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Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers
Remarks on certain two-component systems with peakon solutions
1. | Dipartimento di Matematica e Fisica, Università di Roma Tre, 00146 Roma RM, Italy |
2. | School of Mathematics & Statistics, University of New South Wales, NSW 2052 Sydney, Australia |
3. | School of Mathematics, Loughborough University, Loughborough LE11 3TU, UK |
4. | School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury CT2 7FS, UK |
We consider a Lax pair found by Xia, Qiao and Zhou for a family of two-component analogues of the Camassa-Holm equation, including an arbitrary function $ H $, and show that this apparent freedom can be removed via a combination of a reciprocal transformation and a gauge transformation, which reduces the system to triangular form. The resulting triangular system may or may not be integrable, depending on the choice of $ H $. In addition, we apply the formal series approach of Dubrovin and Zhang to show that scalar equations of Camassa-Holm type with homogeneous nonlinear terms of degree greater than three are not integrable.
References:
[1] |
S. C. Anco, P. L. da Silva and I. L. Freire, A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations, J. Math. Phys., 56 (2015), 091506, 21 pp.
doi: 10.1063/1.4929661. |
[2] |
A. Arsie, P. Lorenzoni and A. Moro,
Integrable viscous conservation laws, Nonlinearity, 28 (2015), 1859-1895.
doi: 10.1088/0951-7715/28/6/1859. |
[3] |
A. Arsie, P. Lorenzoni and A. Moro, On integrable conservation laws, Proc. R. Soc. A, 471 (2015), 20140124, 12 pp.
doi: 10.1098/rspa.2014.0124. |
[4] |
L. E. Barnes and A. N. W. Hone,
Dynamics of conservative peakons in a system of Popowicz, Phys. Lett. A, 383 (2019), 406-413.
doi: 10.1016/j.physleta.2018.11.015. |
[5] |
S. Butler and M. Hay, Simple identification of fake Lax pairs, AIP Conference Proceedings, 1648 (2015), 180006, arXiv: 1311.2406v1. Google Scholar |
[6] |
F. Calogero and M. C. Nucci,
Lax pairs galore, J. Math. Phys., 32 (1991), 72-74.
doi: 10.1063/1.529096. |
[7] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[8] |
X.-K. Chang, X.-B. Hu and J. Szmigielski,
Multipeakons of a two-component modified Camassa-Holm equation and the relation with the finite Kac-van Moerbeke lattice, Adv. Math., 299 (2016), 1-35.
doi: 10.1016/j.aim.2016.05.004. |
[9] |
M. Chen, S.-Q. Liu and Y. J. Zhang,
A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.
doi: 10.1007/s11005-005-0041-7. |
[10] |
C. Cotter, D. Holm, R. Ivanov and J. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation, J. Phys. A: Math. Theor., 44 (2011), 265205.
doi: 10.1088/1751-8113/44/26/265205. |
[11] |
B. Dubrovin and Y. Zhang, Normal forms of hierarchies of integrable PDEs: Frobenius manifolds and Gromov-Witten invariants, preprint, arXiv: math/0108160. Google Scholar |
[12] |
B. Dubrovin, S.-Q. Liu and Y. J. Zhang,
On hamiltonian perturbations of hyperbolic systems of conservation laws. Ⅰ. Quasitriviality of bihamiltonian perturbations, Comm. Pure Appl. Math., 59 (2006), 559-615.
doi: 10.1002/cpa.20111. |
[13] |
B. Dubrovin,
On hamiltonian peturbations of hyperbolic systems of conservation laws Ⅱ. Universality of critical behaviour, Comm. Math. Phys., 267 (2006), 117-139.
doi: 10.1007/s00220-006-0021-5. |
[14] |
B. Dubrovin, Hamiltonian PDEs: Deformations, integrability, solutions, J. Phys. A: Math. Theor., 43 (2010), 434002, 20 pp.
doi: 10.1088/1751-8113/43/43/434002. |
[15] |
G. Falqui,
On a Camassa-Holm type equation with two dependent variables, J. Phys. A: Math. Gen., 39 (2006), 327-342.
doi: 10.1088/0305-4470/39/2/004. |
[16] |
A. S. Fokas,
On a class of physically important integrable equations, Physica D, 87 (1995), 145-150.
doi: 10.1016/0167-2789(95)00133-O. |
[17] |
B. Fuchssteiner and A. S. Fokas,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[18] |
B. Fuchssteiner,
Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243.
doi: 10.1016/0167-2789(96)00048-6. |
[19] |
K. Grayshan and A. A. Himonas,
Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232.
|
[20] |
D. D. Holm, J. T. Ratnanather, A. Trouve and L. Younes, Soliton dynamics in computational anatomy, NeuroImage, 23 (2004), Supplement 1, S170–S178.
doi: 10.1016/j.neuroimage.2004.07.017. |
[21] |
D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: Geometrical aspects, peakons and numerical examples, J. Phys. A: Math. Theor., 43 (2010), 492001, 20 pp.
doi: 10.1088/1751-8113/43/49/492001. |
[22] |
A. N. W. Hone, The associated Camassa-Holm equation and the KdV equation, J. Phys. A: Math. Gen., 32 (1999), L307–L314.
doi: 10.1088/0305-4470/32/27/103. |
[23] |
A. N. W. Hone and J. P. Wang,
Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Probl., 19 (2003), 129-145.
doi: 10.1088/0266-5611/19/1/307. |
[24] |
A. N. W. Hone, V. Novikov and J. P. Wang,
Two-component generalizations of the Camassa-Holm equation, Nonlinearity, 30 (2017), 622-658.
doi: 10.1088/1361-6544/aa5490. |
[25] |
Y. Matsuno,
Parametric representation for the multisoliton solution of the Camassa-Holm equation, J. Phys. Soc. Jpn., 74 (2005), 1983-1987.
doi: 10.1143/JPSJ.74.1983. |
[26] |
H. P. McKean,
Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.
doi: 10.4310/AJM.1998.v2.n4.a10. |
[27] |
A. V. Mikhaĩlov, V. V. Sokolov and A. B. Shabat, The symmetry approach to classification of integrable equations, in What is Integrability? Springer, Ser. Nonlinear Dynam., Springer, Berlin, (1991), 115–184. |
[28] |
A. V. Mikhailov and V. S. Novikov,
Perturbative symmetry approach, J. Phys. A: Math. Gen., 35 (2002), 4775-4790.
doi: 10.1088/0305-4470/35/22/309. |
[29] |
A. V. Mikhailov, V. S. Novikov and J. P. Wang,
Symbolic representation and classification of integrable systems, Algebraic Theory of Differential Equations, London Math. Soc. Lecture Note Ser., Cambridge University Press, 357 (2009), 156-216.
|
[30] |
G. Misiolek,
A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.
doi: 10.1016/S0393-0440(97)00010-7. |
[31] |
V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002, 14 pp.
doi: 10.1088/1751-8113/42/34/342002. |
[32] |
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. |
[33] |
A. Parker,
On the Camassa-Holm equation and a direct method of solution. Ⅱ. Soliton solutions, Proc. Royal Soc. Lond. A Math. Phys. Eng. Sci., 461 (2005), 3611-3632.
doi: 10.1098/rspa.2005.1536. |
[34] |
Z. J. Qiao, New integrable hierarchy, its parametric solutions, cuspons, one-peak solutions, and M/W-shape peak solitons, J. Math. Phys., 48 (2007), 082701, 20 pp.
doi: 10.1063/1.2759830. |
[35] |
S. Y. Sakovich, True and fake Lax pairs: How to distinguish them, preprint, arXiv: nlin/0112027v1. Google Scholar |
[36] |
J. F. Song, C. Z. Qu, and Z. J. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 013503, 9 pp.
doi: 10.1063/1.3530865. |
[37] |
I. A. B. Strachan and B. M. Szablikowski,
Novikov algebras and a classification of multicomponent Camassa-Holm equations, Stud. Appl. Math., 133 (2014), 84-117.
doi: 10.1111/sapm.12040. |
[38] |
B. Q. Xia and Z. J. Qiao, A new two-component integrable system with peakon solutions, Proc. Roy. Soc. Lond. A, 471 (2015), 20140750, 20 pp.
doi: 10.1098/rspa.2014.0750. |
[39] |
B. Q. Xia, Z. J. Qiao and R. G. Zhou,
A synthetical two-component model with peakon solutions, Stud. Appl. Math., 135 (2015), 248-276.
doi: 10.1111/sapm.12085. |
show all references
References:
[1] |
S. C. Anco, P. L. da Silva and I. L. Freire, A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations, J. Math. Phys., 56 (2015), 091506, 21 pp.
doi: 10.1063/1.4929661. |
[2] |
A. Arsie, P. Lorenzoni and A. Moro,
Integrable viscous conservation laws, Nonlinearity, 28 (2015), 1859-1895.
doi: 10.1088/0951-7715/28/6/1859. |
[3] |
A. Arsie, P. Lorenzoni and A. Moro, On integrable conservation laws, Proc. R. Soc. A, 471 (2015), 20140124, 12 pp.
doi: 10.1098/rspa.2014.0124. |
[4] |
L. E. Barnes and A. N. W. Hone,
Dynamics of conservative peakons in a system of Popowicz, Phys. Lett. A, 383 (2019), 406-413.
doi: 10.1016/j.physleta.2018.11.015. |
[5] |
S. Butler and M. Hay, Simple identification of fake Lax pairs, AIP Conference Proceedings, 1648 (2015), 180006, arXiv: 1311.2406v1. Google Scholar |
[6] |
F. Calogero and M. C. Nucci,
Lax pairs galore, J. Math. Phys., 32 (1991), 72-74.
doi: 10.1063/1.529096. |
[7] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[8] |
X.-K. Chang, X.-B. Hu and J. Szmigielski,
Multipeakons of a two-component modified Camassa-Holm equation and the relation with the finite Kac-van Moerbeke lattice, Adv. Math., 299 (2016), 1-35.
doi: 10.1016/j.aim.2016.05.004. |
[9] |
M. Chen, S.-Q. Liu and Y. J. Zhang,
A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.
doi: 10.1007/s11005-005-0041-7. |
[10] |
C. Cotter, D. Holm, R. Ivanov and J. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation, J. Phys. A: Math. Theor., 44 (2011), 265205.
doi: 10.1088/1751-8113/44/26/265205. |
[11] |
B. Dubrovin and Y. Zhang, Normal forms of hierarchies of integrable PDEs: Frobenius manifolds and Gromov-Witten invariants, preprint, arXiv: math/0108160. Google Scholar |
[12] |
B. Dubrovin, S.-Q. Liu and Y. J. Zhang,
On hamiltonian perturbations of hyperbolic systems of conservation laws. Ⅰ. Quasitriviality of bihamiltonian perturbations, Comm. Pure Appl. Math., 59 (2006), 559-615.
doi: 10.1002/cpa.20111. |
[13] |
B. Dubrovin,
On hamiltonian peturbations of hyperbolic systems of conservation laws Ⅱ. Universality of critical behaviour, Comm. Math. Phys., 267 (2006), 117-139.
doi: 10.1007/s00220-006-0021-5. |
[14] |
B. Dubrovin, Hamiltonian PDEs: Deformations, integrability, solutions, J. Phys. A: Math. Theor., 43 (2010), 434002, 20 pp.
doi: 10.1088/1751-8113/43/43/434002. |
[15] |
G. Falqui,
On a Camassa-Holm type equation with two dependent variables, J. Phys. A: Math. Gen., 39 (2006), 327-342.
doi: 10.1088/0305-4470/39/2/004. |
[16] |
A. S. Fokas,
On a class of physically important integrable equations, Physica D, 87 (1995), 145-150.
doi: 10.1016/0167-2789(95)00133-O. |
[17] |
B. Fuchssteiner and A. S. Fokas,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[18] |
B. Fuchssteiner,
Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243.
doi: 10.1016/0167-2789(96)00048-6. |
[19] |
K. Grayshan and A. A. Himonas,
Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232.
|
[20] |
D. D. Holm, J. T. Ratnanather, A. Trouve and L. Younes, Soliton dynamics in computational anatomy, NeuroImage, 23 (2004), Supplement 1, S170–S178.
doi: 10.1016/j.neuroimage.2004.07.017. |
[21] |
D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: Geometrical aspects, peakons and numerical examples, J. Phys. A: Math. Theor., 43 (2010), 492001, 20 pp.
doi: 10.1088/1751-8113/43/49/492001. |
[22] |
A. N. W. Hone, The associated Camassa-Holm equation and the KdV equation, J. Phys. A: Math. Gen., 32 (1999), L307–L314.
doi: 10.1088/0305-4470/32/27/103. |
[23] |
A. N. W. Hone and J. P. Wang,
Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Probl., 19 (2003), 129-145.
doi: 10.1088/0266-5611/19/1/307. |
[24] |
A. N. W. Hone, V. Novikov and J. P. Wang,
Two-component generalizations of the Camassa-Holm equation, Nonlinearity, 30 (2017), 622-658.
doi: 10.1088/1361-6544/aa5490. |
[25] |
Y. Matsuno,
Parametric representation for the multisoliton solution of the Camassa-Holm equation, J. Phys. Soc. Jpn., 74 (2005), 1983-1987.
doi: 10.1143/JPSJ.74.1983. |
[26] |
H. P. McKean,
Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874.
doi: 10.4310/AJM.1998.v2.n4.a10. |
[27] |
A. V. Mikhaĩlov, V. V. Sokolov and A. B. Shabat, The symmetry approach to classification of integrable equations, in What is Integrability? Springer, Ser. Nonlinear Dynam., Springer, Berlin, (1991), 115–184. |
[28] |
A. V. Mikhailov and V. S. Novikov,
Perturbative symmetry approach, J. Phys. A: Math. Gen., 35 (2002), 4775-4790.
doi: 10.1088/0305-4470/35/22/309. |
[29] |
A. V. Mikhailov, V. S. Novikov and J. P. Wang,
Symbolic representation and classification of integrable systems, Algebraic Theory of Differential Equations, London Math. Soc. Lecture Note Ser., Cambridge University Press, 357 (2009), 156-216.
|
[30] |
G. Misiolek,
A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.
doi: 10.1016/S0393-0440(97)00010-7. |
[31] |
V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002, 14 pp.
doi: 10.1088/1751-8113/42/34/342002. |
[32] |
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. |
[33] |
A. Parker,
On the Camassa-Holm equation and a direct method of solution. Ⅱ. Soliton solutions, Proc. Royal Soc. Lond. A Math. Phys. Eng. Sci., 461 (2005), 3611-3632.
doi: 10.1098/rspa.2005.1536. |
[34] |
Z. J. Qiao, New integrable hierarchy, its parametric solutions, cuspons, one-peak solutions, and M/W-shape peak solitons, J. Math. Phys., 48 (2007), 082701, 20 pp.
doi: 10.1063/1.2759830. |
[35] |
S. Y. Sakovich, True and fake Lax pairs: How to distinguish them, preprint, arXiv: nlin/0112027v1. Google Scholar |
[36] |
J. F. Song, C. Z. Qu, and Z. J. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 013503, 9 pp.
doi: 10.1063/1.3530865. |
[37] |
I. A. B. Strachan and B. M. Szablikowski,
Novikov algebras and a classification of multicomponent Camassa-Holm equations, Stud. Appl. Math., 133 (2014), 84-117.
doi: 10.1111/sapm.12040. |
[38] |
B. Q. Xia and Z. J. Qiao, A new two-component integrable system with peakon solutions, Proc. Roy. Soc. Lond. A, 471 (2015), 20140750, 20 pp.
doi: 10.1098/rspa.2014.0750. |
[39] |
B. Q. Xia, Z. J. Qiao and R. G. Zhou,
A synthetical two-component model with peakon solutions, Stud. Appl. Math., 135 (2015), 248-276.
doi: 10.1111/sapm.12085. |
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