Semi-Commutative Differential Operators Associated with the Dirac Opetator and Darboux Transformation

ABSTRACT

In the present paper, the semi-commutative differential oparators associated with the 1-dimensional Dirac operator are constructed. Using this results, the hierarchy of the mKdV (-) polynomials are expressed in terms of the KdV polynomials. These formulas give a new interpretation of the classical Darboux transformation and the Miura transformation. Moreover, the recursion operator associated with the hierarchy of the mKdV (-) polynomials is constructed by the algebraic method.

Cite this paper

M. Matsushima and M. Ohmiya, "Semi-Commutative Differential Operators Associated with the Dirac Opetator and Darboux Transformation,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 209-213. doi: 10.4236/apm.2013.31A029.

M. Matsushima and M. Ohmiya, "Semi-Commutative Differential Operators Associated with the Dirac Opetator and Darboux Transformation,"

References

[1] R. M. Miura, “Korteweg—De Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation,” Journal of Mathematical Physics, Vol. 9, No. 8, 1968, pp. 1202-1204. doi:10.1063/1.1664700

[2] M. Ohmiya, “Spectrum of Darboux Transformation of Differential Operator,” Osaka Journal of Mathematics, Vol. 36, No. 4, 1999, pp. 949-980.

[3] M. Ohmiya, “KdV Polynomials and Λ-Operator,” Osaka Journal of Mathematics, Vol. 32, No. 2, 1995, pp. 409-430.

[4] M. Ohmiya and Y. P. Mishev, “Darboux Transformation and Λ-Operator,” Journal of Mathematics/Tokushima University, Vol. 27, 1993, pp. 1-15.

[5] M. Matsushima and M. Ohmiya, “An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy,” Proceeding of ICNAAM: Numerical Analysis and Applied Mathematics, Vol. 1, 2009, pp. 168-172.

[1] R. M. Miura, “Korteweg—De Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation,” Journal of Mathematical Physics, Vol. 9, No. 8, 1968, pp. 1202-1204. doi:10.1063/1.1664700

[2] M. Ohmiya, “Spectrum of Darboux Transformation of Differential Operator,” Osaka Journal of Mathematics, Vol. 36, No. 4, 1999, pp. 949-980.

[3] M. Ohmiya, “KdV Polynomials and Λ-Operator,” Osaka Journal of Mathematics, Vol. 32, No. 2, 1995, pp. 409-430.

[4] M. Ohmiya and Y. P. Mishev, “Darboux Transformation and Λ-Operator,” Journal of Mathematics/Tokushima University, Vol. 27, 1993, pp. 1-15.

[5] M. Matsushima and M. Ohmiya, “An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy,” Proceeding of ICNAAM: Numerical Analysis and Applied Mathematics, Vol. 1, 2009, pp. 168-172.