TY - JOUR

T1 - Symmetries, Conservation Laws, and Noether's Theorem for Differential-Difference Equations

AU - Peng, Linyu

N1 - Funding Information:
The author is indebted to Cheng Zhang and Da-jun Zhang for their hospitality during his visit at Shanghai University, when part of this work was done. The author is grateful to Peter Hydon, whose comments on led to Theorem. The author would also like to thank Pavlos Xenitidis for insightful discussions on variational principle of DDEs. This work was partially supported by Grant-in-Aid for Scientific Research (16KT0024) and Waseda University Grants for Special Research Projects (2016B-119).

PY - 2017/10

Y1 - 2017/10

N2 - This paper mainly contributes to the extension of Noether's theorem to differential-difference equations. For this purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws, and the Fréchet derivative are also investigated. For nonvariational equations, because Noether's theorem is now available, the self-adjointness method is adapted to the computation of conservation laws for differential-difference equations. Several differential-difference equations are investigated as illustrative examples, including the Toda lattice and semidiscretizations of the Korteweg–de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.

AB - This paper mainly contributes to the extension of Noether's theorem to differential-difference equations. For this purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws, and the Fréchet derivative are also investigated. For nonvariational equations, because Noether's theorem is now available, the self-adjointness method is adapted to the computation of conservation laws for differential-difference equations. Several differential-difference equations are investigated as illustrative examples, including the Toda lattice and semidiscretizations of the Korteweg–de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.

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U2 - 10.1111/sapm.12168

DO - 10.1111/sapm.12168

M3 - Article

AN - SCOPUS:85017399706

VL - 139

SP - 457

EP - 502

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 3

ER -