An analog of the lattice KdV equation of Nijhoff et al. is constructed on a
hexagonal lattice. The resulting system of difference equations exhibits
soliton solutions with interesting local structure: there is a nontrivial phase
shift on moving between adjacent lattice sites, with the magnitude of the shift
tending to zero in the continuum limit.Comment: LaTeX, 9 pages, 2 figures, see warning at top of fil

Schiff, Jeremy

English

arXiv

An analog of the lattice KdV equation of Nijhoff et al. is constructed on a hexagonal lattice. The resulting system of difference equations exhibits soliton solutions with interesting local structure: there is a nontrivial phase shift on moving between adjacent lattice sites, with the magnitude of the shift tending to zero in the continuum limit. Bobenko, Hoffmann and Suris [1] have recently proposed that some effort should be made to understand integrable systems of difference equations on lattices other than Z 2. In addition to being of theoretical interest, it is possible that in certain cases this could have physical significance [2, 3]. This letter examines the analog of Nijhoff et al.’s lattice KdV equation [4, 5] on a hexagonal lattice, which we call “hexaKdV ” for short. The soliton solutions of hexaKdV are constructed. HexaKdV solitons undergo a nontrivial phase shift on moving between adjacent lattice sites, with the magnitude of the shift tending to zero in the continuum limit. Nijhoff et al.’s lattice KdV equation involves a single field b defined at the vertices of the standard lattice Z2. The equation gives a relation between the values of the field at the 4 vertices of each fundamental lattice plaquette. Writing b1 for bn,m, b2 for bn+1,m, b3 for bn+1,m+1 and b4 for bn,m+1 (so b1,b2,b3,b4 are values of the field as we go around a fundamental plaquette), the equation takes the form [5] b1 − b2 − b3 + b4 h −b2 + b3 + b4 − b

Jeremy Schiff

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HexaKdV

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.5216

HexaKdV

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