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On Euler's equation and 'EPDiff'
The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups
1. | Département de Mathématique, CP 218, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles,, Belgium |
2. | Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile |
References:
[1] |
M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-Devries type equations, Inventiones Mathematicae, 50 (1979), 219-248.
doi: 10.1007/BF01410079. |
[2] |
N. Bourbaki, "Algebra I. Chapters 1-3. Elements of Mathematics," Springer-Verlag, Berlin, 1998. |
[3] |
E. E. Demidov, On the Kadomtsev-Petviashvili hierarchy with a noncommutative timespace, Functional Analysis and Its Applications, 29 (1995), 131-133.
doi: 10.1007/BF01080014. |
[4] |
E. E. Demidov, Noncommutative deformation of the Kadomtsev-Petviashvili hierarchy, In "Algebra. 5, Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI)," Moscow, 1995. (Russian), Journal of Mathematical Sciences (New York), 88 (1998), 520-536 (English).
doi: 10.1007/BF02365314. |
[5] |
L. A. Dickey, "Soliton Equations and Hamiltonian Systems," Second Edition, Advanced Series in Mathematical Physics $12$, World Scientific Publ. Co., Singapore, 2003. |
[6] |
L. D. Faddeev, and L. A. Takhtajan, "Hamiltonian Methods in the Theory of Solitons," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin-New York, 1987. |
[7] |
B. A. Khesin and I. Zakharevich, Poisson-Lie groups of pseudodifferential symbols, Communications in Mathematical Physics, 171 (1995), 475-530.
doi: 10.1007/BF02104676. |
[8] |
B. A. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Advances in Mathematics, 176 (2003), 116-144.
doi: 10.1016/S0001-8708(02)00063-4. |
[9] |
B. A. Khesin and R. Wendt, "The Geometry of Infinite-Dimensional Groups," Springer-Verlag, Berlin, 2009. |
[10] |
F. Kubo, Non-commutative Poisson algebra structures on affine Kac-Moody algebras, Journal of Pure and Applied Algebra, 126 (1998), 267-286.
doi: 10.1016/S0022-4049(96)00141-7. |
[11] |
L.-C. Li, Factorization problem on the Hilbert-Schmidt group and the Camassa-Holm equation, Communications on Pure and Applied Mathematics, 61 (2008), 186-209.
doi: 10.1002/cpa.20207. |
[12] |
J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. |
[13] |
J. Mickelsson, "Current Algebras and Groups," Plenum Press, New York and London, 1989. |
[14] |
A. V. Mikhailov, A. B. Shabat and V. V. Sokolov, The symmetry approach to classification of integrable equations, in "What is Integrability?" (ed. V.E. Zakharov), Springer Ser. Nonlinear Dynam., Springer, Berlin, (1991), 115-184. |
[15] |
M. Mulase, Complete integrability of the Kadomtsev-Petvishvili equation, Advances in Mathematics, 54 (1984), 57-66.
doi: 10.1016/0001-8708(84)90036-7. |
[16] |
M. Mulase, Cohomological structure in soliton equations and Jacobian varieties, J. Differential Geom., 19 (1984), 403-430. |
[17] |
M. Mulase, Solvability of the super KP equation and a generalization of the Birkhoff decomposition, Inventiones Mathematicae, 92 (1988), 1-46.
doi: 10.1007/BF01393991. |
[18] |
P. J. Olver, "Applications of Lie Groups to Differential Equations," Second Edition, Springer-Verlag, New York, 1993. |
[19] |
P. J. Olver and V. V. Sokolov, Integrable evolution equations on associative algebras, Communications in Mathematical Physics, 193 (1998) 245-268.
doi: 10.1007/s002200050328. |
[20] |
A. N. Parshin, On a ring of formal pseudo-differential operators, Proc. Steklov Inst. Math. 224 (1999), 266-280. |
[21] |
A. M. Perelomov, "Integrable Systems of Classical Mechanics And Lie Algebras," Birkhäuser Verlag, Berlin, 1990.
doi: 10.1007/978-3-0348-9257-5. |
[22] |
A. Pressley and G.B. Segal, "Loop Groups," Oxford University Press, 1986. |
[23] |
A. G. Reyman and M. A. Semenov-Tian-Shansky, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II, Inventiones mathematicae, 63 (1981), 423-432.
doi: 10.1007/BF01389063. |
[24] |
M. Sakakibara, Factorization methods for noncommutative KP and Toda hierarchy, Journal of Physics A: Mathematical and General, 37 (2004), L599-L604.
doi: 10.1088/0305-4470/37/45/L02. |
[25] |
M. A. Semenov-Tian-Shansky, What is a classical $r$-matrix?, Funct. Anal. Appl., 17 (1983), 259-272. |
[26] |
K. Takasaki, A new approach to the self-dual Yang-Mills equations, Communications in Mathematical Physics, 94 (1984), 35-59.
doi: 10.1007/BF01212348. |
[27] |
K. Takasaki, A new approach to the self-dual Yang-Mills equations II, Saitama Math.J., 3 (1985), 11-40. |
[28] |
K. Takasaki, Dressing operator approach to Moyal algebraic deformation of selfdual gravity, Journal of Geometry and Physics, 14 (1994), 111-120.
doi: 10.1016/0393-0440(94)90003-5. |
[29] |
K. Takasaki, Nonabelian KP hierarchy with Moyal algebraic coefficients, Journal of Geometry and Physics, 14 (1994), 332-364.
doi: 10.1016/0393-0440(94)90040-X. |
[30] |
D. A. Tuganbaev, Laurent series rings and pseudo-differential operator rings, Journal of Mathematical Sciences (NY), 128 (2005), 2843-2893.
doi: 10.1007/s10958-005-0244-6. |
[31] |
Y. Watanabe, Hamiltonian structure of Sato's hierarchy of KP equations and a coadjoint orbit of a certain formal Lie group, Letters in Mathematical Physics, 7 (1983), 99-106.
doi: 10.1007/BF00419926. |
[32] |
Y. Watanabe, Hamiltonian structure of M. Sato's hierarchy of Kadomtsev-Petviashvili equation, Annali di Matematica Pura ed Applicata, 136 (1984), 77-93.
doi: 10.1007/BF01773378. |
[33] |
A. B. Zheglov, On rings of commuting partial differential operators, preprint, arXiv:1106.0765. |
[34] |
A. B. Zheglov, Two dimensional KP systems and their solvability, preprint, arXiv:math-ph/0503067. |
show all references
References:
[1] |
M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-Devries type equations, Inventiones Mathematicae, 50 (1979), 219-248.
doi: 10.1007/BF01410079. |
[2] |
N. Bourbaki, "Algebra I. Chapters 1-3. Elements of Mathematics," Springer-Verlag, Berlin, 1998. |
[3] |
E. E. Demidov, On the Kadomtsev-Petviashvili hierarchy with a noncommutative timespace, Functional Analysis and Its Applications, 29 (1995), 131-133.
doi: 10.1007/BF01080014. |
[4] |
E. E. Demidov, Noncommutative deformation of the Kadomtsev-Petviashvili hierarchy, In "Algebra. 5, Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI)," Moscow, 1995. (Russian), Journal of Mathematical Sciences (New York), 88 (1998), 520-536 (English).
doi: 10.1007/BF02365314. |
[5] |
L. A. Dickey, "Soliton Equations and Hamiltonian Systems," Second Edition, Advanced Series in Mathematical Physics $12$, World Scientific Publ. Co., Singapore, 2003. |
[6] |
L. D. Faddeev, and L. A. Takhtajan, "Hamiltonian Methods in the Theory of Solitons," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin-New York, 1987. |
[7] |
B. A. Khesin and I. Zakharevich, Poisson-Lie groups of pseudodifferential symbols, Communications in Mathematical Physics, 171 (1995), 475-530.
doi: 10.1007/BF02104676. |
[8] |
B. A. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Advances in Mathematics, 176 (2003), 116-144.
doi: 10.1016/S0001-8708(02)00063-4. |
[9] |
B. A. Khesin and R. Wendt, "The Geometry of Infinite-Dimensional Groups," Springer-Verlag, Berlin, 2009. |
[10] |
F. Kubo, Non-commutative Poisson algebra structures on affine Kac-Moody algebras, Journal of Pure and Applied Algebra, 126 (1998), 267-286.
doi: 10.1016/S0022-4049(96)00141-7. |
[11] |
L.-C. Li, Factorization problem on the Hilbert-Schmidt group and the Camassa-Holm equation, Communications on Pure and Applied Mathematics, 61 (2008), 186-209.
doi: 10.1002/cpa.20207. |
[12] |
J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. |
[13] |
J. Mickelsson, "Current Algebras and Groups," Plenum Press, New York and London, 1989. |
[14] |
A. V. Mikhailov, A. B. Shabat and V. V. Sokolov, The symmetry approach to classification of integrable equations, in "What is Integrability?" (ed. V.E. Zakharov), Springer Ser. Nonlinear Dynam., Springer, Berlin, (1991), 115-184. |
[15] |
M. Mulase, Complete integrability of the Kadomtsev-Petvishvili equation, Advances in Mathematics, 54 (1984), 57-66.
doi: 10.1016/0001-8708(84)90036-7. |
[16] |
M. Mulase, Cohomological structure in soliton equations and Jacobian varieties, J. Differential Geom., 19 (1984), 403-430. |
[17] |
M. Mulase, Solvability of the super KP equation and a generalization of the Birkhoff decomposition, Inventiones Mathematicae, 92 (1988), 1-46.
doi: 10.1007/BF01393991. |
[18] |
P. J. Olver, "Applications of Lie Groups to Differential Equations," Second Edition, Springer-Verlag, New York, 1993. |
[19] |
P. J. Olver and V. V. Sokolov, Integrable evolution equations on associative algebras, Communications in Mathematical Physics, 193 (1998) 245-268.
doi: 10.1007/s002200050328. |
[20] |
A. N. Parshin, On a ring of formal pseudo-differential operators, Proc. Steklov Inst. Math. 224 (1999), 266-280. |
[21] |
A. M. Perelomov, "Integrable Systems of Classical Mechanics And Lie Algebras," Birkhäuser Verlag, Berlin, 1990.
doi: 10.1007/978-3-0348-9257-5. |
[22] |
A. Pressley and G.B. Segal, "Loop Groups," Oxford University Press, 1986. |
[23] |
A. G. Reyman and M. A. Semenov-Tian-Shansky, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II, Inventiones mathematicae, 63 (1981), 423-432.
doi: 10.1007/BF01389063. |
[24] |
M. Sakakibara, Factorization methods for noncommutative KP and Toda hierarchy, Journal of Physics A: Mathematical and General, 37 (2004), L599-L604.
doi: 10.1088/0305-4470/37/45/L02. |
[25] |
M. A. Semenov-Tian-Shansky, What is a classical $r$-matrix?, Funct. Anal. Appl., 17 (1983), 259-272. |
[26] |
K. Takasaki, A new approach to the self-dual Yang-Mills equations, Communications in Mathematical Physics, 94 (1984), 35-59.
doi: 10.1007/BF01212348. |
[27] |
K. Takasaki, A new approach to the self-dual Yang-Mills equations II, Saitama Math.J., 3 (1985), 11-40. |
[28] |
K. Takasaki, Dressing operator approach to Moyal algebraic deformation of selfdual gravity, Journal of Geometry and Physics, 14 (1994), 111-120.
doi: 10.1016/0393-0440(94)90003-5. |
[29] |
K. Takasaki, Nonabelian KP hierarchy with Moyal algebraic coefficients, Journal of Geometry and Physics, 14 (1994), 332-364.
doi: 10.1016/0393-0440(94)90040-X. |
[30] |
D. A. Tuganbaev, Laurent series rings and pseudo-differential operator rings, Journal of Mathematical Sciences (NY), 128 (2005), 2843-2893.
doi: 10.1007/s10958-005-0244-6. |
[31] |
Y. Watanabe, Hamiltonian structure of Sato's hierarchy of KP equations and a coadjoint orbit of a certain formal Lie group, Letters in Mathematical Physics, 7 (1983), 99-106.
doi: 10.1007/BF00419926. |
[32] |
Y. Watanabe, Hamiltonian structure of M. Sato's hierarchy of Kadomtsev-Petviashvili equation, Annali di Matematica Pura ed Applicata, 136 (1984), 77-93.
doi: 10.1007/BF01773378. |
[33] |
A. B. Zheglov, On rings of commuting partial differential operators, preprint, arXiv:1106.0765. |
[34] |
A. B. Zheglov, Two dimensional KP systems and their solvability, preprint, arXiv:math-ph/0503067. |
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