Conservation Laws of Discrete Evolution Equations by Symmetries and Adjoint Symmetries

A direct approach is proposed for constructing conservation laws of discrete evolution equations, regardless of the existence of a Lagrangian. The approach utilizes pairs of symmetries and adjoint symmetries, in which adjoint symmetries make up for the disadvantage of non-Lagrangian structures in presenting a correspondence between symmetries and conservation laws. Applications are made for the construction of conservation laws of the Volterra lattice equation.

[1]  V. K. Chandrasekar,et al.  Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second-order differential equations , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  Vito Volterra,et al.  Leçons sur la théorie mathématique de la lutte pour la vie , 1931 .

[3]  Wenxiu Ma,et al.  Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations , 1992 .

[4]  E. Nissimov,et al.  Bäcklund and Darboux Transformations. The Geometry of Solitons , 2022 .

[5]  Risto Korhonen,et al.  Symmetries and Integrability of Difference Equations , 2007 .

[6]  Wenxiu Ma,et al.  Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras , 2006 .

[7]  Tu Gui-Zhang,et al.  On Liouville integrability of zero-curvature equations and the Yang hierarchy , 1989 .

[8]  C. Morawetz.  Variations on conservation laws for the wave equation , 2000 .

[9]  W. Ma.  The algebraic structures of isospectral Lax operators and applications to integrable equations , 1992 .

[10]  Adjoint symmetries, separability, and volume forms , 2000 .

[11]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[12]  Dengyuan Chen,et al.  The conservation laws of some discrete soliton systems , 2002 .

[13]  N. Ibragimov.  A new conservation theorem , 2007 .

[14]  Franco Magri,et al.  A Simple model of the integrable Hamiltonian equation , 1978 .

[15]  Xianguo Geng,et al.  Backlund Transformations of Soliton Systems from Symmetry Constraints , 2001 .

[16]  K. M. Tamizhmani,et al.  Integrability properties of the differential-difference Kadomtsev–Petviashvili hierarchy and continuum limits , 2012, 1211.3585.

[17]  秦 孟兆,et al.  RELATIONSHIP BETWEEN SYMMETRIES AND CONSERVATION LAWS OF NONLINEAR EVOLUTION EQUATIONS , 1979 .

[18]  G. Bluman,et al.  Direct Construction of Conservation Laws from Field Equations , 1997 .

[19]  A. Fordy.  APPLICATIONS OF LIE GROUPS TO DIFFERENTIAL EQUATIONS (Graduate Texts in Mathematics) , 1987 .

[20]  Wenxiu Ma,et al.  Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations , 1998, solv-int/9809009.

[21]  Athanassios S. Fokas,et al.  Symplectic structures, their B?acklund transformation and hereditary symmetries , 1981 .

[22]  T. Koikawa,et al.  Canonical structure of soliton equations. I , 1982 .

[23]  Wenxiu Ma.  Lie Algebra Structures Associated with Zero Curvature Equations and Generalized Zero Curvature Equations , 2013 .

[24]  Nail H. Ibragimov,et al.  Nonlinear self-adjointness and conservation laws , 2011, 1107.4877.

[25]  John Stillwell,et al.  Symmetry , 2000, Am. Math. Mon..

[26]  J. Meinhardt,et al.  Symmetries and differential equations , 1981 .

[27]  Wen-Xiu Ma,et al.  An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems , 1994 .

[28]  TRANSFORMATION GROUPS APPLIED TO MATHEMATICAL PHYSICS (Mathematics and Its Applications (Soviet Series)) , 1986 .

[29]  Ruguang Zhou,et al.  Adjoint Symmetry Constraints Leading to Binary Nonlinearization , 2002 .