TY - JOUR

T1 - Nonholonomic deformation of generalized KdV-type equations

AU - Guha, Partha

PY - 2009

Y1 - 2009

N2 - Karasu-Kalkani et al (2008 J. Math. Phys. 49 073516) recently derived a new sixth-order wave equation KdV6, which was shown by Kupershmidt (2008 Phys. Lett. 372A 2634) to have an infinite commuting hierarchy with a common infinite set of conserved densities. Incidentally, this equation was written for the first time by Calogero and is included in the book by Calogero and Degasperis (1982 Lecture Notes in Computer Science vol 144 (Amsterdam: North-Holland) p 516). In this paper, we give a geometric insight into the KdV6 equation. Using Kirillov's theory of coadjoint representation of the Virasoro algebra, we show how to obtain a large class of KdV6-type equations equivalent to the original equation. Using a semidirect product extension of the Virasoro algebra, , we propose the nonholonomic deformation of the Ito equation. We also show that the Adler-Kostant-Symes scheme provides a geometrical method for constructing nonholonomic deformed integrable systems. Applying the Adler-Kostant-Symes scheme to loop algebra, we construct a new nonholonomic deformation of the coupled KdV equation.

AB - Karasu-Kalkani et al (2008 J. Math. Phys. 49 073516) recently derived a new sixth-order wave equation KdV6, which was shown by Kupershmidt (2008 Phys. Lett. 372A 2634) to have an infinite commuting hierarchy with a common infinite set of conserved densities. Incidentally, this equation was written for the first time by Calogero and is included in the book by Calogero and Degasperis (1982 Lecture Notes in Computer Science vol 144 (Amsterdam: North-Holland) p 516). In this paper, we give a geometric insight into the KdV6 equation. Using Kirillov's theory of coadjoint representation of the Virasoro algebra, we show how to obtain a large class of KdV6-type equations equivalent to the original equation. Using a semidirect product extension of the Virasoro algebra, , we propose the nonholonomic deformation of the Ito equation. We also show that the Adler-Kostant-Symes scheme provides a geometrical method for constructing nonholonomic deformed integrable systems. Applying the Adler-Kostant-Symes scheme to loop algebra, we construct a new nonholonomic deformation of the coupled KdV equation.

UR - http://www.scopus.com/inward/record.url?scp=70449589832&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/42/34/345201

DO - 10.1088/1751-8113/42/34/345201

M3 - Article

AN - SCOPUS:70449589832

SN - 1751-8113

VL - 42

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 34

M1 - 345201

ER -