The complete symmetry group of the generalised hyperladder problem

Abstract

We further consider the n-dimensional ladder system, that is the homogeneous quadratic system of first-order differential equations of the formẋ i = x i n j =1 a ij x j , i = 1, n, where (a ij ) = (i + 1 − j), i, j = 1, n introduced by Imai and Hirata (nlin.SI/0212007). We establish the most general system of first-order ordinary differential equations invariant under the algebra which characterises the ladder system of Imai and Hirata and the algebra of minimal dimension required to specify completely this most general system. We provide the complete symmetry group of the generalised hyperladder system and discuss its integrability.  2004 Elsevier Inc. All rights reserved. Ladder systemsIn two recent papers Imai and Hirata developed a necessary condition for the existence of Lie point symmetries in n-dimensional systems of first-order ordinary differential equations [5] and applied the ideas developed there to establish a new integrable family in the class of Lotka-Volterra systems [6]. Of the infinite number of Lie symmetries that such a

Symmetry-based solution of a model for a combination of a risky investment and a riskless investment

“…Hence we have that the symmetry consistent with both (4.1) and (4.2) is the one-parameter generator Λ a = x∂ x + γ w∂ w , (4.8) where we drop the single parameter, a 2 , as being inessential since it is a common multiplier. 3 We have a single Lie point symmetry, (4.8), consistent with both the partial differential equation (4.1) and the associated conditions (4.2). We determine the invariants for the reduction from a 2 + 1 equation to a 1 + 1 equation from the associated Lagrange's system dt…”

“…Some further applications are found in [4,29,35] and [36]. Recently Myeni and Leach [26,27] extended recent investigations of Complete Symmetry Groups in the area of ordinary differential equations [1][2][3]32] to some of the evolution equations of the type found in Financial Mathematics. It would be a fair comment to remark that the symmetry analysis of the equations obtained in the mathematical modelling of various financial instruments has only just begun.…”

Symmetry and singularity properties of the generalised Kummer–Schwarz and related equations

“…3 The concept was introduced by Krause in 1994 [14,15], its theory developed by Andriopoulos et al [2,3] and applied in a number of instances [4,17,18,26,27]. One recalls that more than one representation for a given equation is possible and that, even with the proviso of minimality of dimension, the group is not unique.…”

“…One recalls that more than one representation for a given equation is possible and that, even with the proviso of minimality of dimension, the group is not unique. 4 Given a set of symmetries as a candidate for a representation of a Complete Symmetry Group one proceeds as follows.…”

“…For Σ 1 , Σ 2 and Σ 3 in turn the equations satisfied by η are 3 aeq systems to which that which follows is applicable mutatis mutandis. 4 Here we treat the Complete Symmetry Group of ordinary differential equations, but the concept applies also to partial differential equations [24,25]. 5 Here we confine our attention to nonlocal symmetries.…”