Liouville Correspondence Between the Modified KdV Hierarchy and Its Dual Integrable Hierarchy

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• 作者：Jing Kang ; Xiaochuan Liu ; Peter J. Olver ; Changzheng Qu
• 关键词：Liouville transformation ; Modified Camassa–Holm hierarchy ; Modified KdV hierarchy ; Tri ; Hamiltonian duality ; Hamiltonian conservation law ; Local conservation law ; Scaling homogeneity ; 37K05 ; 37K10
• 刊名：Journal of Nonlinear Science
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：26
• 期：1
• 页码：141-170
• 全文大小：706 KB
• 参考文献：Beals, R., Sattinger, D.H., Szmigielski, J.: Acoustic scattering and the extended Korteweg de Vries hierarchy. Adv. Math. 140, 190–206 (1998)MathSciNet CrossRef MATH
  Beals, R., Sattinger, D.H., Szmigielski, J.: Multipeakons and the classical moment problem. Adv. Math. 154, 229–257 (2000)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef
  Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MathSciNet CrossRef MATH
  Camassa, R., Holm, D.D., Hyman, J.: An new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)CrossRef
  Cao, C.W., Geng, X.G.: A nonconfocal generator of involute systems and three associated soliton hierarchies. J. Math. Phys. 32, 2323–2328 (1991)MathSciNet CrossRef MATH
  Chou, K.S., Qu, C.Z.: Integrable equations arising from motions of plane curves I. Phys. D 162, 9–33 (2002)MathSciNet CrossRef MATH
  Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier 50, 321–362 (2000)MathSciNet CrossRef MATH
  Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. Roy. Soc. Lond. Ser. A 457, 953–970 (2001)MathSciNet CrossRef MATH
  Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNet CrossRef MATH
  Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa 26, 303–328 (1998)MathSciNet MATH
  Constantin, A., Gerdjikov, V., Ivanov, R.: Inverse scattering transform for the Camassa–Holm equation. Inverse Prob. 22, 2197–2207 (2006)MathSciNet CrossRef MATH
  Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)MathSciNet CrossRef MATH
  Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999)MathSciNet CrossRef
  Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)MathSciNet CrossRef MATH
  El Dika, K., Molinet, L.: Stability of multipeakons. Ann. Inst. Henri Poincaré Anal. Nonlinéaire 18, 1517–1532 (2009)CrossRef
  Fisher, M., Schiff, J.: The Camassa–Holm equation: conserved quantities and the initial value problem. Phys. Lett. A 259, 371–376 (1999)MathSciNet CrossRef MATH
  Fokas, A.S.: The Korteweg–de Vries equation and beyond. Acta Appl. Math. 39, 295–305 (1995a)
  Fokas, A.S.: On a class of physically important integrable equations. Phys. D 87, 145–150 (1995b)
  Fokas, A.S., Fuchssteiner, B.: Bäcklund transformations for hereditary symmetries. Nonlinear Anal. 5, 423–432 (1981)MathSciNet CrossRef MATH
  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)
  Fuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation. Phys. D 95, 229–243 (1996)MathSciNet CrossRef MATH
  Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D, 4, 47–66 (1981/1982)
  Gui, G.L., Liu, Y., Olver, P.J., Qu, C.Z.: Wave
  eaking and peakons for a modified Camassa–Holm equation. Commun. Math. Phys. 319, 731–759 (2013)MathSciNet CrossRef MATH
  Hernandez-Heredero, R., Reyes, E.: Geometric integrability of the Camassa–Holm equation II. Int. Math. Res. Not. 2012, 3089–3125 (2012)MathSciNet MATH
  Johnson, R.S.: Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)MathSciNet CrossRef MATH
  Johnson, R.S.: On solutions of the Camassa–Holm equation. Proc. Roy. Soc. Lond. Ser. A 459, 1687–1708 (2003)CrossRef MATH
  Kouranbaeva, S.: The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40, 857–868 (1999)MathSciNet CrossRef MATH
  Lenells, J.: The correspondence between KdV and Camassa–Holm. Int. Math. Res. Not. 71, 3797–3811 (2004)MathSciNet CrossRef
  Lenells, J.: Conservation laws of the Camassa–Holm equation. J. Phys. A: Math. Gen. 38, 869–880 (2005)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef MATH
  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)
  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)MathSciNet CrossRef
  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)MathSciNet CrossRef MATH
  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)MathSciNet CrossRef MATH
  Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978)MathSciNet CrossRef MATH
  McKean, H.P.: The Liouville correspondence between the Korteweg–de Vries and the Camassa–Holm hierarchies. Commun. Pure Appl. Math. 56, 998–1015 (2003)MathSciNet CrossRef MATH
  Milson, R.: Liouville transformation and exactly solvable Schrödinger equations. Int. J. Theor. Phys. 37, 1735–1752 (1998)MathSciNet CrossRef MATH
  Newell, A.C.: Solitons in mathematics and physics. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 48, SIAM, Philadelphia (1985)
  Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)
  Olver, P.J.: Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18, 1212–1215 (1977)MathSciNet CrossRef MATH
  Olver, P.J.: Applications of Lie groups to differential equations. Graduate Text in Mathematics, vol. 107, 2nd edn. Springer, New York (1993)
  Olver, P.J.: Nonlocal symmetries and ghosts. In: Shabat, A.B., et al. (eds.) New Trends in Integrability and Partial Solvability, pp. 199–215. Kluwer, Dordrecht (2004)CrossRef
  Olver, P.J., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)MathSciNet CrossRef
  Olver, P.J., Sanders, J., Wang, J.P.: Ghost symmetries. J. Nonlinear Math. Phys. 9(Suppl. 1), 164–172 (2002)MathSciNet CrossRef
  Qiao, Z.: A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math. Phys. 47, 112701 (2006)
  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)MathSciNet CrossRef MATH
  Reyes, E.: Geometric integrability of the Camassa–Holm equation. Lett. Math. Phys. 59, 117–131 (2002)MathSciNet CrossRef MATH
  Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations. Cambridge University Press, Cambridge (2002)CrossRef MATH
  Rosenau, P.: On solitons, compactons, and Lagrange maps. Phys. Lett. A 211, 265–275 (1996)MathSciNet CrossRef MATH
  Schiff, J.: Zero curvature formulations of dual hierarchies. J. Math. Phys. 37, 1928–1938 (1996)MathSciNet CrossRef MATH
  Schiff, J.: The Camassa–Holm equation: a loop group approach. Phys. D 121, 24–43 (1998)MathSciNet CrossRef MATH
  
• 作者单位：Jing Kang (1)
  Xiaochuan Liu (1)
  Peter J. Olver (2)
  Changzheng Qu (3)
  
  1. Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi’an, 710069, People’s Republic of China
  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA
  3. Department of Mathematics, Ningbo University, Ningbo, 315211, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Mathematical and Computational Physics
  Mechanics
  Applied Mathematics and Computational Methods of Engineering
  Economic Theory
  
• 出版者：Springer New York
• ISSN：1432-1467

文摘

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. Keywords Liouville transformation Modified Camassa–Holm hierarchy Modified KdV hierarchy Tri-Hamiltonian duality Hamiltonian conservation law Local conservation law Scaling homogeneity