Reciprocal link for 2 + 1-dimensional extensions of shallow water equations

Applied Mathematics Letters

Volumn 13, Issue 3, 2000, Pages 37-42

  Hone, A N W ^a  

^a UNIVERSITY OF ADELAIDE   (Australia)

Author keywords

Breaking solitons; Conservation laws; Non isospectral lax pairs; Partial differential equations; Reciprocal transformations

Indexed keywords

EID: 0039840642     PISSN: 08939659     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0893-9659(99)00183-4     Document Type: Article

References (23)

1. J.G. Kingston and C. Rogers, Reciprocal Bäcklund transformations of conservation laws, Phys. Lett. A 92, 261-264, (1982).
2. C. Rogers, Reciprocal transformations and their applications, In Nonlinear Evolutions, (Edited by J. Leon), pp. 109-123, World Scientific, Singapore, (1988).
3. C. Rogers, Bäcklund transformations in soliton theory, In Soliton Theory: A Survey of Results, (Edited by A.P. Fordy), pp. 97-130, University Press, Manchester, (1990).
4. A.S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D 4, 47-66, (1981).
5. R. Camassa and D.D. Holm, A completely integrable dispersive shallow water equation with peaked solitons, Phys. Rev. Lett. 71, 1661-1664, (1993).
6. C. Gilson and A. Pickering, Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations, J. Phys. A 28, 2871-2888, (1995).
7. V. Marinakis and T.C. Bountis, On the integrability of a new class of water wave equations, In Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology, (Edited by D.B. Duncan and J.C. Eilbeck), Heriot-Watt University, Edinburgh, published on WWW: http://www.ma.hw.ac.uk /solitons/procs, (1995).
8. B. Fuchssteiner, Some tricks from the symmetry toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Physica D 95, 229-243, (1996).
9. J. Schiff, The Camassa-Holm equation: A loop group approach, Physica D 121, 24-43, (1998).
10. P.A. Clarkson, P.R. Gordoa and A. Pickering, Multi-component equations associated to non-isospectral scattering problems, Inverse Problems 13, 1463-1476, (1997).
11. S.-J. Yu, K. Toda, N. Sasa and T. Fukuyama, N soliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3 + 1) dimensions, solv-int/9801003, (preprint).
12. O.I. Bogoyavlenskii, Overturning solitons in new two-dimensional integrable equations, Math. USSR Izvestiya 34, 245-259, (1990).
13. O.I. Bogoyavlenskii, Breaking solitons II, Math. USSR Izvestiya 35, 245-248, (1990).
14. O.I. Bogoyavlenskii, Breaking solitons III, Math. USSR Izvestiya 36, 129-137, (1991).
15. J. Schiff, Integrability of Chern-Simons-Higgs vortex equations and a reduction of the self-dual Yang-Mills equations to three dimensions, In Painlevé Transcendents, (Edited by D. Levi and P. Winternitz), NATO ASI Series B, Volume 278, pp. 393-405, Plenum Press, (1992).
16. J. Tafel, 2-dimensional reductions of the self-dual Yang-Mills equations, In Nonlinear Evolution Equations and Dynamical Systems, Proceedings NEEDS '91, (Edited by M. Boiti, L. Martina and F. Pempinelli), pp. 188-195, World Scientific, (1992).
17. F. Calogero, A method to generate solvable nonlinear evolution equations, Lett. Nuovo Cimento 14, 443-448, (1975).
18. F. Calogero and A. Degasperis, Spectral Transform and Solitons I, 227-233, North-Holland, (1982).
19. A.K. Common and M. Musette, Two discretisations of the Ermakov-Pinney equation, Phys. Lett. A 235, 574-580, (1997).
20. M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform Fourier analysis for nonlinear equations, Stud. Appl. Math. 8, 249-315, (1974).
21. A. Pickering, private communication.
22. A.N.W. Hone, Exact discretization of the Ermakov-Pinney dquation, Physics Letters A (to appear).
23. M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Commun. Math. Phys. 61, 1-30, (1978).