Bicomplex formulation and Moyal deformation of (2 + 1 )-dimensional Fordy-Kulish systems

Journal of Physics A: Mathematical and General

Volumn 34, Issue 12, 2001, Pages 2571-2581

  Dimakis, A ^a   Muller Hoissen F ^b  

^a UNIVERSITY OF THE AEGEAN   (Greece)

^b MAX PLANCK INSTITUTE FOR DYNAMICS AND SELF ORGANIZATION   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035971033     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/34/12/305     Document Type: Article

References (43)

1. Dimakis A and Müller-Hoissen F 2000 Bi-differential calculi and integrable models J. Phys. A: Math. Gen. 33 957-74 Dimakis A and Müller-Hoissen F 2000 Bicomplexes and integrable models J. Phys. A: Math. Gen. 33 6579-91
2. Dimakis A and Müller-Hoissen F 2000 Bi-differential calculi and integrable models J. Phys. A: Math. Gen. 33 957-74 Dimakis A and Müller-Hoissen F 2000 Bicomplexes and integrable models J. Phys. A: Math. Gen. 33 6579-91
3. Fordy A P and Kulish P P 1983 Nonlinear Schrödinger equations and simple Lie algebras Commun. Math. Phys. 89 427-13
4. Oh P and Park Q-H 1996 More on generalized Heisenberg ferromagnet models Phys. Lett. B 383 333-8 Terng C L and Uhlenbeck K 1999 Schrödinger flows on Grassmannians Preprint math.DG/9901086
5. Oh P and Park Q-H 1996 More on generalized Heisenberg ferromagnet models Phys. Lett. B 383 333-8 Terng C L and Uhlenbeck K 1999 Schrödinger flows on Grassmannians Preprint math.DG/9901086
6. Fordy A P 1984 Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces J. Phys. A: Math. Gen. 17 1235-45 Olver P J and Sokolov V V 1998 Non-Abelian integrable systems of the nonlinear Schrödinger type Inverse Problems 14 L5-8 Tsuchida T and Wadati M 1999 Complete integrability of derivative nonlinear Schrödinger equations Inverse Problems 15 1363-73 Porsezian K 1997 Completely integrable nonlinear Schrödinger type equations on moving space curves Phys. Rev. E 55 3785-8 Porsezian K 1998 Nonlinear Schrödinger family on moving space curves: Lax pairs, soliton solution and equivalent spin chains Chaos Solitons Fractals 9 1709-22
7. Fordy A P 1984 Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces J. Phys. A: Math. Gen. 17 1235-45 Olver P J and Sokolov V V 1998 Non-Abelian integrable systems of the nonlinear Schrödinger type Inverse Problems 14 L5-8 Tsuchida T and Wadati M 1999 Complete integrability of derivative nonlinear Schrödinger equations Inverse Problems 15 1363-73 Porsezian K 1997 Completely integrable nonlinear Schrödinger type equations on moving space curves Phys. Rev. E 55 3785-8 Porsezian K 1998 Nonlinear Schrödinger family on moving space curves: Lax pairs, soliton solution and equivalent spin chains Chaos Solitons Fractals 9 1709-22
8. Fordy A P 1984 Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces J. Phys. A: Math. Gen. 17 1235-45 Olver P J and Sokolov V V 1998 Non-Abelian integrable systems of the nonlinear Schrödinger type Inverse Problems 14 L5-8 Tsuchida T and Wadati M 1999 Complete integrability of derivative nonlinear Schrödinger equations Inverse Problems 15 1363-73 Porsezian K 1997 Completely integrable nonlinear Schrödinger type equations on moving space curves Phys. Rev. E 55 3785-8 Porsezian K 1998 Nonlinear Schrödinger family on moving space curves: Lax pairs, soliton solution and equivalent spin chains Chaos Solitons Fractals 9 1709-22
9. Fordy A P 1984 Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces J. Phys. A: Math. Gen. 17 1235-45 Olver P J and Sokolov V V 1998 Non-Abelian integrable systems of the nonlinear Schrödinger type Inverse Problems 14 L5-8 Tsuchida T and Wadati M 1999 Complete integrability of derivative nonlinear Schrödinger equations Inverse Problems 15 1363-73 Porsezian K 1997 Completely integrable nonlinear Schrödinger type equations on moving space curves Phys. Rev. E 55 3785-8 Porsezian K 1998 Nonlinear Schrödinger family on moving space curves: Lax pairs, soliton solution and equivalent spin chains Chaos Solitons Fractals 9 1709-22
10. Fordy A P 1984 Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces J. Phys. A: Math. Gen. 17 1235-45 Olver P J and Sokolov V V 1998 Non-Abelian integrable systems of the nonlinear Schrödinger type Inverse Problems 14 L5-8 Tsuchida T and Wadati M 1999 Complete integrability of derivative nonlinear Schrödinger equations Inverse Problems 15 1363-73 Porsezian K 1997 Completely integrable nonlinear Schrödinger type equations on moving space curves Phys. Rev. E 55 3785-8 Porsezian K 1998 Nonlinear Schrödinger family on moving space curves: Lax pairs, soliton solution and equivalent spin chains Chaos Solitons Fractals 9 1709-22
11. Athorne C and Fordy A 1987 Integrable equations in (2 + 1) dimensions associated with symmetric and homogeneous spaces J. Math. Phys. 28 2018-24
12. Ishimori Y 1984 Multi-vortex solutions of a two-dimensional nonlinear wave equation Prog. Theor. Phys. 72 33-7 Cheng Y, Li Y-S and Tang G-X 1990 The gauge equivalence of the Davey-Stewartson equation and (2 + 1)- dimensional continuous Heisenberg ferromagnetic model J. Phys. A: Math. Gen. 23 L473-7 Konopelchenko B G 1993 Solitons in Multidimensions: Inverse Spectral Transform Method (Singapore: World Scientific) Chakravarty S, Kent S L and Newman E T 1995 Some reductions of the self-dual Yang-Mills equations to integrable systems in 2 + 1 dimensions J. Math. Phys. 36 763-72 Radha R and Lakshmanan M 1999 Generalized dromions in the (2 + 1) dimensional long dispersive wave (2LDW) and scalar nonlinear Schrödinger (NLS) equations Chaos Solitons Fractals 10 1821-4
13. Ishimori Y 1984 Multi-vortex solutions of a two-dimensional nonlinear wave equation Prog. Theor. Phys. 72 33-7 Cheng Y, Li Y-S and Tang G-X 1990 The gauge equivalence of the Davey-Stewartson equation and (2 + 1)-dimensional continuous Heisenberg ferromagnetic model J. Phys. A: Math. Gen. 23 L473-7 Konopelchenko B G 1993 Solitons in Multidimensions: Inverse Spectral Transform Method (Singapore: World Scientific) Chakravarty S, Kent S L and Newman E T 1995 Some reductions of the self-dual Yang-Mills equations to integrable systems in 2 + 1 dimensions J. Math. Phys. 36 763-72 Radha R and Lakshmanan M 1999 Generalized dromions in the (2 + 1) dimensional long dispersive wave (2LDW) and scalar nonlinear Schrödinger (NLS) equations Chaos Solitons Fractals 10 1821-4
14. Ishimori Y 1984 Multi-vortex solutions of a two-dimensional nonlinear wave equation Prog. Theor. Phys. 72 33-7 Cheng Y, Li Y-S and Tang G-X 1990 The gauge equivalence of the Davey-Stewartson equation and (2 + 1)- dimensional continuous Heisenberg ferromagnetic model J. Phys. A: Math. Gen. 23 L473-7 Konopelchenko B G 1993 Solitons in Multidimensions: Inverse Spectral Transform Method (Singapore: World Scientific) Chakravarty S, Kent S L and Newman E T 1995 Some reductions of the self-dual Yang-Mills equations to integrable systems in 2 + 1 dimensions J. Math. Phys. 36 763-72 Radha R and Lakshmanan M 1999 Generalized dromions in the (2 + 1) dimensional long dispersive wave (2LDW) and scalar nonlinear Schrödinger (NLS) equations Chaos Solitons Fractals 10 1821-4
15. Ishimori Y 1984 Multi-vortex solutions of a two-dimensional nonlinear wave equation Prog. Theor. Phys. 72 33-7 Cheng Y, Li Y-S and Tang G-X 1990 The gauge equivalence of the Davey-Stewartson equation and (2 + 1)- dimensional continuous Heisenberg ferromagnetic model J. Phys. A: Math. Gen. 23 L473-7 Konopelchenko B G 1993 Solitons in Multidimensions: Inverse Spectral Transform Method (Singapore: World Scientific) Chakravarty S, Kent S L and Newman E T 1995 Some reductions of the self-dual Yang-Mills equations to integrable systems in 2 + 1 dimensions J. Math. Phys. 36 763-72 Radha R and Lakshmanan M 1999 Generalized dromions in the (2 + 1) dimensional long dispersive wave (2LDW) and scalar nonlinear Schrödinger (NLS) equations Chaos Solitons Fractals 10 1821-4
16. Ishimori Y 1984 Multi-vortex solutions of a two-dimensional nonlinear wave equation Prog. Theor. Phys. 72 33-7 Cheng Y, Li Y-S and Tang G-X 1990 The gauge equivalence of the Davey-Stewartson equation and (2 + 1)- dimensional continuous Heisenberg ferromagnetic model J. Phys. A: Math. Gen. 23 L473-7 Konopelchenko B G 1993 Solitons in Multidimensions: Inverse Spectral Transform Method (Singapore: World Scientific) Chakravarty S, Kent S L and Newman E T 1995 Some reductions of the self-dual Yang-Mills equations to integrable systems in 2 + 1 dimensions J. Math. Phys. 36 763-72 Radha R and Lakshmanan M 1999 Generalized dromions in the (2 + 1) dimensional long dispersive wave (2LDW) and scalar nonlinear Schrödinger (NLS) equations Chaos Solitons Fractals 10 1821-4
17. Myrzakulov R, Vijayalakshmi S, Syzdykova R N and Lakshmanan M 1998 On the simplest (2 + 1) dimensional integrable spin systems and their equivalent Schrödinger equations J. Math. Phys. 39 2122-40 Myrzakulov R, Nugmanova G N and Syzdykova R N 1998 Gauge equivalence between (2 + 1)-dimensional continuous Heisenberg ferromagnetic models and nonlinear Schrödinger-type equations J. Phys. A: Math. Gen. 31 9535-45 Ding Q 1999 The gauge equivalence of the NLS and the Schrödinger flow of maps in 2 + 1 dimensions J. Phys. A: Math. Gen. 32 5087-96
18. Myrzakulov R, Vijayalakshmi S, Syzdykova R N and Lakshmanan M 1998 On the simplest (2 + 1) dimensional integrable spin systems and their equivalent Schrödinger equations J. Math. Phys. 39 2122-40 Myrzakulov R, Nugmanova G N and Syzdykova R N 1998 Gauge equivalence between (2 + 1)-dimensional continuous Heisenberg ferromagnetic models and nonlinear Schrödinger-type equations J. Phys. A: Math. Gen. 31 9535-45 Ding Q 1999 The gauge equivalence of the NLS and the Schrödinger flow of maps in 2 + 1 dimensions J. Phys. A: Math. Gen. 32 5087-96
19. Myrzakulov R, Vijayalakshmi S, Syzdykova R N and Lakshmanan M 1998 On the simplest (2 + 1) dimensional integrable spin systems and their equivalent Schrödinger equations J. Math. Phys. 39 2122-40 Myrzakulov R, Nugmanova G N and Syzdykova R N 1998 Gauge equivalence between (2 + 1)-dimensional continuous Heisenberg ferromagnetic models and nonlinear Schrödinger-type equations J. Phys. A: Math. Gen. 31 9535-45 Ding Q 1999 The gauge equivalence of the NLS and the Schrödinger flow of maps in 2 + 1 dimensions J. Phys. A: Math. Gen. 32 5087-96
20. Zakharov V E 1980 The inverse scattering method Solitons ed R K Bullough and P J Caudrey (Berlin: Springer) pp 243-85 Strachan I A B 1993 Some integrable hierarchies in (2 + 1) dimensions and their twistor description J. Math. Phys. 34 243-59 Myrzakulov R, Vijayalakshmi S, Nugmanova G N and Lakshmanan M 1997 A (2 + 1) dimensional integrable spin model: geometrical and gauge equivalent counterpart, solitons and localized coherent structures Phys. Lett. A 233 391-6
21. Zakharov V E 1980 The inverse scattering method Solitons ed R K Bullough and P J Caudrey (Berlin: Springer) pp 243-85 Strachan I A B 1993 Some integrable hierarchies in (2 + 1) dimensions and their twistor description J. Math. Phys. 34 243-59 Myrzakulov R, Vijayalakshmi S, Nugmanova G N and Lakshmanan M 1997 A (2 + 1) dimensional integrable spin model: geometrical and gauge equivalent counterpart, solitons and localized coherent structures Phys. Lett. A 233 391-6
22. Zakharov V E 1980 The inverse scattering method Solitons ed R K Bullough and P J Caudrey (Berlin: Springer) pp 243-85 Strachan I A B 1993 Some integrable hierarchies in (2 + 1) dimensions and their twistor description J. Math. Phys. 34 243-59 Myrzakulov R, Vijayalakshmi S, Nugmanova G N and Lakshmanan M 1997 A (2 + 1) dimensional integrable spin model: geometrical and gauge equivalent counterpart, solitons and localized coherent structures Phys. Lett. A 233 391-6
23. Bayen F, Flato M, Fronsdal C, Lichnerowicz A and Sternheimer D 1978 Deformation theory and quantization I, II Ann. Phys., NY 111 61-151
24. Takasaki K 2001 Anti-self dual Yang-Mills equations on noncommutative space-time J. Geom. Phys. 37 291-306 (Takasaki K 2000 Preprint hep-th/0005194)
25. Takasaki K 2001 Anti-self dual Yang-Mills equations on noncommutative space-time J. Geom. Phys. 37 291-306 (Takasaki K 2000 Preprint hep-th/0005194)
26. Dimakis A and Müller-Hoissen F 2000 A noncommutative version of the nonlinear Schrödinger equation Preprint hep-th/0007015
27. Dimakis A and Müller-Hoissen F 2000 Bicomplexes, integrable models, and noncommutative geometry Int. J. Mod. Phys. B 14 2455-60 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0006005) Dimakis A and Müller-Hoissen F 2000 The Korteweg-de-Vries equation on a noncommutative space-time Phys. Lett. A 278 139-45 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007074) Dimakis A and Müller-Hoissen F 2000 Moyal deformation, Seiberg-Witten map, and integrable models Lett. Math. Phys. 195 157-78 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007160)
28. Dimakis A and Müller-Hoissen F 2000 Bicomplexes, integrable models, and noncommutative geometry Int. J. Mod. Phys. B 14 2455-60 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0006005) Dimakis A and Müller-Hoissen F 2000 The Korteweg-de-Vries equation on a noncommutative space-time Phys. Lett. A 278 139-45 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007074) Dimakis A and Müller-Hoissen F 2000 Moyal deformation, Seiberg-Witten map, and integrable models Lett. Math. Phys. 195 157-78 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007160)
29. Dimakis A and Müller-Hoissen F 2000 Bicomplexes, integrable models, and noncommutative geometry Int. J. Mod. Phys. B 14 2455-60 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0006005) Dimakis A and Müller-Hoissen F 2000 The Korteweg-de-Vries equation on a noncommutative space-time Phys. Lett. A 278 139-45 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007074) Dimakis A and Müller-Hoissen F 2000 Moyal deformation, Seiberg-Witten map, and integrable models Lett. Math. Phys. 195 157-78 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007160)
30. Dimakis A and Müller-Hoissen F 2000 Bicomplexes, integrable models, and noncommutative geometry Int. J. Mod. Phys. B 14 2455-60 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0006005) Dimakis A and Müller-Hoissen F 2000 The Korteweg-de-Vries equation on a noncommutative space-time Phys. Lett. A 278 139-45 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007074) Dimakis A and Müller-Hoissen F 2000 Moyal deformation, Seiberg-Witten map, and integrable models Lett. Math. Phys. 195 157-78 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007160)
31. Dimakis A and Müller-Hoissen F 2000 Bicomplexes, integrable models, and noncommutative geometry Int. J. Mod. Phys. B 14 2455-60 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0006005) Dimakis A and Müller-Hoissen F 2000 The Korteweg-de-Vries equation on a noncommutative space-time Phys. Lett. A 278 139-45 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007074) Dimakis A and Müller-Hoissen F 2000 Moyal deformation, Seiberg-Witten map, and integrable models Lett. Math. Phys. 195 157-78 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007160)
32. Dimakis A and Müller-Hoissen F 2000 Bicomplexes, integrable models, and noncommutative geometry Int. J. Mod. Phys. B 14 2455-60 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0006005) Dimakis A and Müller-Hoissen F 2000 The Korteweg-de-Vries equation on a noncommutative space-time Phys. Lett. A 278 139-45 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007074) Dimakis A and Müller-Hoissen F 2000 Moyal deformation, Seiberg-Witten map, and integrable models Lett. Math. Phys. 195 157-78 (Dimakis A and Müller-Hoissen F 2000 Preprint hep-th/0007160)
33. Langer J and Perline R 2000 Geometric Realizations of Fordy-Kulish systems Pacific J. Math. at press
34. Lakshmanan M, Ruijgrok Th W and Thompson C J 1976 On the dynamics of a continuum spin system Physica A 84 577-90
35. Zakharov V E and Takhtajan L A 1979 Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet Theor. Math. Phys. 38 17-23
36. Faddeev L D and Takhtajan L A 1987 Hamiltonian Methods in the Theory of Solitons (Berlin: Springer)
37. Da Rios L S 1906 Sul moto d'un liquido indefinito con un filetto vorticoso di forma qualunque Rend. Circ. Mat. Palermo 22 117-35
38. Ricca R L 1991 Rediscovery of Da Rios equations Nature 352 561-2
39. Langer J 1999 Recursion in curve geometry J. Math., NY 5 25-51
40. Hasimoto H 1972 A soliton on a vortex filament J. Fluid Mech. 51 477-85
41. Seiberg N and Witten E 1999 String theory and noncommutative geometry J. High Energy Phys. 9 32
42. Crampin M, Sarlet W and Thompson G 2000 Bi-differential calculi and bi-Hamiltonian systems J. Phys. A: Math. Gen. 33 L177-80 Crampin M, Sarlet W and Thompson G 2000 Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors J. Phys. A: Math. Gen. 33 8755-70
43. Crampin M, Sarlet W and Thompson G 2000 Bi-differential calculi and bi-Hamiltonian systems J. Phys. A: Math. Gen. 33 L177-80 Crampin M, Sarlet W and Thompson G 2000 Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors J. Phys. A: Math. Gen. 33 8755-70