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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

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), 219-248. 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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