American Institute of Mathematical Sciences

Advanced
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.  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

[1]

Philippe Gravejat. Asymptotics of the solitary waves for the generalized Kadomtsev-Petviashvili equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 835-882. doi: 10.3934/dcds.2008.21.835
[2]

Pedro Isaza, Juan López, Jorge Mejía. Cauchy problem for the fifth order Kadomtsev-Petviashvili (KPII) equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 887-905. doi: 10.3934/cpaa.2006.5.887
[3]

Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483
[4]

Pedro Isaza, Jorge Mejía. On the support of solutions to the Kadomtsev-Petviashvili (KP-II) equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1239-1255. doi: 10.3934/cpaa.2011.10.1239
[5]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247
[6]

Christian Klein, Ralf Peter. Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1689-1717. doi: 10.3934/dcdsb.2014.19.1689
[7]

Yuanhong Wei, Yong Li, Xue Yang. On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1095-1106. doi: 10.3934/dcdss.2017059
[8]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
[9]

Daniele Mundici. The Haar theorem for lattice-ordered abelian groups with order-unit. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 537-549. doi: 10.3934/dcds.2008.21.537
[10]

Brandon Seward. Krieger's finite generator theorem for actions of countable groups Ⅱ. Journal of Modern Dynamics, 2019, 15: 1-39. doi: 10.3934/jmd.2019012
[11]

Anwar Ja'afar Mohamad Jawad, Mohammad Mirzazadeh, Anjan Biswas. Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1155-1164. doi: 10.3934/dcdss.2015.8.1155
[12]

Qiang Li. A kind of generalized transversality theorem for $C^r$ mapping with parameter. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1043-1050. doi: 10.3934/dcdss.2017055
[13]

Lizhi Zhang, Congming Li, Wenxiong Chen, Tingzhi Cheng. A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1721-1736. doi: 10.3934/dcds.2016.36.1721
[14]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351
[15]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517
[16]

Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 0 (0) : 0-0. doi: 10.3934/jgm.2020007
[17]

Daniel T. Wise. Research announcement: The structure of groups with a quasiconvex hierarchy. Electronic Research Announcements, 2009, 16: 44-55. doi: 10.3934/era.2009.16.44
[18]

Mauro Fabrizio, Jaime Munõz Rivera. An integration model for two different ethnic groups. Evolution Equations & Control Theory, 2014, 3 (2) : 277-286. doi: 10.3934/eect.2014.3.277
[19]

Sangye Lungten, Wil H. A. Schilders, Joseph M. L. Maubach. Sparse inverse incidence matrices for Schilders' factorization applied to resistor network modeling. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 227-239. doi: 10.3934/naco.2014.4.227
[20]

Habibulla Akhadkulov, Akhtam Dzhalilov, Konstantin Khanin. Notes on a theorem of Katznelson and Ornstein. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4587-4609. doi: 10.3934/dcds.2017197

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences