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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

R. CamassaD. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. Google Scholar

[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. Google Scholar

[19]

A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perturbation Theory, Rome, 1998, World Sci. Publishing, River Edge, NJ, 1999, 23–37. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[43]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar

[44]

G. B. Whitham, Linear and Nonlinear Waves, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9781118032954. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[49]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

R. CamassaD. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. Google Scholar

[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. Google Scholar

[19]

A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perturbation Theory, Rome, 1998, World Sci. Publishing, River Edge, NJ, 1999, 23–37. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[43]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar

[44]

G. B. Whitham, Linear and Nonlinear Waves, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9781118032954. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[49]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. Google Scholar

[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. Google Scholar

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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