December  2019, 11(4): 561-573. doi: 10.3934/jgm.2019028

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

* Corresponding author: Andrew Hone; permanent address: SMSAS, University of Kent, Canterbury, UK

This article is dedicated to Darryl Holm on his 70th birthday.

Received  April 2018 Revised  June 2019 Published  November 2019

Fund Project: The second author is supported by EPSRC fellowship EP/M004333/1. The fourth author is supported by EPSRC grant EP/P012698/1

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.

Citation: Mike Hay, Andrew N. W. Hone, Vladimir S. Novikov, Jing Ping Wang. Remarks on certain two-component systems with peakon solutions. Journal of Geometric Mechanics, 2019, 11 (4) : 561-573. doi: 10.3934/jgm.2019028
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.  Google Scholar

[2]

A. ArsieP. Lorenzoni and A. Moro, Integrable viscous conservation laws, Nonlinearity, 28 (2015), 1859-1895.  doi: 10.1088/0951-7715/28/6/1859.  Google Scholar

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S. Butler and M. Hay, Simple identification of fake Lax pairs, AIP Conference Proceedings, 1648 (2015), 180006, arXiv: 1311.2406v1. Google Scholar

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F. Calogero and M. C. Nucci, Lax pairs galore, J. Math. Phys., 32 (1991), 72-74.  doi: 10.1063/1.529096.  Google Scholar

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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.  Google Scholar

[8]

X.-K. ChangX.-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.  Google Scholar

[9]

M. ChenS.-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.  Google Scholar

[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.  Google Scholar

[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. DubrovinS.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

K. Grayshan and A. A. Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. N. W. HoneV. Novikov and J. P. Wang, Two-component generalizations of the Camassa-Holm equation, Nonlinearity, 30 (2017), 622-658.  doi: 10.1088/1361-6544/aa5490.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

A. V. MikhailovV. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

B. Q. XiaZ. 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.  Google Scholar

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.  Google Scholar

[2]

A. ArsieP. Lorenzoni and A. Moro, Integrable viscous conservation laws, Nonlinearity, 28 (2015), 1859-1895.  doi: 10.1088/0951-7715/28/6/1859.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

X.-K. ChangX.-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.  Google Scholar

[9]

M. ChenS.-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.  Google Scholar

[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.  Google Scholar

[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. DubrovinS.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

K. Grayshan and A. A. Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. N. W. HoneV. Novikov and J. P. Wang, Two-component generalizations of the Camassa-Holm equation, Nonlinearity, 30 (2017), 622-658.  doi: 10.1088/1361-6544/aa5490.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

A. V. MikhailovV. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

B. Q. XiaZ. 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.  Google Scholar

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