TY - JOUR

T1 - Infinitely many conservation laws for the discrete kdv equation

AU - Rasin, Alexander G.

AU - Schiff, Jeremy

PY - 2009

Y1 - 2009

N2 - Rasin and Hydon (2007 J. Phys. A: Math. Theor. 40 12763-73) suggested a way to construct an infinite number of conservation laws for the discrete KdV equation (dKdV), by repeated application of a certain symmetry to a known conservation law. It was not decided, however, whether the resulting conservation laws were distinct and nontrivial. In this paper we obtain the following results: (1) we give an alternative method to construct an infinite number of conservation laws using a discrete version of the Gardner transformation. (2) We give a direct proof that the conservation laws obtained by the method of Rasin and Hydon are indeed distinct and nontrivial. (3) We consider a continuum limit in which the dKdV equation becomes a first-order eikonal equation. In this limit the two sets of conservation laws become the same, and are evidently distinct and nontrivial. This proves the nontriviality of the conservation laws constructed by the Gardner method, and gives an alternative proof of the nontriviality of the conservation laws constructed by the method of Rasin and Hydon.

AB - Rasin and Hydon (2007 J. Phys. A: Math. Theor. 40 12763-73) suggested a way to construct an infinite number of conservation laws for the discrete KdV equation (dKdV), by repeated application of a certain symmetry to a known conservation law. It was not decided, however, whether the resulting conservation laws were distinct and nontrivial. In this paper we obtain the following results: (1) we give an alternative method to construct an infinite number of conservation laws using a discrete version of the Gardner transformation. (2) We give a direct proof that the conservation laws obtained by the method of Rasin and Hydon are indeed distinct and nontrivial. (3) We consider a continuum limit in which the dKdV equation becomes a first-order eikonal equation. In this limit the two sets of conservation laws become the same, and are evidently distinct and nontrivial. This proves the nontriviality of the conservation laws constructed by the Gardner method, and gives an alternative proof of the nontriviality of the conservation laws constructed by the method of Rasin and Hydon.

UR - http://www.scopus.com/inward/record.url?scp=67650914305&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/42/17/175205

DO - 10.1088/1751-8113/42/17/175205

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:67650914305

SN - 1751-8113

VL - 42

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 17

M1 - 175205

ER -