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January  2020, 40(1): 529-548. doi: 10.3934/dcds.2020021

Compacton equations and integrability: The rosenau-hyman and Cooper-Shepard-Sodano equations

1. 

Departamento de Matemática Aplicada a las TIC, ETSIST. Universidad Politécnica de Madrid, C. Nikola Tesla s/n, 28031 Madrid, Spain

2. 

Department of Mathematics, Jinan University, Guangzhou 510632, China

3. 

Centro Internacional de Ciencias, Av. Universidad s/n, Colonia Chamilpa, 62210 Cuernavaca, Morelos, Mexico

4. 

Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile

* Corresponding author: R. Hernández Heredero

Received  April 2019 Revised  June 2019 Published  October 2019

We study integrability –in the sense of admitting recursion operators– of two nonlinear equations which are known to possess compacton solutions: the
$ K(m, n) $
equation introduced by Rosenau and Hyman
$ D_t(u) + D_x(u^m) + D_x^3(u^n) = 0 \; , $
and the CSS equation introduced by Coooper, Shepard, and Sodano,
$ D_t(u) + u^{l-2}D_x(u) + \alpha p D_x (u^{p-1} u_x^2) + 2\alpha D_x^2(u^p u_x) = 0 \; . $
We obtain a full classification of integrable
$ K(m, n) $
and CSS equations
; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of CSS.
Citation: Rafael Hernández Heredero, Marianna Euler, Norbert Euler, Enrique G. Reyes. Compacton equations and integrability: The rosenau-hyman and Cooper-Shepard-Sodano equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 529-548. doi: 10.3934/dcds.2020021
References:
[1]

M. Adler, On the trace functional for formal pseudodifferential operators and the symplectic structure of the KdV type equations, Inventiones Math., 50 (1979), 219-248.  doi: 10.1007/BF01410079.  Google Scholar

[2]

D. M. Ambrose and J. D. Wright, Preservation of support and positivity for solutions of degenerate evolution equations, Nonlinearity, 23 (2010), 607-620.  doi: 10.1088/0951-7715/23/3/010.  Google Scholar

[3]

D. M. Ambrose and J. D. Wright, Dispersion versus anti-diffusion:well-posedness in variable coefficient and quasilinear equations of KdV type, Indiana Univ. Math. J., 62 (2013), 1237-1281.  doi: 10.1512/iumj.2013.62.5049.  Google Scholar

[4]

D. M. AmbroseG. SimpsonJ. D. Wright and D. G. Yang, Ill-posedness of degenerate dispersive equations, Nonlinearity, 25 (2012), 2655-2680.  doi: 10.1088/0951-7715/25/9/2655.  Google Scholar

[5]

I. M. Bakirov, On the Symmetries of some System of Evolution Equations, Technical Report, Institute of Mathematics, Russian Academy of Sciences, Ufa, 1991. Google Scholar

[6]

F. BeukersJ. A. Sanders and J. P. Wang, One symmetry does not imply integrability, J. Diff. Eq., 146 (1998), 251-260.  doi: 10.1006/jdeq.1998.3426.  Google Scholar

[7]

A. H. Bilge, A system with a recursion operator but one higher local symmetry, Lie Groups Appl., 1 (1994), 132–139. (Also available as arXiv: 1904.01291).  Google Scholar

[8]

F. Calogero, Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?, In: What Is Integrability?, 1–62. Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991.  Google Scholar

[9] F. Calogero, Isochronous Systems, Oxford University Press, Oxford, UK, 2008.   Google Scholar
[10]

F. Calogero and A. Degasperis, Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform, J. Math. Phys., 22 (1981), 22-31.  doi: 10.1063/1.524750.  Google Scholar

[11]

F. Calogero and M. Mariani, A modified Schwarzian Korteweg-de Vries equation in 2+1 dimensions with lots of isochronous solutions, Yadernaya Fiz., 68 (2005), 1710-1717.  doi: 10.1134/1.2121911.  Google Scholar

[12]

F. CalogeroM. Euler and N. Euler, New evolution PDEs with many isochronous solutions, J. Math. Anbal. Appl., 353 (2009), 481-488.  doi: 10.1016/j.jmaa.2008.12.038.  Google Scholar

[13]

F. CooperM. Shepard and P. Sodano, Solitary waves in a class of generalized Korteweg-de Vries equations, Phys. Rev. E, 48 (1993), 4027-4032.  doi: 10.1103/PhysRevE.48.4027.  Google Scholar

[14]

D. K. Demskoy and V. V. Sokolov, On recursion operators for elliptic models, Nonlinearity, 21 (2008), 1253-1264.  doi: 10.1088/0951-7715/21/6/006.  Google Scholar

[15]

B. Dey and A. Khare, Stability of compacton solutions, Physical Review E, 58 (1998), R2741–R2744. doi: 10.1103/PhysRevE.58.R2741.  Google Scholar

[16]

A. S. Fokas, A symmetry approach to exactly solvable evolution equations, J. Math. Physics, 21 (1980), 1318-1325.  doi: 10.1063/1.524581.  Google Scholar

[17]

R. Hernández Heredero, Integrable quasilinear equations, Teoret. Mat. Fiz., 133 (2002), 233-246.   Google Scholar

[18]

R. Hernández Heredero, Classification of fully nonlinear integrable evolution equations of third order, J. Nonlinear Math. Phys., 12 (2005), 567-585.  doi: 10.2991/jnmp.2005.12.4.10.  Google Scholar

[19]

R. Hernández Heredero and E.G. Reyes, Nonlocal symmetries, compacton equations and integrability, Int. J. Geometric Methods in Modern Physics, 10 (2013), 1350046 (24 pages). doi: 10.1142/S0219887813500461.  Google Scholar

[20]

R. Hernández Heredero, V. V. Sokolov and S. I. Svinolupov, Why are there so many integrable equations of third order?, in Proceedings of NEEDS'94, Los Alamos, ed. E.V. Zhakarov, A.E. Bishop, D.D. Holm (World Scientific, 1994), 1995, 42–53.  Google Scholar

[21]

A. Khare and F. Cooper, One-parameter family of soliton solutions with compact support in a class of generalized Korteweg-de Vries equations., Physical Review E, 48 (1993), 4843-4844.  doi: 10.1103/PhysRevE.48.4843.  Google Scholar

[22]

A. Ludu and J. P. Draayer, Patterns on liquid surfaces: Cnoidal waves, compactons and scaling, Physica D, 123 (1998), 82-91.  doi: 10.1016/S0167-2789(98)00113-4.  Google Scholar

[23]

Y. A. Li, P. J. Olver and P. Rosenau, Non-analytic solutions of nonlinear wave models., In Nonlinear Theory of Generalized Functions (Vienna, 1997), 129–145, Chapman & Hall/CRC Res. Notes Math., 401, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[24]

M. Mariani and F. Calogero, Isochronous PDEs, Yadernaya Fiz., 68 (2005), 935-944.  doi: 10.1134/1.1935022.  Google Scholar

[25]

A. V. Mikhailov, A. B. Shabat and V. V. Sokolov, The symmetry approach to classification of integrable equations, in What is Integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991,115–184.  Google Scholar

[26]

A. V. Mikhailov and V. V. Sokolov, Symmetries of differential equations and the problem of integrability, Lect. Notes Phys., 767 (2009), 19-88.  doi: 10.1007/978-3-540-88111-7_2.  Google Scholar

[27]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Springer-Verlag, New York, 1993.  Google Scholar

[28]

N. PeterssonN. Euler and M. Euler, Recursion operators for a class of integrable third order equations, Studies in Applied Mathematics, 112 (2004), 201-225.  doi: 10.1111/j.0022-2526.2004.01511.x.  Google Scholar

[29]

P. Rosenau and A. Zilburg, Compactons, J. Phys. A: Math. Theor., 51 (2018), 343001 (136pp). doi: 10.1088/1751-8121/aabff5.  Google Scholar

[30]

P. Rosenau, What is $\dots$ a Compacton?, Notices of the AMS, 52 (2005), 738-739.   Google Scholar

[31]

P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelength, Phys. Rev. Lett., 70 (1993), 564-567.   Google Scholar

[32]

J. A. Sanders and J.-P. Wang, On the integrability of homogeneous scalar evolution equations, J. Differential Equations, 147 (1998), 410-434.  doi: 10.1006/jdeq.1998.3452.  Google Scholar

[33]

J. A. Sanders and J.-P. Wang, On the integrability of non-polynomial scalar evolution equations, J. Differential Equations, 166 (2000), 132-150.  doi: 10.1006/jdeq.2000.3782.  Google Scholar

[34]

V. V. Sokolov and V. V. Shabat, Classification of integrable evolution equations, Sov. Sci. Rev. C, 4 (1984), 221-280.   Google Scholar

[35]

S. I. Svinolupov and V. V. Sokolov, Weak nonlocalities in evolution equations, Mathematical Notes, 48 (1990), 1234-1239.  doi: 10.1007/BF01240266.  Google Scholar

[36]

J. Vodová, A complete list of conservation laws for non-integrable compacton equations of $K(m, m)$ type, Nonlinearity, 26 (2013), 757-762.  doi: 10.1088/0951-7715/26/3/757.  Google Scholar

[37]

A. Zilburg and P. Rosenau, Loss of regularity in the $K(m, n)$ equations, Nonlinearity, 31 (2018), 2651-2665.  doi: 10.1088/1361-6544/aab58b.  Google Scholar

show all references

References:
[1]

M. Adler, On the trace functional for formal pseudodifferential operators and the symplectic structure of the KdV type equations, Inventiones Math., 50 (1979), 219-248.  doi: 10.1007/BF01410079.  Google Scholar

[2]

D. M. Ambrose and J. D. Wright, Preservation of support and positivity for solutions of degenerate evolution equations, Nonlinearity, 23 (2010), 607-620.  doi: 10.1088/0951-7715/23/3/010.  Google Scholar

[3]

D. M. Ambrose and J. D. Wright, Dispersion versus anti-diffusion:well-posedness in variable coefficient and quasilinear equations of KdV type, Indiana Univ. Math. J., 62 (2013), 1237-1281.  doi: 10.1512/iumj.2013.62.5049.  Google Scholar

[4]

D. M. AmbroseG. SimpsonJ. D. Wright and D. G. Yang, Ill-posedness of degenerate dispersive equations, Nonlinearity, 25 (2012), 2655-2680.  doi: 10.1088/0951-7715/25/9/2655.  Google Scholar

[5]

I. M. Bakirov, On the Symmetries of some System of Evolution Equations, Technical Report, Institute of Mathematics, Russian Academy of Sciences, Ufa, 1991. Google Scholar

[6]

F. BeukersJ. A. Sanders and J. P. Wang, One symmetry does not imply integrability, J. Diff. Eq., 146 (1998), 251-260.  doi: 10.1006/jdeq.1998.3426.  Google Scholar

[7]

A. H. Bilge, A system with a recursion operator but one higher local symmetry, Lie Groups Appl., 1 (1994), 132–139. (Also available as arXiv: 1904.01291).  Google Scholar

[8]

F. Calogero, Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?, In: What Is Integrability?, 1–62. Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991.  Google Scholar

[9] F. Calogero, Isochronous Systems, Oxford University Press, Oxford, UK, 2008.   Google Scholar
[10]

F. Calogero and A. Degasperis, Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform, J. Math. Phys., 22 (1981), 22-31.  doi: 10.1063/1.524750.  Google Scholar

[11]

F. Calogero and M. Mariani, A modified Schwarzian Korteweg-de Vries equation in 2+1 dimensions with lots of isochronous solutions, Yadernaya Fiz., 68 (2005), 1710-1717.  doi: 10.1134/1.2121911.  Google Scholar

[12]

F. CalogeroM. Euler and N. Euler, New evolution PDEs with many isochronous solutions, J. Math. Anbal. Appl., 353 (2009), 481-488.  doi: 10.1016/j.jmaa.2008.12.038.  Google Scholar

[13]

F. CooperM. Shepard and P. Sodano, Solitary waves in a class of generalized Korteweg-de Vries equations, Phys. Rev. E, 48 (1993), 4027-4032.  doi: 10.1103/PhysRevE.48.4027.  Google Scholar

[14]

D. K. Demskoy and V. V. Sokolov, On recursion operators for elliptic models, Nonlinearity, 21 (2008), 1253-1264.  doi: 10.1088/0951-7715/21/6/006.  Google Scholar

[15]

B. Dey and A. Khare, Stability of compacton solutions, Physical Review E, 58 (1998), R2741–R2744. doi: 10.1103/PhysRevE.58.R2741.  Google Scholar

[16]

A. S. Fokas, A symmetry approach to exactly solvable evolution equations, J. Math. Physics, 21 (1980), 1318-1325.  doi: 10.1063/1.524581.  Google Scholar

[17]

R. Hernández Heredero, Integrable quasilinear equations, Teoret. Mat. Fiz., 133 (2002), 233-246.   Google Scholar

[18]

R. Hernández Heredero, Classification of fully nonlinear integrable evolution equations of third order, J. Nonlinear Math. Phys., 12 (2005), 567-585.  doi: 10.2991/jnmp.2005.12.4.10.  Google Scholar

[19]

R. Hernández Heredero and E.G. Reyes, Nonlocal symmetries, compacton equations and integrability, Int. J. Geometric Methods in Modern Physics, 10 (2013), 1350046 (24 pages). doi: 10.1142/S0219887813500461.  Google Scholar

[20]

R. Hernández Heredero, V. V. Sokolov and S. I. Svinolupov, Why are there so many integrable equations of third order?, in Proceedings of NEEDS'94, Los Alamos, ed. E.V. Zhakarov, A.E. Bishop, D.D. Holm (World Scientific, 1994), 1995, 42–53.  Google Scholar

[21]

A. Khare and F. Cooper, One-parameter family of soliton solutions with compact support in a class of generalized Korteweg-de Vries equations., Physical Review E, 48 (1993), 4843-4844.  doi: 10.1103/PhysRevE.48.4843.  Google Scholar

[22]

A. Ludu and J. P. Draayer, Patterns on liquid surfaces: Cnoidal waves, compactons and scaling, Physica D, 123 (1998), 82-91.  doi: 10.1016/S0167-2789(98)00113-4.  Google Scholar

[23]

Y. A. Li, P. J. Olver and P. Rosenau, Non-analytic solutions of nonlinear wave models., In Nonlinear Theory of Generalized Functions (Vienna, 1997), 129–145, Chapman & Hall/CRC Res. Notes Math., 401, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[24]

M. Mariani and F. Calogero, Isochronous PDEs, Yadernaya Fiz., 68 (2005), 935-944.  doi: 10.1134/1.1935022.  Google Scholar

[25]

A. V. Mikhailov, A. B. Shabat and V. V. Sokolov, The symmetry approach to classification of integrable equations, in What is Integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991,115–184.  Google Scholar

[26]

A. V. Mikhailov and V. V. Sokolov, Symmetries of differential equations and the problem of integrability, Lect. Notes Phys., 767 (2009), 19-88.  doi: 10.1007/978-3-540-88111-7_2.  Google Scholar

[27]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Springer-Verlag, New York, 1993.  Google Scholar

[28]

N. PeterssonN. Euler and M. Euler, Recursion operators for a class of integrable third order equations, Studies in Applied Mathematics, 112 (2004), 201-225.  doi: 10.1111/j.0022-2526.2004.01511.x.  Google Scholar

[29]

P. Rosenau and A. Zilburg, Compactons, J. Phys. A: Math. Theor., 51 (2018), 343001 (136pp). doi: 10.1088/1751-8121/aabff5.  Google Scholar

[30]

P. Rosenau, What is $\dots$ a Compacton?, Notices of the AMS, 52 (2005), 738-739.   Google Scholar

[31]

P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelength, Phys. Rev. Lett., 70 (1993), 564-567.   Google Scholar

[32]

J. A. Sanders and J.-P. Wang, On the integrability of homogeneous scalar evolution equations, J. Differential Equations, 147 (1998), 410-434.  doi: 10.1006/jdeq.1998.3452.  Google Scholar

[33]

J. A. Sanders and J.-P. Wang, On the integrability of non-polynomial scalar evolution equations, J. Differential Equations, 166 (2000), 132-150.  doi: 10.1006/jdeq.2000.3782.  Google Scholar

[34]

V. V. Sokolov and V. V. Shabat, Classification of integrable evolution equations, Sov. Sci. Rev. C, 4 (1984), 221-280.   Google Scholar

[35]

S. I. Svinolupov and V. V. Sokolov, Weak nonlocalities in evolution equations, Mathematical Notes, 48 (1990), 1234-1239.  doi: 10.1007/BF01240266.  Google Scholar

[36]

J. Vodová, A complete list of conservation laws for non-integrable compacton equations of $K(m, m)$ type, Nonlinearity, 26 (2013), 757-762.  doi: 10.1088/0951-7715/26/3/757.  Google Scholar

[37]

A. Zilburg and P. Rosenau, Loss of regularity in the $K(m, n)$ equations, Nonlinearity, 31 (2018), 2651-2665.  doi: 10.1088/1361-6544/aab58b.  Google Scholar

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