Abstract
We investigate the behaviour of solutions of the recently proposed extended Volterra lattice. A variety of methods are used to determine the effects of the new terms on small amplitude equations, and, following approximation of the partial differential delay equations by PDEs we also determine similarity reductions.
Highlights
The integrability of the basic or standard Volterra lattice utðn; tÞ 1⁄4 uðn; tÞ1⁄2uðn þ 1; tÞ À uðn À 1; tÞ has been known for some time, and important properties such as Miura maps, modified systems, Hamiltonian structures, Lax pairs, Bäcklund transformations, Hirota bilinear form and N-soliton solutions have been derived [8,12,16,5,20,9,10,6]
In [2,3] an integrable non-isospectral ð2 þ 1Þ-dimensional extension of the Volterra lattice hierarchy was constructed, this consisting of a sequence of equations in uðn; t; yÞ with n being discrete and t and y continuous
In this paper we investigate this extended ð1 þ 1Þ-dimensional Volterra lattice hierarchy, to determine the types of solution it supports, and how these compare with those supported by the basic Volterra lattice
Summary
The integrability of the basic or standard Volterra lattice utðn; tÞ 1⁄4 uðn; tÞ1⁄2uðn þ 1; tÞ À uðn À 1; tÞ has been known for some time, and important properties such as Miura maps, modified systems, Hamiltonian structures, Lax pairs, Bäcklund transformations, Hirota bilinear form and N-soliton solutions have been derived [8,12,16,5,20,9,10,6]. The consideration of reductions to ordinary difference equations allowed the derivation of discrete Painlevé hierarchies Other reductions of this ð2 þ 1Þ-dimensional hierarchy included an extended ð1 þ 1Þ-dimensional Volterra lattice hierarchy, the members of which are evolution equations in uðx; tÞ, both x and t being continuous but with the equations involving derivatives with respect to x as well as shifts in x. Pickering et al / Commun Nonlinear Sci Numer Simulat 19 (2014) 589–600 which approximate the extended Volterra lattice in certain continuum limits [17,18,19] The behaviour of these approximating PDEs is described and discussed. One equation arising in this context is a nonisospectral extended KdV equation, whose solutions merit further investigation These are detailed, where we explain connections to integrable systems.
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