zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A two-component generalization of the Camassa-Holm equation and its solutions. (English) Zbl 1105.35102
The authors propose a two-component generalization of the Camassa-Holm equation. They establish a reciprocal transformation between this system and the first negative flow of the AKNS (Ablowitz-Kaup-Newell-Segur) hierarchy.

35Q53KdV-like (Korteweg-de Vries) equations
37K35Lie-Bäcklund and other transformations
Full Text: DOI
[1] Abenta S., Grava T. Modulation of Camassa--Holm equation and reciprocal transformations. math-ph/0506042
[2] Ablowitz M.J., Kaup D.J., Newell A.C., Segur H. (1974). The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53:249--315 · Zbl 0408.35068
[3] Alber M.S., Camassa R., Fedorov Yu.N., Holm D.D., Marsden J.E. (2001). The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDEs of shallow water and Dym type. Comm. Math. Phys. 221:197--227 · Zbl 1001.37062 · doi:10.1007/PL00005573
[4] Antonowicz M., Fordy A.P. (1987). Coupled K dV equations with multi-Hamiltonian structures. Physica D28:345--357 · Zbl 0638.35079 · doi:10.1016/0167-2789(87)90023-6
[5] Antonowicz M., Fordy A.P. (1988). Coupled Harry Dym equations with multi-Hamiltonian structures. J. Phys. A21:L269--L275 · Zbl 0673.35088 · doi:10.1088/0305-4470/21/5/001
[6] Antonowicz M., Fordy A.P. (1989). Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems. Comm. Math. Phys. 124:465--486 · Zbl 0696.35172 · doi:10.1007/BF01219659
[7] Beals R., Sattinger D.H., Szmigielski J. (2000). Multipeakons and the classical moment problem. Adv. Math. 154:229--257 · Zbl 0968.35008 · doi:10.1006/aima.1999.1883
[8] Camassa R., Holm D.D. (1993). An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71:1661--1664 · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[9] Camassa R., Holm D.D., Hyman J.M. (1994). A new integrable shallow water equation. Adv. Appl. Mech. 31:1--33 · Zbl 0808.76011 · doi:10.1016/S0065-2156(08)70254-0
[10] Constantin A. (2001). On the scattering problem for the Camassa--Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457:953--970 · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[11] Constantin A., McKean H.P. (1999). A shallow water equation on the circle. Comm. Pure Appl. Math. 52:949--982 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[12] Constantin A., Strauss W.A. (2002). Stability of the Camassa--Holm solitons. J. Nonlinear Sci. 12:415--422 · Zbl 1022.35053 · doi:10.1007/s00332-002-0517-x
[13] Constantin A., Strauss W.A. (2000). Stability of peakons. Comm. Pure Appl. Math. 53:603--610 · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[14] Degasperis, A., Holm, D.D., Hone, A.N.W.: Integrable and non-integrable equations with peakons. In: Nonlinear physics: theory and experiment, II (Gallipoli, 2002), pp. 37--43, World Scientific, River Edge, (2003). · Zbl 1053.37039
[15] Dubrovin, B., Zhang, Y.: Normal forms of integrable PDEs, Frobenius manifolds and Gromov--Witten invariants. math.DG/0108160
[16] Dubrovin, B., Liu, S.Q., Zhang, Y.: On Hamiltonian perturbations of hyperbolic systems of conservation laws, I: quasi-triviality of bi-Hamiltonian perturbations. Commun. Pure Appl. Math. (to appear) math.DG/0410027 · Zbl 1108.35112
[17] Falqui, G., On a two-component generalization of the CH equation. In: Talk given at the conference ”Analytic and geometric theory of the Camassa--Holm equation and Integrable systems”, Bologna (2004)
[18] Fokas A.S. (1995). On a class of physically important integrable equations. Physica D87:145--150 · Zbl 1194.35363
[19] Fuchssteiner B. (1996). Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa--Holm equation. Physica D95:229--243 · Zbl 0900.35345 · doi:10.1016/0167-2789(96)00048-6
[20] Fuchssteiner B., Fokas A.S. (1981). Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D4:47--66 · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[21] Gu C.H., Zhou Z.X. (1987). On the Darboux matrices of Bäcklund transformations for AKNS systems. Lett. Math. Phys. 13:179--187 · Zbl 0645.35086 · doi:10.1007/BF00423444
[22] Hone A.N.W. (1999). The associated Camassa--Holm equation and the K dV equation. J. Phys. A 32:L307--L314 · Zbl 0989.37065 · doi:10.1088/0305-4470/32/27/103
[23] Jaulent M., Jean C. (1976). The inverse problem for the one-dimensional Schrödinger equation with an energy-dependent potential. I. Ann. Inst. H. Poincar Sect. A (N.S.) 25:105--118 · Zbl 0357.34018
[24] Johnson R.S. (2003). On solutions of the Camassa--Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459:1687--1708 · Zbl 1039.76006 · doi:10.1098/rspa.2002.1078
[25] Li Y.S., Zhang J.E. (2004). The multiple-soliton solution of the Camassa--Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 460:2617--2627 · Zbl 1068.35109 · doi:10.1098/rspa.2004.1331
[26] Li, Y.S., Zhang, J.E.: Analytical multiple-soliton solution of the Camassa--Holm equation. Preprint 2004, J. Nonlinear Math. Phys. (to appear) · Zbl 1068.35109
[27] Liu S.Q., Zhang Y. (2005). Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 54:427--453 · Zbl 1079.37058 · doi:10.1016/j.geomphys.2004.11.003
[28] Martĺnez Alonso L. (1980). Schrödinger spectral problems with energy-dependent potentials as sources of nonlinear Hamiltonian evolution equations. J. Math. Phys. 21:2342--2349 · Zbl 0455.35111 · doi:10.1063/1.524690
[29] Matveev V.B., Salle M.A. (1991). Darboux transformations and solitons In: Springer series in nonlinear dynamics. Springer, Berlin Heildelberg New York · Zbl 0744.35045
[30] McKean H. (2003). The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56:998--1015 · Zbl 1037.37030
[31] McKean H. (2004). Breakdown of the Camassa--Holm equation. Comm. Pure Appl. Math. 57:416--418 · Zbl 1052.35130 · doi:10.1002/cpa.20003
[32] Schiff J. (1998). The Camassa--Holm equation: a loop group approach. Physica D 121(1--2):24--43 · Zbl 0943.37034 · doi:10.1016/S0167-2789(98)00099-2