×

zbMATH — the first resource for mathematics

Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation. (English) Zbl 0900.35345
Summary: The main subject of the paper is to give a survey and to present new methods on how integrability results (i.e. results for symmetry groups, inverse scattering formulations, action-angle transformations and the like) can be transferred from one equation to others in case the equations are not related by Bäcklund transformations. As a main example the so-called Camassa-Holm equations chosen for which the relevant results are obtained by having a look on the Korteweg de Vries (KdV) equation. The Camassa-Holm equations turns out to be a different-factorization equation of the KdV, it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations. We follow here an old method already marginally mentioned in Fuchssteiner and Fokas (1981), and Fuchssteiner (1983) and recently applied by others (Olver and Rosenau, 1995). The method allows an immediate recovery of the recursion operator for the Camassa-Holm equation from the invariance structure of the KdV, although both equations are not related by Bäcklund transformations. However, in addition, and different from other approaches, from there by use of the squared eigenfunction relation for the KdV equation, the Lax pair formulation for the different-factorization equation is derived.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benjamin, T.B; Bona, J.L; Mahony, J.J, Model equations for long waves in nonlinear dispersive systems, J. philos. trans. roy. soc. London A, 272, 47-78, (1972) · Zbl 0229.35013
[2] Bona, J; Smith, R, Existence of solutions to the Korteweg-de Vries initial value problem, (), 179-180
[3] Camassa, R; Holm, D.D, An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 1661, (1993) · Zbl 0972.35521
[4] Camassa, R; Holm, D.D; Hyman, M, A Bihamiltonian shallow water equation with peaked solitons, preprint, center for nonlinear studies, (1993), Los Alamos National Laboratory Los Alamos, NM 87545
[5] Carillo, S; Fuchssteiner, B, The abundant symmetry structure of hierarchies of nonlinear equations obtained by reciprocal links, J. math. phys., 30, 1606-1613, (1989) · Zbl 0693.35137
[6] Fokas, A.S; Fuchssteiner, B, Bäcklund transformations for hereditary symmetries, Nonlinear anal. TMA, 5, 423-432, (1981) · Zbl 0491.35007
[7] Fuchssteiner, B, Application of hereditary symmetries to nonlinear evolution equations, Nonlinear anal. TMA, 3, 849-862, (1979) · Zbl 0419.35049
[8] Fuchssteiner, B, The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems, Progr. theoret. phys., 68, 1082-1104, (1982) · Zbl 1098.37540
[9] Fuchssteiner, B, Mastersymmetries, higher-order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. theoret. phys., 70, 1508-1522, (1983) · Zbl 1098.37536
[10] Fuchssteiner, B, Distribution algebras and elementary shock wave analysis, (), 469-475
[11] Fuchssteiner, B, Hamiltonian structure and integrability, (), 211-256 · Zbl 0742.58025
[12] Fuchssteiner, B; Carillo, S, The action-angle transformation for soliton equations, Physica A, 166, 651-675, (1990) · Zbl 0717.35074
[13] Fuchssteiner, B; Fokas, A.S, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4, 47-66, (1981) · Zbl 1194.37114
[14] Fuchssteiner, B; Schiavo, M.Lo, Nonlinear PDE’s and recursive flows: theory, Appl. math. lett., 6, 97-100, (1993) · Zbl 0770.35059
[15] Fuchssteiner, B; Schiavo, M.Lo, Nonlinear PDE’s and recursive flows: applications, Appl. math. lett., 6, 101-104, (1993) · Zbl 0770.35058
[16] Fuchssteiner, B; Oevel, G, Geometry and action-angle variables of multisoliton systems, Rev. math. phys., 1, 415-479, (1990) · Zbl 0727.35123
[17] Gelfand, I.M; Dorfman, I.Y, Hamiltonian operators and algebraic structures related to them, Funktsional. anal. i prilozhen., 13, 13-30, (1974) · Zbl 0428.58009
[18] Gelfand, I.M; Dorfman, I.Y, The Schouten bracket and Hamiltonian operators, Funktsional. anal. i prilozhen., 14, 71-74, (1980) · Zbl 0444.58010
[19] Gelfand, I.M; Dorfman, I.Y, Hamiltonian operators and infinite-dimensional Lie-algebras, Funktsional. anal i prilozhen., 15, 23-40, (1981) · Zbl 0478.58013
[20] Green, A.E; Naghdi, P.M, A derivation of equations for wave propagation in water of variable depth, J. fluid mech., 78, 237-246, (1976) · Zbl 0351.76014
[21] Ince, E.L, Ordinary differential equations, (1956), Dover New York, reprint · Zbl 0063.02971
[22] P.J. Olver and P. Rosenau, Tri-hamiltonian soliton-compacton duality, Phys. Rev. E. to appear.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.