Nonlocal Symmetries of the Camassa-Holm Type Equations

Abstract

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.

Keywords

Nonlocal symmetry / Recursion operator / Camassa-Holm equation / Modified Camassa-Holm equation / Novikov equation / Degasperis-Procesi equation / Liouville correspondence

References

[1] Akhatov I S, Gazizov R K, Ibragimov N K. Nonlocal symmetries, Heuristic approach. J. Sov. Math., 1991, 55: 1401-1450

[2] Bluman G W, Kumei S. Symmetry-based algorithms to relate partial differential equations, Linearization by nonlocal symmetries. Eur. J. Appl. Math., 1990, 1: 217-213

[3] Bluman G W, Kumei S, Reid G J. New classes of symmetries for partial differential equations. J. Math. Phys., 1988, 29: 806-811

[4] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 1993, 71: 1661-1664

[5] Camassa R, Holm D D, Hyman J. An new integrable shallow water equation. Adv. Appl. Mech., 1994, 31: 1-33

[6] Caudrey P J, Dodd R K, Gibbon J D. A new hierarchy of Korteweg-de Vries equations. Proc. Roy. Soc. London Ser. A, 1976, 351: 407-422

[7] Constantin A, Gerdjikov V, Ivanov R. Inverse scattering transform for the Camassa-Holm equation. Inverse Problems, 2006, 22: 2197-2207

[8] Constantin A, Lannes D. The hydrodynamical relevance of the Camassa-Holm and Degasperis- Procesi equations. Arch. Ration. Mech. Anal., 2009, 192: 165-186

[9] Degasperis A, Holm D D, Hone A N W. A new integrable equation with peakon solutions. Theor. Math. Phys., 2002, 133: 1463-1474

[10] Degasperis A, Procesi M. Asymptotic integrability, Symmetry and perturbation theory, 23, World Sci. Publ., River Edge, NJ, 1999

[11] Fokas A S, Fuchssteiner B. B¨acklund transformations for hereditary symmetries. Nonlinear Anal., 1981, 5: 423-432

[12] Fokas A S, Olver P J, Rosenau P. A plethora of integrable bi-Hamiltonian equations, Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., 26, Birkh¨auser Boston, Boston, MA, 1997 93-101

[13] Fuchssteiner B. Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation. Physica D, 1996, 95: 229-243

[14] Fuchssteiner B, Oevel W. The bi-Hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants. J. Math. Phys., 1982, 23: 358-363

[15] Gui G L, Liu Y, Olver P J, Qu C Z. Wave-breaking and peakons for a modified Camassa-Holm equation. Commun. Math. Phys., 2013, 319: 731-759

[16] Guthrie G A. Recursion operators and nonlocal symmetries. Proc. R. Soc. London. Ser. A, 1994, 446: 107-114

[17] Guthrie G A, Hickman M S. Nonlocal symmetries of the KdV equation. J. Math. Phys., 1993, 34: 193-205

[18] Han P, Lou SY. Abundant symmetry structure of the Kaup-Kupershmidt hierarchies. Chin. Phys. Lett., 1993, 10: 257-260

[19] Hernndez-Heredero R, Reyes E G. Nonlocal symmetries and a Darboux transformation for the Camassa-Holm equation. J. Phys. A, 2009, 42: 182002

[20] Hone A N W, Wang J P. Integrable peakon equations with cubic nonlinearity. J. Phys. A: Math. Theor., 2008, 41: 372002

[21] Huang L L, Chen Y. Nonlocal symmetry and similarity reductions for the Drinfeld-Sokolov-Satsuma- Hirota system. Appl. Math. Lett., 2017, 64: 177-184

[22] Huang L L, Chen Y. Nonlocal symmetry and similarity reductions for a (2+1)-dimensional Korteweg-de Vries equation. Nonlinear Dyn., 2018, 92: 221-234

[23] Kang J, Liu X C, Olver P J, Qu C Z. Liouville correspondence between the modified KdV hierarchy and its dual integrable hierarchy. J. Nonlinear Sci., 2016, 26: 141-170

[24] Kang J, Liu X C, Olver P J, Qu CZ. Liouville correspondences between integrable hierarchies. SIGMA, 2017, 13: 26

[25] Kaup D J. On the inverse scattering problem for cubic eigenvalue problems of the class. Stud. Appl. Math., 1980, 62: 189-216

[26] Lenells J. The correspondence between KdV and Camassa-Holm. Int. Math. Res. Not., 2004, 71: 3797-3811

[27] Lou S Y. Twelve sets of symmetries of the Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Phys. Lett. A, 1993, 175: 23-26

[28] Lou S Y. Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations. J. Math. Phys., 1994, 35: 2390-2396

[29] Lou S Y, Hu X B. Non-local symmetries via Darboux transformations. J. Phys. A, 1997, 30: 95-100

[30] Magri F. A simple model of the integrable Hamiltonian equation. J. Math. Phys., 1978, 19: 1156-1162

[31] McKean H P. The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies. Commun. Pure Appl. Math., 2003, 56: 998-1015

[32] Novikov V. Generalizations of the Camassa-Holm equation. J. Phys. A: Math. Theor., 2009, 42: 342002

[33] Olver P J. Evolution equations possessing infinitely many symmetries. J. Math. Phys., 1977, 18: 1212-1215

[34] Olver P J. Applications of Lie Groups to Differential Equations, 2nd ed., Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1993

[35] Olver P J. Shabat A B. Nonlocal Symmetries and Ghosts, New Trends in Integrability and Partial Solvability. Kluwet Academic, Dordrecht, The Netherlands, 2004 199-215

[36] Olver P J, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E, 1996, 53: 1900-1906

[37] Olver P J, Sanders J, Wang J P. Ghost symmetries. J. Nonlin. Math. Phys., 2002, 9: 164-172

[38] Qu C Z. Potential symmetries to systems of nonlinear diffusion equations. J. Phys. A, 2007, 40: 1757-1773

[39] Reyes E G. Geometric integrability of the Camassa-Holm equation. Lett. Math. Phys., 2002, 59: 117-131

[40] Reyes E G. Nonlocal symmetries and Kaup-Kupershmidt equation. J. Math. Phys., 2005, 46: 073507

[41] Reyes E G. On nonlocal symmetries of some shallow water equations. J. Phys. A, 2007, 40: 4467-4476

[42] Sawada K, Kotera T. A method for finding N-soliton solutions of the KdV equation and KdV-like equation. Prog. Theor. Phys., 1974, 51: 1355-1367

[43] Sergyeyev A. Why nonlocal recursion operators produce local symmetries: new results and applications. J. Phys. A, 2005, 38: 3397-3407

[44] Shi Z H, Kang J, Yan L. Nonlocal symmetries and conservation laws of nonlocal Camassa-Holm equations. Acta Math. Appl. Sinica, English Series, 2015, 31: 909-920

[45] Vinogradov A M, Krasilshchik I S. A method of calculating higher-symmetries of nonlinear evolutionary equations, and nonlocal symmtries. Dokl. Akad. Nauk. SSSR, 1980, 253: 1289-1293