# American Institute of Mathematical Sciences

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

 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.
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

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##### 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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