Superposition principles associated with the Moutard transformation: An integrable discretization of a (2 + l)-dimensional sine-Gordon system

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 453, Issue 1957, 1997, Pages 255-279

  Nimmof, J J C ^a   Schief, W K ^b  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

^b UNIVERSITY OF GLASGOW   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33746752207     PISSN: 13645021     EISSN: None     Source Type: Journal    
DOI: 10.1098/rspa.1997.0015     Document Type: Article

References (35)

1. Athorne, C. & Nimmo, J. J. C. 1991 On the Moutard transformation for integrable partial differential equations. Inverse Prob. 7, 809-826.
2. Beutler, R. 1993 Positon solutions of the sine-Gordon equation. J. Math. Phys. 34, 3098-3109.
3. Bianchi, L. 1899 Vorlesungen über Differenzialgeometrie. Leipzig: Teubner.
4. Boiti, M., Leon, J. J. P. fe Pempinelli, F. 1987 Integrable two-dimensional generalization of the sine- and sinh-Gordon equations. Inverse Prob. 3, 37-49.
5. Clarkson, P. A., Mansfield, E. L. & Milne, A. E. 1996 Symmetries and exact solutions of a 2 + 1-dimensional sine-Gordon system. Phil. Trans. R. Soc. Land. A 354, 1807-1835.
6. Date, E., Jimbo, M., Kashiwara, M. & Miwa, T. 1982 Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type. Physica D 4, 343-365.
7. Date, E., Jimbo, M. & Miwa, T. 1982 Method for generating discrete soliton equations. I. J. Phys. Jpn 51, 4110-4127.
8. Date, E., Jimbo, M. & Miwa, T. 1983 Method for generating discrete soliton equations. V. J. Phys. Jpn 53, 766-771.
9. Hirota, R. 1977 Nonlinear partial difference equations. 3. Discrete sine-Gordon equation. J. Phys. Jpn 43, 2079-2086.
10. Hirota, R. 1981 Discrete analogue of a generalized Toda equation. J. Phys. Jpn 50, 3785-3791.
11. Konopelchenko, B. G. & Dubrovsky, V. G. 1984 Some new integrable nonlinear evolution equations in 2 + 1 dimensions. Phys. Lett. A102, 15-17.
12. Konopelchenko, B. G. & Dubrovsky, V. G. 1993 A 2 + 1 dimensional integrable generalization of the sine-Gordon equation. 1. 9-9-dressing and the initial-value problem. Stud. Appl. Math. 90, 189-223.
13. Konopelchenko, B. G. & Rogers, C. 1991 On (2 + l)-dimensional nonlinear systems of Loewnertype. Phys. Lett. A158, 391-397.
14. Konopelchenko, B. G. 1982 Elementary Bäcklund transformations, nonlinear superposition principle and solutions of the integrable equations. Phys. Lett. A 87, 445-448.
15. Konopelchenko, B. G., Schief, W. K. & Rogers, C. 1992 A (2 + l)-dimensional sine-Gordon system. Its auto-Bäcklund transformation. Phys. Lett. A172, 39-48.
16. Lamb, G. L. 1974 Bäcklund transformations for certain evolution equations. J. Math. Phys. 15, 2157-2165.
17. Levi, D. & Benguria, R. 1980 Bäcklund transformations and nonlinear differential differenceequations. Proc. Nain. Acad. Sci. US 77, 5025-5027.
18. Levi, D. 1981 Non-linear differential difference-equations as Bäcklund transformations. J. Phys. A14, 1082-1098.
19. Matveev, V. B. & Salle, M. A. 1991 Darboux transformations and solitons. Berlin: Springer.
20. Miwa, T. 1982 On Hirota difference-equations. Proc. Jpn Acad. A 58, 9-12.
21. Moutard, T. 1878 Sur la construction des équations de la forme QZQ = A(x,y), qui admettent une intégrale générale explicite. J. Ecole Polytechnique 45, 1-11.
22. Nijhoff, F. & Capel, H. 1995 The discrete Korteweg-de Vries equation. In KdV '95 (ed. M. Hazewinkel, H. W. Capel fe E. M. de Jager). Dordrecht: Kluwer.
23. Nijhoff, F. W., Capel, H. W., Wiersma, G. L. & Quispel, G. R. W. 1984 Bäcklund transformations and three-dimensional lattice equations. Phys. Lett. A 105, 267-272.
24. Nijhoff, F. W., Capel, H. W. & Wiersma, G. L. 1985 Integrable lattice systems in two and three dimensions. In Geometric aspects of the Einstein equations and integrable systems, vol. 293. Lecture notes in physics (ed. R. Martini), pp. 263-302. Berlin: Springer.
25. Nimmo, J. J. C. & Schief, W. K. 1997 An integrable discretization of a (2 + l)-dimensional sine-Gordon equation. (In preparation.)
26. Nimmo, J. J. C. 1990 The Crum transformation for a third order scattering problem. Proc. R. Soc. Land. A 431, 361-369.
27. Nimmo, J. J. C. 1992 A class of solutions of the Konopelchenko-Rogers equation. Phys. Lett. A 168, 113-119.
28. Nimmo, J. J. C. 1993 Darboux transformations in (2 + l)-dimensions. In Applications of analytic and geometric methods to nonlinear differential equations (cd. P. Clarkson). Dordrecht: Kluwer.
29. Nizhnik, L. P. 1980 Integration of multidimensional nonlinear equations by the inverse problem method. Dokl Akad. Nauk SSSR 254, 332-335.
30. Oevel, W. & Schief, W. K. 1994 Squared eigenfunctions of the (modified) KP hierarchy and scattering problems of Loewner type. Rev. Math. Phys. 6, 1301-1338.
31. Ohta, Y., Hirota, R., Tsujimota, S. & Imai, T. 1993 Casorati and discrete Gram type determinant representation of solutions to the discrete KP hierarchy. J. Phys. Jpn 62, 1872-1886.
32. Quispel, G. R. W., Nijhoff, F. W., Capel, H. W. & van der Linden, J. 1984 Linear integral equations and nonlinear difference-difference equations. Physica A 125, 344-380.
33. Schief, W. K. 1994 On a 2 + 1-dimensional integrable Ernst-type equation. Proc. R. Soc. Land. A 446, 381-398.
34. Veselov, A. P. & Novikov, S. P. 1984 Finite-zone two-dimensional potential Schrödinger operators. Explicit formulae and evolution equations. Dokl. Akad. Nauk SSSR 279, 20-24.
35. Zabusky, N. J. & Kruskal, M. D. 1965 Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240-243.