm hierarchy, $N$-dimensional integrable systems, and algebrogeometric solution on a symplectic submanifold, Commu. Math. Phys., 239 (2003), 309-341.
doi: 10.1007/s00220-003-0880-y. |
[50] |
Z. Qiao,
Integrable hierarchy, $3× 3$ constrained systems, and parametric and stationary solutions, Acta Appl. Math., 83 (2004), 199-220.
doi: 10.1023/B:ACAP.0000038872.88367.dd. |
[51] |
L. Wei,
Conservation laws for a modified lubrication equation, Nonlinear Analysis: RWA, 26 (2015), 44-55.
doi: 10.1016/j.nonrwa.2015.04.005. |
[52] |
L. Wei and J. Zhang,
Self-adjointness and conservation laws for Kadomtsev-Petviashvili-Burgers equation, Nonlinear Analysis: RWA, 23 (2015), 123-128.
doi: 10.1016/j.nonrwa.2014.11.008. |
[53] |
X. Wu and Z. Yin,
Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Sup. Pisa CI. Sci., 11 (2012), 707-727.
|
[54] |
Z. Xin and P. Zhang,
On the weak solution 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. |
[55] |
W. Yan, Y. Li and Y. Zhang,
The Cauchy problem for the integrable Novikov equation, J. Differential Equations, 253 (2012), 298-318.
doi: 10.1016/j.jde.2012.03.015. |
[56] |
Z. Yin,
On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.
|
[57] |
S. Zhou and C. Mu,
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