Recursion Operators for a Class of Integrable Third‐Order Evolution Equations

Abstract

We consider u t = u α u xxx +n(u)u x u xx +m(u)ux +q(u)u x +s(u) with α = 0 and α = 3, for those functional forms of m, n, p, q, r, s for which the equation is integrable in the sense of an infinite number of Lie-Bäcklund symmetries. Local xand t-independent recursion operators that generate these infinite sets of symmetries are obtained for the equations. A combination of potential forms, hodograph transformations and x-generalised hodograph transformations are applied to the obtained equations.

Об Обратимых Преобразованиях Беклунда Для Автономных Эволюционных Уравнений

“…Отметим, что это уравнение представляет собой специальный случай полулинейных эволюционных уравнений, которые допускают рекурсионный интегро-дифференци-альный оператор, изученный в работе [16]. Случай 1.3.…”

A Novel Riccati Sequence

“…Following a classification of linearisable evolution partial differential equations by Petersson et al [14] particular attention was paid to the corresponding ordinary differential equations of two of these evolution partial differential equations [8]. The base equations were the Riccati equation [16] and the Ermakov-Pinney equation [7,15].…”

A multidimensional superposition principle: numerical simulation and analysis of soliton invariant manifolds I

“…The main problem is to find the recursion operator for a given system or to show that an infinite number of Lie-Bäcklund symmetries exists or that it does not exist (the latter being the more demanding task). Since the recursion operator can in general contain nonlocal variables even for equations linearisable by a (nonlocal) coordinate transformation (Petersson et al [22]), the procedure is not an easy one especially for higher order equations and for systems.…”

“…In their paper Petersson et al [22] report the classification of the symmetry integrable class of equations u t = u α u xxx + n(u)u x u xx + m(u)u The equation…”

“…(λ's are arbitrary constants) is a member of the class (1.1) [22]. Equation (1.2) is of interest to us in the context of ordinary differential equations due to its reduction to an Ermakov-Pinney equation.…”