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]

Inventiones Mathematicae, 50 (1979), 219-248. doi: 10.1007/BF01410079.  Google Scholar

[2]

Springer-Verlag, Berlin, 1998.  Google Scholar

[3]

Functional Analysis and Its Applications, 29 (1995), 131-133. doi: 10.1007/BF01080014.  Google Scholar

[4]

Journal of Mathematical Sciences (New York), 88 (1998), 520-536 (English). doi: 10.1007/BF02365314.  Google Scholar

[5]

Advanced Series in Mathematical Physics $12$, World Scientific Publ. Co., Singapore, 2003.  Google Scholar

[6]

Springer Series in Soviet Mathematics, Springer-Verlag, Berlin-New York, 1987.  Google Scholar

[7]

Communications in Mathematical Physics, 171 (1995), 475-530. doi: 10.1007/BF02104676.  Google Scholar

[8]

Advances in Mathematics, 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[9]

Springer-Verlag, Berlin, 2009.  Google Scholar

[10]

Journal of Pure and Applied Algebra, 126 (1998), 267-286. doi: 10.1016/S0022-4049(96)00141-7.  Google Scholar

[11]

Communications on Pure and Applied Mathematics, 61 (2008), 186-209. doi: 10.1002/cpa.20207.  Google Scholar

[12]

Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999.  Google Scholar

[13]

Plenum Press, New York and London, 1989.  Google Scholar

[14]

in "What is Integrability?" (ed. V.E. Zakharov), Springer Ser. Nonlinear Dynam., Springer, Berlin, (1991), 115-184.  Google Scholar

[15]

Advances in Mathematics, 54 (1984), 57-66. doi: 10.1016/0001-8708(84)90036-7.  Google Scholar

[16]

J. Differential Geom., 19 (1984), 403-430.  Google Scholar

[17]

Inventiones Mathematicae, 92 (1988), 1-46. doi: 10.1007/BF01393991.  Google Scholar

[18]

Second Edition, Springer-Verlag, New York, 1993.  Google Scholar

[19]

Communications in Mathematical Physics, 193 (1998) 245-268. doi: 10.1007/s002200050328.  Google Scholar

[20]

Proc. Steklov Inst. Math. 224 (1999), 266-280.  Google Scholar

[21]

Birkhäuser Verlag, Berlin, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar

[22]

Oxford University Press, 1986. Google Scholar

[23]

Inventiones mathematicae, 63 (1981), 423-432. doi: 10.1007/BF01389063.  Google Scholar

[24]

Journal of Physics A: Mathematical and General, 37 (2004), L599-L604. doi: 10.1088/0305-4470/37/45/L02.  Google Scholar

[25]

Funct. Anal. Appl., 17 (1983), 259-272. Google Scholar

[26]

Communications in Mathematical Physics, 94 (1984), 35-59. doi: 10.1007/BF01212348.  Google Scholar

[27]

Saitama Math.J., 3 (1985), 11-40.  Google Scholar

[28]

Journal of Geometry and Physics, 14 (1994), 111-120. doi: 10.1016/0393-0440(94)90003-5.  Google Scholar

[29]

Journal of Geometry and Physics, 14 (1994), 332-364. doi: 10.1016/0393-0440(94)90040-X.  Google Scholar

[30]

Journal of Mathematical Sciences (NY), 128 (2005), 2843-2893. doi: 10.1007/s10958-005-0244-6.  Google Scholar

[31]

Letters in Mathematical Physics, 7 (1983), 99-106. doi: 10.1007/BF00419926.  Google Scholar

[32]

Annali di Matematica Pura ed Applicata, 136 (1984), 77-93. 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]

Inventiones Mathematicae, 50 (1979), 219-248. doi: 10.1007/BF01410079.  Google Scholar

[2]

Springer-Verlag, Berlin, 1998.  Google Scholar

[3]

Functional Analysis and Its Applications, 29 (1995), 131-133. doi: 10.1007/BF01080014.  Google Scholar

[4]

Journal of Mathematical Sciences (New York), 88 (1998), 520-536 (English). doi: 10.1007/BF02365314.  Google Scholar

[5]

Advanced Series in Mathematical Physics $12$, World Scientific Publ. Co., Singapore, 2003.  Google Scholar

[6]

Springer Series in Soviet Mathematics, Springer-Verlag, Berlin-New York, 1987.  Google Scholar

[7]

Communications in Mathematical Physics, 171 (1995), 475-530. doi: 10.1007/BF02104676.  Google Scholar

[8]

Advances in Mathematics, 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[9]

Springer-Verlag, Berlin, 2009.  Google Scholar

[10]

Journal of Pure and Applied Algebra, 126 (1998), 267-286. doi: 10.1016/S0022-4049(96)00141-7.  Google Scholar

[11]

Communications on Pure and Applied Mathematics, 61 (2008), 186-209. doi: 10.1002/cpa.20207.  Google Scholar

[12]

Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999.  Google Scholar

[13]

Plenum Press, New York and London, 1989.  Google Scholar

[14]

in "What is Integrability?" (ed. V.E. Zakharov), Springer Ser. Nonlinear Dynam., Springer, Berlin, (1991), 115-184.  Google Scholar

[15]

Advances in Mathematics, 54 (1984), 57-66. doi: 10.1016/0001-8708(84)90036-7.  Google Scholar

[16]

J. Differential Geom., 19 (1984), 403-430.  Google Scholar

[17]

Inventiones Mathematicae, 92 (1988), 1-46. doi: 10.1007/BF01393991.  Google Scholar

[18]

Second Edition, Springer-Verlag, New York, 1993.  Google Scholar

[19]

Communications in Mathematical Physics, 193 (1998) 245-268. doi: 10.1007/s002200050328.  Google Scholar

[20]

Proc. Steklov Inst. Math. 224 (1999), 266-280.  Google Scholar

[21]

Birkhäuser Verlag, Berlin, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar

[22]

Oxford University Press, 1986. Google Scholar

[23]

Inventiones mathematicae, 63 (1981), 423-432. doi: 10.1007/BF01389063.  Google Scholar

[24]

Journal of Physics A: Mathematical and General, 37 (2004), L599-L604. doi: 10.1088/0305-4470/37/45/L02.  Google Scholar

[25]

Funct. Anal. Appl., 17 (1983), 259-272. Google Scholar

[26]

Communications in Mathematical Physics, 94 (1984), 35-59. doi: 10.1007/BF01212348.  Google Scholar

[27]

Saitama Math.J., 3 (1985), 11-40.  Google Scholar

[28]

Journal of Geometry and Physics, 14 (1994), 111-120. doi: 10.1016/0393-0440(94)90003-5.  Google Scholar

[29]

Journal of Geometry and Physics, 14 (1994), 332-364. doi: 10.1016/0393-0440(94)90040-X.  Google Scholar

[30]

Journal of Mathematical Sciences (NY), 128 (2005), 2843-2893. doi: 10.1007/s10958-005-0244-6.  Google Scholar

[31]

Letters in Mathematical Physics, 7 (1983), 99-106. doi: 10.1007/BF00419926.  Google Scholar

[32]

Annali di Matematica Pura ed Applicata, 136 (1984), 77-93. 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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