Discrete Chebyshev nets and a universal permutability theorem

Journal of Physics A: Mathematical and Theoretical

Volumn 40, Issue 18, 2007, Pages 4775-4801

  Schief, W K ^a  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77953505971     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/40/18/007     Document Type: Article

References (29)

1. Lund F and Regge T 1976 Unified approach to strings and vortices with soliton solutions Phys. Rev. D 14 1524-35
2. Ablowitz M J, Kaup D J, Newell A C and Segur H 1973 Nonlinear evolution equations of physical significance Phys. Rev. Lett. 31 125-7
3. Ablowitz M J and Segur H 1981 Solitons and the Inverse Scattering Transform (SIAM Studies in Applied Mathematics vol 4) (Philadelphia: SIAM)
4. Pohlmeyer K 1976 Integrable Hamiltonian systems and interactions through quadratic constraints Commun. Math. Phys. 46 207-21
5. Lund F 1977 Note on the geometry of the nonlinear σ model in two dimensions Phys. Rev. D 15 1540-3
6. Sym A 1985 Soliton surfaces and their applications Geometric Aspects of the Einstein Equations and Integrable Systems ed R Martini (Berlin: Springer)
7. Rogers C and Schief W K 2002 Darboux and Bäcklund Transformations. Geometry and Modern Applications in Soliton Theory (Cambridge Texts in Applied Mathematics) (Cambridge: Cambridge University Press)
8. Bobenko A and Pinkall U 1996 Discrete surfaces with constant negative Gaussian curvature and the Hirota equation J. Diff. Geom. 43 527-611
9. Sauer R 1950 Parallelogrammgitter als Modelle pseudosphärischer Flächen Math. Z. 52 611-22
10. Wunderlich W 1951 Zur Differenzengeometrie der Flächen konstanter negativer Krümmung Österreich. Akad. Wiss. Math.-Nat. Kl. S.-B. II 160 39-77
11. Hirota R 1977 Nonlinear partial difference equations: III. Discrete sine-Gordon equation J. Phys. Soc. Japan 43 2079-86
12. Bobenko A I and Seiler R (ed) 1999 Discrete Integrable Geometry and Physics (Oxford: Clarendon)
13. do Carmo M P 1976 Differential Geometry of Curves of Surfaces (Englewood Cliffs, NJ: Prentice-Hall)
14. Chebyshev P L 1955 On the cutting of our clothes Collected Works vol 5 (Moscow: Fizmatgiz) pp 165-70
15. Eisenhart L P 1960 A Treatise on the Differential Geometry of Curves and Surfaces (New York: Dover)
16. Nappi C R 1980 Some properties of an analog of the chiral model Phys. Rev. D 21 418-20
17. Burtsev S P, Zakharov V E and Mikhailov A V 1987 Inverse scattering method with variable spectral parameter Teoret. Mat. Fiz. 70 323-41
18. Kramer D 1982 Equivalence of various pseudopotential approaches for Einstein-Maxwell fields J. Phys. A: Math. Gen. 15 2201-7
19. Hoffmann T 2000 On the equivalence of the discrete nonlinear Schrödinger equation and the discrete isotropic Heisenberg magnet Phys. Lett. A 265 62-7
20. Nijhoff F W and Walker A J 2001 The discrete and continuous Painlevé hierarchy and the Garnier system Glasgow Math. J. 43A 109-23
21. Bobenko A I and Suris Yu B 2002 Integrable systems on quad-graphs Int. Math. Res. Not. 2002 573-611
22. Nijhoff F W 2002 Lax pair for the Adler (lattice Krichever-Novikov) system Phys. Lett. A 297 49-58
23. Adler V E, Bobenko A I and Suris Yu B 2003 Classification of integrable equations on quad-graphs. The consistency approach Commun. Math. Phys. 233 513-43
24. Zakharov V E and Manakov S V 1985 Construction of multidimensional nonlinear integrable systems and their solutions Funct. Anal. Pril. 19 11-25
25. Darboux G 1910 Leçons sur les systèmes orthogonaux et les coordonnées curvilignes (Paris: Gauthier-Villars)
26. Bogdanov L V and Konopelchenko B G 1995 Lattice and q-difference Darboux-Zakharov-Manakov systems via method J. Phys. A: Math. Gen. 28 L173-8
27. Boiti M, Laddomada C and Pempinelli F 1981 An equivalent real form of the nonlinear Schrödinger equation and the permutability for Bäcklund transformations Nuovo Cimento B 62 315-26
28. Schief W K 2001 Isothermic surfaces in spaces of arbitrary dimension: integrability, discretization and Bäcklund transformations. A discrete Calapso equation Stud. Appl. Math. 106 85-137
29. Calapso P 1903 Sulla superficie a linee di curvatura isoterme Rend. Circ. Mat. Palermo 17 275-86