Noncommutative Burgers equation

Journal of Physics A: Mathematical and General

Volumn 36, Issue 48, 2003, Pages 11981-11998

  Hamanaka, Masashi ^a   Toda, Kouichi ^b  

^a UNIVERSITY OF TOKYO   (Japan)

^b TOYAMA PREFECTURAL UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0346895558     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/36/48/006     Document Type: Article

References (69)

1. Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 957 (Preprint math-ph/9908015) Dimakis A and Muller-Hoissen F 2000 Int. J. Mod. Phys. B 14 2455 (Preprint hep-th/0006005) Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 6579 (Preprint nlin.si/0006029) Dimakis A and Muller-Hoissen F 2000 A noncommutative version of the nonlinear Schroedinger equation Preprint hep-th/0007015 Dimakis A and Muller-Hoissen F 2000 Lett. Math. Phys. 54 123 (Preprint hep-th/0007160) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 2571 (Preprint nlin.si/0008016) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 9163 (Preprint nlin.si/0104071)
2. Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 957 (Preprint math-ph/9908015) Dimakis A and Muller-Hoissen F 2000 Int. J. Mod. Phys. B 14 2455 (Preprint hep-th/0006005) Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 6579 (Preprint nlin.si/0006029) Dimakis A and Muller-Hoissen F 2000 A noncommutative version of the nonlinear Schroedinger equation Preprint hep-th/0007015 Dimakis A and Muller-Hoissen F 2000 Lett. Math. Phys. 54 123 (Preprint hep-th/0007160) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 2571 (Preprint nlin.si/0008016) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 9163 (Preprint nlin.si/0104071)
3. Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 957 (Preprint math-ph/9908015) Dimakis A and Muller-Hoissen F 2000 Int. J. Mod. Phys. B 14 2455 (Preprint hep-th/0006005) Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 6579 (Preprint nlin.si/0006029) Dimakis A and Muller-Hoissen F 2000 A noncommutative version of the nonlinear Schroedinger equation Preprint hep-th/0007015 Dimakis A and Muller-Hoissen F 2000 Lett. Math. Phys. 54 123 (Preprint hep-th/0007160) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 2571 (Preprint nlin.si/0008016) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 9163 (Preprint nlin.si/0104071)
4. Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 957 (Preprint math-ph/9908015) Dimakis A and Muller-Hoissen F 2000 Int. J. Mod. Phys. B 14 2455 (Preprint hep-th/0006005) Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 6579 (Preprint nlin.si/0006029) Dimakis A and Muller-Hoissen F 2000 A noncommutative version of the nonlinear Schroedinger equation Preprint hep-th/0007015 Dimakis A and Muller-Hoissen F 2000 Lett. Math. Phys. 54 123 (Preprint hep-th/0007160) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 2571 (Preprint nlin.si/0008016) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 9163 (Preprint nlin.si/0104071)
5. Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 957 (Preprint math-ph/9908015) Dimakis A and Muller-Hoissen F 2000 Int. J. Mod. Phys. B 14 2455 (Preprint hep-th/0006005) Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 6579 (Preprint nlin.si/0006029) Dimakis A and Muller-Hoissen F 2000 A noncommutative version of the nonlinear Schroedinger equation Preprint hep-th/0007015 Dimakis A and Muller-Hoissen F 2000 Lett. Math. Phys. 54 123 (Preprint hep-th/0007160) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 2571 (Preprint nlin.si/0008016) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 9163 (Preprint nlin.si/0104071)
6. Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 957 (Preprint math-ph/9908015) Dimakis A and Muller-Hoissen F 2000 Int. J. Mod. Phys. B 14 2455 (Preprint hep-th/0006005) Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 6579 (Preprint nlin.si/0006029) Dimakis A and Muller-Hoissen F 2000 A noncommutative version of the nonlinear Schroedinger equation Preprint hep-th/0007015 Dimakis A and Muller-Hoissen F 2000 Lett. Math. Phys. 54 123 (Preprint hep-th/0007160) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 2571 (Preprint nlin.si/0008016) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 9163 (Preprint nlin.si/0104071)
7. Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 957 (Preprint math-ph/9908015) Dimakis A and Muller-Hoissen F 2000 Int. J. Mod. Phys. B 14 2455 (Preprint hep-th/0006005) Dimakis A and Muller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 6579 (Preprint nlin.si/0006029) Dimakis A and Muller-Hoissen F 2000 A noncommutative version of the nonlinear Schroedinger equation Preprint hep-th/0007015 Dimakis A and Muller-Hoissen F 2000 Lett. Math. Phys. 54 123 (Preprint hep-th/0007160) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 2571 (Preprint nlin.si/0008016) Dimakis A and Muller-Hoissen F 2001 J. Phys. A: Math. Gen. 34 9163 (Preprint nlin.si/0104071)
8. Dimakis A and Muller-Hoissen F 2000 Phys. Lett. A 278 139 (Preprint hep-th/0007074)
9. Kapustin A, Kuznetsov A and Orlov D 2001 Commun. Math. Phys. 221 385 (Preprint hep-th/0002193)
10. Takasaki K 2001 J. Geom. Phys. 37 291 (Preprint hep-th/0005194)
11. Gross D J and Nekrasov N A 2000 J. High Energy Phys. JHEP07(2000)034 (Preprint hep-th/0005204)
12. Nekrasov N A 2000 Noncommutative instantons revisited Preprint hep-th/0010017
13. Legare M 2000 Noncommutative generalized NS and super matrix KdV systems from a noncommutative version of (anti-)self-dual Yang-Mills equations Preprint hep-th/0012077 Legare M 2002 J. Phys. A: Math. Gen. 35 5489
14. Legare M 2000 Noncommutative generalized NS and super matrix KdV systems from a noncommutative version of (anti-)self-dual Yang-Mills equations Preprint hep-th/0012077 Legare M 2002 J. Phys. A: Math. Gen. 35 5489
15. Lechtenfeld O, Popov A D and Spendig B 2001 J. High Energy Phys. JHEP06(2001)011 (Preprint hep-th/0103196)
16. Correa D H, Lozano G S, Moreno E F and Schaposnik F A 2001 Phys. Lett. B 515 206 (Preprint hep-th/0105085)
17. Paniak L D 2001 Exact noncommutative KP and KdV multi-solitons Preprint hep-th/0105185
18. Lechtenfeld O and Popov A D 2001 J. High Energy Phys. JHEP11(2001)040 (Preprint hep-th/0106213) Lechtenfeld O and Popov A D 2001 Phys. Lett. B 523 178 (Preprint hep-th/0108118) Lechtenfeld O and Popov A D 2002 J. High Energy Phys. JHEP03(2002)040 (Preprint hep-th/0109209)
19. Lechtenfeld O and Popov A D 2001 J. High Energy Phys. JHEP11(2001)040 (Preprint hep-th/0106213) Lechtenfeld O and Popov A D 2001 Phys. Lett. B 523 178 (Preprint hep-th/0108118) Lechtenfeld O and Popov A D 2002 J. High Energy Phys. JHEP03(2002)040 (Preprint hep-th/0109209)
20. Lechtenfeld O and Popov A D 2001 J. High Energy Phys. JHEP11(2001)040 (Preprint hep-th/0106213) Lechtenfeld O and Popov A D 2001 Phys. Lett. B 523 178 (Preprint hep-th/0108118) Lechtenfeld O and Popov A D 2002 J. High Energy Phys. JHEP03(2002)040 (Preprint hep-th/0109209)
21. Hannabuss K C 2001 Lett. Math. Phys. 58 153 (Preprint hep-th/0108228)
22. Grisaru M T and Penati S 2003 The noncommutative sine-Gordon system Nucl. Phys. B 655 250
23. Bieling S 2002 J. Phys. A: Math. Gen. 35 6281 (Preprint hep-th/0203269)
24. Wolf M 2002 J. High Energy Phys. JHEP06(2002)055 (Preprint hep-th/0204185)
25. Toda K 2002 Extensions of soliton equations to non-commutative (2 + 1) dimensions JHEP Proc. Workshop on Integrable Theories, Solitons and Duality (Sao Paulo, Brazil, 1-6 July 2002) (JHEP PRHEP-unesp2002/038)
26. Franco-Sollova F and Ivanova T 2003 J. Phys. A: Math. Gen. 36 4207 (Preprint hep-th/0209153)
27. Horváth Z, Lechtenfeld O and Wolf M 2002 J. High Energy Phys. JHEP12(2002)060 (Preprint hep-th/0211041)
28. Furuta K, Inami T and Yamamoto M 2002 Topics in nonlinear sigma models in D = 3 Preprint hep-th/0211129
29. Hamanaka M and Toda K 2002 Phys. Lett. A 316 77
30. Cabrera-Carnero I and Moriconi M 2002 Noncommutative integrable field theories in 2D Preprint hep-th/0211193
31. Ihl M and Uhlmann S 2002 Noncommutative extended waves and soliton-like configurations in N = 2 string theory Preprint hep-th/0211263
32. Wang N and Wadati M 2003 J. Phys. Soc. Japan 72 1366 Wang N and Wadati M 2003 J. Phys. Soc. Japan 72 1881
33. Wang N and Wadati M 2003 J. Phys. Soc. Japan 72 1366 Wang N and Wadati M 2003 J. Phys. Soc. Japan 72 1881
34. Curtright T, Fairlie D and Zachos C K 1997 Phys. Lett. B 405 37 (Preprint hep-th/9704037) Zachos C K, Fairlie D and Curtright T 1997 Matrix membranes and integrability Preprint hep-th/9709042
35. Curtright T, Fairlie D and Zachos C K 1997 Phys. Lett. B 405 37 (Preprint hep-th/9704037) Zachos C K, Fairlie D and Curtright T 1997 Matrix membranes and integrability Preprint hep-th/9709042
36. Douglas M R and Nekrasov N A 2002 Rev. Mod. Phys. 73 977 (Preprint hep-th/0106048) Szabo R J 2003 Phys. Rept. 378 207 (Preprint hep-th/0109162) and references therein
37. Douglas M R and Nekrasov N A 2002 Rev. Mod. Phys. 73 977 (Preprint hep-th/0106048) Szabo R J 2003 Phys. Rept. 378 207 (Preprint hep-th/0109162) and references therein
38. 4 Preprint hep-th/0010006 Nekrasov N A 2002 Trieste lectures on solitons in noncommutative gauge theories Preprint hep-th/0011095 Konechny A and Schwarz A 2002 Phys. Rep. 360 353 (Preprints hep-th/0012145, hep-th/0107251) Hamanaka M 2003 Noncommutative solitons and D-branes PhD Thesis (Preprint hep-th/0303256) and references therein
39. 4 Preprint hep-th/0010006 Nekrasov N A 2002 Trieste lectures on solitons in noncommutative gauge theories Preprint hep-th/0011095 Konechny A and Schwarz A 2002 Phys. Rep. 360 353 (Preprints hep-th/0012145, hep-th/0107251) Hamanaka M 2003 Noncommutative solitons and D-branes PhD Thesis (Preprint hep-th/0303256) and references therein
40. 4 Preprint hep-th/0010006 Nekrasov N A 2002 Trieste lectures on solitons in noncommutative gauge theories Preprint hep-th/0011095 Konechny A and Schwarz A 2002 Phys. Rep. 360 353 (Preprints hep-th/0012145, hep-th/0107251) Hamanaka M 2003 Noncommutative solitons and D-branes PhD Thesis (Preprint hep-th/0303256) and references therein
41. 4 Preprint hep-th/0010006 Nekrasov N A 2002 Trieste lectures on solitons in noncommutative gauge theories Preprint hep-th/0011095 Konechny A and Schwarz A 2002 Phys. Rep. 360 353 (Preprints hep-th/0012145, hep-th/0107251) Hamanaka M 2003 Noncommutative solitons and D-branes PhD Thesis (Preprint hep-th/0303256) and references therein
42. Nekrasov N and Schwarz A 1998 Commun. Math. Phys. 198 689 (Preprint hep-th/9802068)
43. Atiyah M F, Hitchin N J, Drinfeld V G and Manin Y I 1978 Phys. Lett. A 65 185
44. Korteweg D J and de Vries G 1895 Phil. Mag. 39 422
45. Burgers J M 1948 Adv. Appl. Mech. 1 171
46. Hopf E 1950 Comm. Pure Appl. Math. 3 201 Cole J D 1951 Q. Appl. Math. 9 225
47. Hopf E 1950 Comm. Pure Appl. Math. 3 201 Cole J D 1951 Q. Appl. Math. 9 225
48. Hirota R 1971 Phys. Rev. Lett. 27 1192
49. Martina L and Pashaev O K 2003 'Burgers' equation in non-commutative space-time Preprint hep-th/0302055
50. Moyal J E 1949 Proc. Camb. Phil. Soc. 45 99 Groenewold H J 1946 Physica 12 405
51. Moyal J E 1949 Proc. Camb. Phil. Soc. 45 99 Groenewold H J 1946 Physica 12 405
52. Zakharov V 1991 What is Integrability? (Berlin: Springer)
53. Faddeev L D and Takhtajan L A 1987 Hamiltonian Methods in the Theory of Solitons (Berlin: Springer)
54. Ablowitz M J and Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge: Cambridge University Press)
55. Toda K and Yu S-J 2000 J. Math. Phys. 41 4747 Toda K and Yu S-J 2001 J. Nonlinear Math. Phys. Suppl. 8 272 Toda K and Yu S-J 2001 Inverse Prob. 17 1053 and references therein
56. Toda K and Yu S-J 2000 J. Math. Phys. 41 4747 Toda K and Yu S-J 2001 J. Nonlinear Math. Phys. Suppl. 8 272 Toda K and Yu S-J 2001 Inverse Prob. 17 1053 and references therein
57. Toda K and Yu S-J 2000 J. Math. Phys. 41 4747 Toda K and Yu S-J 2001 J. Nonlinear Math. Phys. Suppl. 8 272 Toda K and Yu S-J 2001 Inverse Prob. 17 1053 and references therein
58. Mason L J and Woodhouse N M 1996 Integrability, Self-Duality, and TwistorTheory (Oxford: Oxford University Press)
59. Ward R S 1985 Phil. Trans. R. Soc. A 315 451 Ward R S 1986 Multidimensional integrable systems Lectrure Notes in Physics vol 280 (Berlin: Springer) p 106 Ward R S 1990 Integrable systems in twistor theory Twistors in Mathematics and Physics (Cambridge: Cambridge University Press) p 246
60. Ward R S 1985 Phil. Trans. R. Soc. A 315 451 Ward R S 1986 Multidimensional integrable systems Lectrure Notes in Physics vol 280 (Berlin: Springer) p 106 Ward R S 1990 Integrable systems in twistor theory Twistors in Mathematics and Physics (Cambridge: Cambridge University Press) p 246
61. Ward R S 1985 Phil. Trans. R. Soc. A 315 451 Ward R S 1986 Multidimensional integrable systems Lectrure Notes in Physics vol 280 (Berlin: Springer) p 106 Ward R S 1990 Integrable systems in twistor theory Twistors in Mathematics and Physics (Cambridge: Cambridge University Press) p 246
62. Olver P J and Sokolov V V 1998 Commun. Math. Phys. 193 245
63. Sato M and Sato Y 1982 Soliton equations as dynamical systems on infinite dimensional Grassmann manifold Nonlinear Partial Differential Equations in Applied Sciences (Amsterdam: North-Holland) p 259
64. Miwa T, Jimbo M and Date E 2000 Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras translated by M Reid (Cambridge: Cambridge University Press)
65. Błaszak M 1998 Multi-Hamiltonian Theory of Dynamical Systems (Berlin: Springer)
66. Kupershmidt B 2000 KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems (Providence, RI: American Mathematical Society)
67. Hamanaka M and Toda K Towards noncommutative integrable equations Preprint hep-th/0309265
68. Tsuchida T 2002 J. Phys. A: Math. Gen. 35 7827 (Preprint nlin.si/0105053)
69. Harvey J A 2001 Komaba lectures on noncommutative solitons and D-branes Preprint hep-th/0102076 and references therein