×

Liouville correspondence between the modified KdV hierarchy and its dual integrable hierarchy. (English) Zbl 1344.37075

Authors’ abstract: We study an explicit correspondence between the integrable modified KdV hierarchy and its dual integrable modified Camassa-Holm hierarchy. A Liouville transformation between the isospectral problems of the two hierarchies also relates their respective recursion operators and serves to establish the Liouville correspondence between their flows and Hamiltonian conservation laws. In addition, a novel transformation mapping the modified Camassa-Holm equation to the Camassa-Holm equation is found. Furthermore, it is shown that the Hamiltonian conservation laws in the negative direction of the modified Camassa-Holm hierarchy are both local in the field variables and homogeneous under rescaling.
Reviewer: Ma Wen-Xiu (Tampa)

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beals, R., Sattinger, D.H., Szmigielski, J.: Acoustic scattering and the extended Korteweg de Vries hierarchy. Adv. Math. 140, 190-206 (1998) · Zbl 0919.35118 · doi:10.1006/aima.1998.1768
[2] Beals, R., Sattinger, D.H., Szmigielski, J.: Multipeakons and the classical moment problem. Adv. Math. 154, 229-257 (2000) · Zbl 0968.35008 · doi:10.1006/aima.1999.1883
[3] Bies, P.M., Gorka, P., Reyes, E.: The dual modified Korteweg-de Vries-Fokas-Qiao equation: geometry and local analysis. J. Math. Phys. 53, 073710 (2012) · Zbl 1297.37032 · doi:10.1063/1.4736845
[4] Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[5] Camassa, R., Holm, D.D., Hyman, J.: An new integrable shallow water equation. Adv. Appl. Mech. 31, 1-33 (1994) · Zbl 0808.76011 · doi:10.1016/S0065-2156(08)70254-0
[6] Cao, C.W., Geng, X.G.: A nonconfocal generator of involute systems and three associated soliton hierarchies. J. Math. Phys. 32, 2323-2328 (1991) · Zbl 0737.35083 · doi:10.1063/1.529156
[7] Chou, K.S., Qu, C.Z.: Integrable equations arising from motions of plane curves I. Phys. D 162, 9-33 (2002) · Zbl 0987.35139 · doi:10.1016/S0167-2789(01)00364-5
[8] Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier 50, 321-362 (2000) · Zbl 0944.35062 · doi:10.5802/aif.1757
[9] Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. Roy. Soc. Lond. Ser. A 457, 953-970 (2001) · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[10] Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229-243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[11] Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa 26, 303-328 (1998) · Zbl 0918.35005
[12] Constantin, A., Gerdjikov, V., Ivanov, R.: Inverse scattering transform for the Camassa-Holm equation. Inverse Prob. 22, 2197-2207 (2006) · Zbl 1105.37044 · doi:10.1088/0266-5611/22/6/017
[13] Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 192, 165-186 (2009) · Zbl 1169.76010 · doi:10.1007/s00205-008-0128-2
[14] Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949-982 (1999) · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[15] Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53, 603-610 (2000) · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[16] El Dika, K., Molinet, L.: Stability of multipeakons. Ann. Inst. Henri Poincaré Anal. Nonlinéaire 18, 1517-1532 (2009) · Zbl 1171.35459 · doi:10.1016/j.anihpc.2009.02.002
[17] Fisher, M., Schiff, J.: The Camassa-Holm equation: conserved quantities and the initial value problem. Phys. Lett. A 259, 371-376 (1999) · Zbl 0936.35166 · doi:10.1016/S0375-9601(99)00466-1
[18] Fokas, A.S.: The Korteweg-de Vries equation and beyond. Acta Appl. Math. 39, 295-305 (1995a) · Zbl 0842.58045
[19] Fokas, A.S.: On a class of physically important integrable equations. Phys. D 87, 145-150 (1995b) · Zbl 1194.35363
[20] Fokas, A.S., Fuchssteiner, B.: Bäcklund transformations for hereditary symmetries. Nonlinear Anal. 5, 423-432 (1981) · Zbl 0491.35007 · doi:10.1016/0362-546X(81)90025-0
[21] Fokas, A.S., Olver, P.J., Rosenau, P.: A plethora of integrable bi-Hamiltonian equations. In: Algebraic Aspects of Integrable Systems, pp. 93-101. Progr. Nonlinear Differential Equations Appl., vol. 26, Birkhäuser Boston, Boston, MA, (1997) · Zbl 0865.35121
[22] Fuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. Phys. D 95, 229-243 (1996) · Zbl 0900.35345 · doi:10.1016/0167-2789(96)00048-6
[23] Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D, 4, 47-66 (1981/1982) · Zbl 1194.37114
[24] Gui, G.L., Liu, Y., Olver, P.J., Qu, C.Z.: Wave-breaking and peakons for a modified Camassa-Holm equation. Commun. Math. Phys. 319, 731-759 (2013) · Zbl 1263.35186 · doi:10.1007/s00220-012-1566-0
[25] Hernandez-Heredero, R., Reyes, E.: Geometric integrability of the Camassa-Holm equation II. Int. Math. Res. Not. 2012, 3089-3125 (2012) · Zbl 1251.35126
[26] Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63-82 (2002) · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[27] Johnson, R.S.: On solutions of the Camassa-Holm equation. Proc. Roy. Soc. Lond. Ser. A 459, 1687-1708 (2003) · Zbl 1039.76006 · doi:10.1098/rspa.2002.1078
[28] Kouranbaeva, S.: The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40, 857-868 (1999) · Zbl 0958.37060 · doi:10.1063/1.532690
[29] Lenells, J.: The correspondence between KdV and Camassa-Holm. Int. Math. Res. Not. 71, 3797-3811 (2004) · Zbl 1082.35134 · doi:10.1155/S1073792804142451
[30] Lenells, J.: Conservation laws of the Camassa-Holm equation. J. Phys. A: Math. Gen. 38, 869-880 (2005) · Zbl 1076.35100 · doi:10.1088/0305-4470/38/4/007
[31] Li, Y.A., Olver, P.J.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ. 162, 27-63 (2000) · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683
[32] Li, Y.A., Olver, P.J., Rosenau, P.: Non-analytic solutions of nonlinear wave models. In: Grosser, M., Höormann, G., Kunzinger, M., Oberguggenberger, M. (eds.) Nonlinear Theory of Generalized Functions, pp. 129-145. Research Notes in Mathematics, vol. 401, Chapman and Hall/CRC, New York (1999) · Zbl 0940.35176
[33] Liu, X.C., Liu, Y., Olver, P.J., Qu, C.Z.: Orbital stability of peakons for a generalization of the modified Camassa-Holm equation. Nonlinearity 22, 2297-2319 (2014) · Zbl 1307.35035 · doi:10.1088/0951-7715/27/9/2297
[34] Liu, X.C., Liu, Y., Qu, C.Z.: Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation. Adv. Math. 255, 1-37 (2014) · Zbl 1288.35063 · doi:10.1016/j.aim.2013.12.032
[35] Liu, Y., Olver, P.J., Qu, C.Z., Zhang, S.H.: On the blow-up of solutions to the integrable modified Camassa-Holm equation. Anal. Appl. 12, 355-368 (2014) · Zbl 1302.35074 · doi:10.1142/S0219530514500274
[36] Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156-1162 (1978) · Zbl 0383.35065 · doi:10.1063/1.523777
[37] McKean, H.P.: The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies. Commun. Pure Appl. Math. 56, 998-1015 (2003) · Zbl 1037.37030 · doi:10.1002/cpa.10083
[38] Milson, R.: Liouville transformation and exactly solvable Schrödinger equations. Int. J. Theor. Phys. 37, 1735-1752 (1998) · Zbl 0929.34075 · doi:10.1023/A:1026696709617
[39] Newell, A.C.: Solitons in mathematics and physics. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 48, SIAM, Philadelphia (1985) · Zbl 0565.35003
[40] Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974) · Zbl 0303.41035
[41] Olver, P.J.: Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18, 1212-1215 (1977) · Zbl 0348.35024 · doi:10.1063/1.523393
[42] Olver, P.J.: Applications of Lie groups to differential equations. Graduate Text in Mathematics, vol. 107, 2nd edn. Springer, New York (1993) · Zbl 0785.58003
[43] Olver, PJ; Shabat, AB (ed.); etal., Nonlocal symmetries and ghosts, 199-215 (2004), Dordrecht · Zbl 1072.35193 · doi:10.1007/978-94-007-1023-8_8
[44] Olver, P.J., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900-1906 (1996) · doi:10.1103/PhysRevE.53.1900
[45] Olver, P.J., Sanders, J., Wang, J.P.: Ghost symmetries. J. Nonlinear Math. Phys. 9(Suppl. 1), 164-172 (2002) · doi:10.2991/jnmp.2002.9.s1.14
[46] Qiao, Z.: A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math. Phys. 47, 112701 (2006) · Zbl 1112.37063
[47] Qu, C.Z., Liu, X.C., Liu, Y.: Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity. Commun. Math. Phys. 322, 967-997 (2013) · Zbl 1307.35264 · doi:10.1007/s00220-013-1749-3
[48] Reyes, E.: Geometric integrability of the Camassa-Holm equation. Lett. Math. Phys. 59, 117-131 (2002) · Zbl 0997.35081 · doi:10.1023/A:1014933316169
[49] Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations. Cambridge University Press, Cambridge (2002) · Zbl 1019.53002 · doi:10.1017/CBO9780511606359
[50] Rosenau, P.: On solitons, compactons, and Lagrange maps. Phys. Lett. A 211, 265-275 (1996) · Zbl 1059.35524 · doi:10.1016/0375-9601(95)00933-7
[51] Schiff, J.: Zero curvature formulations of dual hierarchies. J. Math. Phys. 37, 1928-1938 (1996) · Zbl 0863.35093 · doi:10.1063/1.531486
[52] Schiff, J.: The Camassa-Holm equation: a loop group approach. Phys. D 121, 24-43 (1998) · Zbl 0943.37034 · doi:10.1016/S0167-2789(98)00099-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.