The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups

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September 2013, 5(3): 345-364. doi: 10.3934/jgm.2013.5.345

The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups

Anahita Eslami Rad ^1, and Enrique G. Reyes ^2,

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

Received March 2013 Revised August 2013 Published September 2013

Abstract
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We present explicit formal solutions to the systems of equations in two independent variables $t_m$, $x$, $m =1,2,\dots$, of the Kadomtsev-Petviashvili hierarchy. The main tools used are a Birkhoff-like factorization of formal Lie groups due to M. Mulase, and the classical theory of A.G. Reyman and M.A. Semenov-Tian-Shansky on the integration of Hamiltonian systems on coadjoint orbits using $r$-matrices. Our paper also contains full proofs of Mulase's results.

Keywords: $r$-matrices, Mulase factorization theorem, formal Lie groups, Kadomtsev-Petviashvili hierarchy, Reyman-Semenov Tian Shansky integration theorem..

Mathematics Subject Classification: Primary: 37K10, 37K30; Secondary: 17B80, 22E6.

Citation: Anahita Eslami Rad, Enrique G. Reyes. The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups. Journal of Geometric Mechanics, 2013, 5 (3) : 345-364. doi: 10.3934/jgm.2013.5.345

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. doi: 10.1007/BF01410079. Google Scholar
[2]	N. Bourbaki, "Algebra I. Chapters 1-3. Elements of Mathematics,", Springer-Verlag, (1998). Google Scholar
[3]	E. E. Demidov, On the Kadomtsev-Petviashvili hierarchy with a noncommutative timespace,, Functional Analysis and Its Applications, 29 (1995), 131. doi: 10.1007/BF01080014. Google Scholar
[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. doi: 10.1007/BF02365314. Google Scholar
[5]	L. A. Dickey, "Soliton Equations and Hamiltonian Systems," Second Edition,, Advanced Series in Mathematical Physics $12$, (2003). Google Scholar
[6]	L. D. Faddeev, and L. A. Takhtajan, "Hamiltonian Methods in the Theory of Solitons,", Springer Series in Soviet Mathematics, (1987). Google Scholar
[7]	B. A. Khesin and I. Zakharevich, Poisson-Lie groups of pseudodifferential symbols,, Communications in Mathematical Physics, 171 (1995), 475. doi: 10.1007/BF02104676. Google Scholar
[8]	B. A. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits,, Advances in Mathematics, 176 (2003), 116. doi: 10.1016/S0001-8708(02)00063-4. Google Scholar
[9]	B. A. Khesin and R. Wendt, "The Geometry of Infinite-Dimensional Groups,", Springer-Verlag, (2009). Google Scholar
[10]	F. Kubo, Non-commutative Poisson algebra structures on affine Kac-Moody algebras,, Journal of Pure and Applied Algebra, 126 (1998), 267. doi: 10.1016/S0022-4049(96)00141-7. Google Scholar
[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. doi: 10.1002/cpa.20207. Google Scholar
[12]	J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,", Second edition, (1999). Google Scholar
[13]	J. Mickelsson, "Current Algebras and Groups,", Plenum Press, (1989). Google Scholar
[14]	A. V. Mikhailov, A. B. Shabat and V. V. Sokolov, The symmetry approach to classification of integrable equations,, in, (1991), 115. Google Scholar
[15]	M. Mulase, Complete integrability of the Kadomtsev-Petvishvili equation,, Advances in Mathematics, 54 (1984), 57. doi: 10.1016/0001-8708(84)90036-7. Google Scholar
[16]	M. Mulase, Cohomological structure in soliton equations and Jacobian varieties,, J. Differential Geom., 19 (1984), 403. Google Scholar
[17]	M. Mulase, Solvability of the super KP equation and a generalization of the Birkhoff decomposition,, Inventiones Mathematicae, 92 (1988), 1. doi: 10.1007/BF01393991. Google Scholar
[18]	P. J. Olver, "Applications of Lie Groups to Differential Equations,", Second Edition, (1993). Google Scholar
[19]	P. J. Olver and V. V. Sokolov, Integrable evolution equations on associative algebras,, Communications in Mathematical Physics, 193 (1998), 245. doi: 10.1007/s002200050328. Google Scholar
[20]	A. N. Parshin, On a ring of formal pseudo-differential operators,, Proc. Steklov Inst. Math. 224* (1999), 224* (1999), 266. Google Scholar
[21]	A. M. Perelomov, "Integrable Systems of Classical Mechanics And Lie Algebras,", Birkhäuser Verlag, (1990). doi: 10.1007/978-3-0348-9257-5. Google Scholar
[22]	A. Pressley and G.B. Segal, "Loop Groups,", Oxford University Press, (1986). Google Scholar
[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. doi: 10.1007/BF01389063. Google Scholar
[24]	M. Sakakibara, Factorization methods for noncommutative KP and Toda hierarchy,, Journal of Physics A: Mathematical and General, 37 (2004). doi: 10.1088/0305-4470/37/45/L02. Google Scholar
[25]	M. A. Semenov-Tian-Shansky, What is a classical $r$-matrix?,, Funct. Anal. Appl., 17 (1983), 259. Google Scholar
[26]	K. Takasaki, A new approach to the self-dual Yang-Mills equations,, Communications in Mathematical Physics, 94 (1984), 35. doi: 10.1007/BF01212348. Google Scholar
[27]	K. Takasaki, A new approach to the self-dual Yang-Mills equations II,, Saitama Math.J., 3 (1985), 11. Google Scholar
[28]	K. Takasaki, Dressing operator approach to Moyal algebraic deformation of selfdual gravity,, Journal of Geometry and Physics, 14 (1994), 111. doi: 10.1016/0393-0440(94)90003-5. Google Scholar
[29]	K. Takasaki, Nonabelian KP hierarchy with Moyal algebraic coefficients,, Journal of Geometry and Physics, 14 (1994), 332. doi: 10.1016/0393-0440(94)90040-X. Google Scholar
[30]	D. A. Tuganbaev, Laurent series rings and pseudo-differential operator rings,, Journal of Mathematical Sciences (NY), 128 (2005), 2843. doi: 10.1007/s10958-005-0244-6. Google Scholar
[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. doi: 10.1007/BF00419926. Google Scholar
[32]	Y. Watanabe, Hamiltonian structure of M. Sato's hierarchy of Kadomtsev-Petviashvili equation,, Annali di Matematica Pura ed Applicata, 136 (1984), 77. doi: 10.1007/BF01773378. Google Scholar
[33]	A. B. Zheglov, On rings of commuting partial differential operators, preprint,, , (). Google Scholar
[34]	A. B. Zheglov, Two dimensional KP systems and their solvability, preprint,, , (). Google Scholar

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. doi: 10.1007/BF01410079. Google Scholar
[2]	N. Bourbaki, "Algebra I. Chapters 1-3. Elements of Mathematics,", Springer-Verlag, (1998). Google Scholar
[3]	E. E. Demidov, On the Kadomtsev-Petviashvili hierarchy with a noncommutative timespace,, Functional Analysis and Its Applications, 29 (1995), 131. doi: 10.1007/BF01080014. Google Scholar
[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. doi: 10.1007/BF02365314. Google Scholar
[5]	L. A. Dickey, "Soliton Equations and Hamiltonian Systems," Second Edition,, Advanced Series in Mathematical Physics $12$, (2003). Google Scholar
[6]	L. D. Faddeev, and L. A. Takhtajan, "Hamiltonian Methods in the Theory of Solitons,", Springer Series in Soviet Mathematics, (1987). Google Scholar
[7]	B. A. Khesin and I. Zakharevich, Poisson-Lie groups of pseudodifferential symbols,, Communications in Mathematical Physics, 171 (1995), 475. doi: 10.1007/BF02104676. Google Scholar
[8]	B. A. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits,, Advances in Mathematics, 176 (2003), 116. doi: 10.1016/S0001-8708(02)00063-4. Google Scholar
[9]	B. A. Khesin and R. Wendt, "The Geometry of Infinite-Dimensional Groups,", Springer-Verlag, (2009). Google Scholar
[10]	F. Kubo, Non-commutative Poisson algebra structures on affine Kac-Moody algebras,, Journal of Pure and Applied Algebra, 126 (1998), 267. doi: 10.1016/S0022-4049(96)00141-7. Google Scholar
[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. doi: 10.1002/cpa.20207. Google Scholar
[12]	J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,", Second edition, (1999). Google Scholar
[13]	J. Mickelsson, "Current Algebras and Groups,", Plenum Press, (1989). Google Scholar
[14]	A. V. Mikhailov, A. B. Shabat and V. V. Sokolov, The symmetry approach to classification of integrable equations,, in, (1991), 115. Google Scholar
[15]	M. Mulase, Complete integrability of the Kadomtsev-Petvishvili equation,, Advances in Mathematics, 54 (1984), 57. doi: 10.1016/0001-8708(84)90036-7. Google Scholar
[16]	M. Mulase, Cohomological structure in soliton equations and Jacobian varieties,, J. Differential Geom., 19 (1984), 403. Google Scholar
[17]	M. Mulase, Solvability of the super KP equation and a generalization of the Birkhoff decomposition,, Inventiones Mathematicae, 92 (1988), 1. doi: 10.1007/BF01393991. Google Scholar
[18]	P. J. Olver, "Applications of Lie Groups to Differential Equations,", Second Edition, (1993). Google Scholar
[19]	P. J. Olver and V. V. Sokolov, Integrable evolution equations on associative algebras,, Communications in Mathematical Physics, 193 (1998), 245. doi: 10.1007/s002200050328. Google Scholar
[20]	A. N. Parshin, On a ring of formal pseudo-differential operators,, Proc. Steklov Inst. Math. 224* (1999), 224* (1999), 266. Google Scholar
[21]	A. M. Perelomov, "Integrable Systems of Classical Mechanics And Lie Algebras,", Birkhäuser Verlag, (1990). doi: 10.1007/978-3-0348-9257-5. Google Scholar
[22]	A. Pressley and G.B. Segal, "Loop Groups,", Oxford University Press, (1986). Google Scholar
[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. doi: 10.1007/BF01389063. Google Scholar
[24]	M. Sakakibara, Factorization methods for noncommutative KP and Toda hierarchy,, Journal of Physics A: Mathematical and General, 37 (2004). doi: 10.1088/0305-4470/37/45/L02. Google Scholar
[25]	M. A. Semenov-Tian-Shansky, What is a classical $r$-matrix?,, Funct. Anal. Appl., 17 (1983), 259. Google Scholar
[26]	K. Takasaki, A new approach to the self-dual Yang-Mills equations,, Communications in Mathematical Physics, 94 (1984), 35. doi: 10.1007/BF01212348. Google Scholar
[27]	K. Takasaki, A new approach to the self-dual Yang-Mills equations II,, Saitama Math.J., 3 (1985), 11. Google Scholar
[28]	K. Takasaki, Dressing operator approach to Moyal algebraic deformation of selfdual gravity,, Journal of Geometry and Physics, 14 (1994), 111. doi: 10.1016/0393-0440(94)90003-5. Google Scholar
[29]	K. Takasaki, Nonabelian KP hierarchy with Moyal algebraic coefficients,, Journal of Geometry and Physics, 14 (1994), 332. doi: 10.1016/0393-0440(94)90040-X. Google Scholar
[30]	D. A. Tuganbaev, Laurent series rings and pseudo-differential operator rings,, Journal of Mathematical Sciences (NY), 128 (2005), 2843. doi: 10.1007/s10958-005-0244-6. Google Scholar
[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. doi: 10.1007/BF00419926. Google Scholar
[32]	Y. Watanabe, Hamiltonian structure of M. Sato's hierarchy of Kadomtsev-Petviashvili equation,, Annali di Matematica Pura ed Applicata, 136 (1984), 77. doi: 10.1007/BF01773378. Google Scholar
[33]	A. B. Zheglov, On rings of commuting partial differential operators, preprint,, , (). Google Scholar
[34]	A. B. Zheglov, Two dimensional KP systems and their solvability, preprint,, , (). Google Scholar

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