Behaviour of the extended Volterra lattice

Non-isospectral extension of the Volterra lattice hierarchy, and Hankel determinants

“…is integrable, in the sense of possessing a Lax pair, for any choice of functions α k , β k of t [4,30], also see Remark 2.1 below. The first non-autonomous equation of the extended hierarchy, du n dτ 1 = u n [(n + 1) u n+1 + u n − (n − 2) u n−1 ] , (1.6) already appeared in [11,26,41]. Quite surprisingly, this equation is deeply related to the problem of computing the coefficients in a continued fraction expansion of a given function, Thiele's expansion formula.…”

Behaviour of the extended Toda lattice

“…Transformations of equations of the form (2.18) to the KdV equation have been given many times previously, for example, in [5,8,9].…”

“…At leading order, and using (4.3), both components of equation (4.1) become 5) and φ remains undetermined. From (4.5) 6) and integrating with respect to z yields…”

“…Both (4.18) and (4.19) can be mapped onto the standard KdV equation using transformations such as those described by Popovych and Vaneeva [8] and Pickering et al [5].…”

“…The autonomous versions of such equations were placed within a suitable modification of the usual algebraic structure associated with completely integrable evolution equations in [4]. The first member of the (1 + 1)-dimensional differential-delay Volterra lattice hierarchy was studied in [5], where we considered various amplitudes for parameters, and obtained a number of asymptotic reductions to generalizations of the Korteweg-de Vries (KdV) equation, amongst others. In the present paper, for the first member of the extended Toda lattice hierarchy, again by considering various magnitudes for the parameters involved, we derive reduced equations related to the KdV and potential Boussinesq equations.…”