A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

Abstract

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear PDE to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1 + 1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.

New exact solutions of a generalized shallow water wave equation

“…where the generating functional F (z(x, t)) satisfies the constraint (5). Note that the inclusion of negative powers of F , in general, creates pole of u ′ triggering the presence of singular solutions in some situations.…”

A Generalization of the Wadati-Konno-Ichikawa Soliton Hierarchy and its Liouville Integrability

“…This implies that the operator Φ ¼ Ψ y , defined by (3.7) and (3.8), is a common hereditary recursion operator for the whole generalized soliton hierarchy (2.17). We remark that various recursion operators can be found through Lax representations or by computer algebra systems for partial differential equations (see, e.g., [26,27]), and that there exist direct computer algorithms for constructing symmetries of differential and/or differential-difference equations (see, e.g., [28]). …”

Construction of a hierarchy of negative‐order integrable Burgers equations of higher orders

“…The recursion operator Φ( u ) for the Burgers equation is defined as where ∂ x denotes the total derivative with respect to x , and is its integration operator. The recursion operator is the theme to generate integrable hierarchies and higher order symmetries for integrable equations.…”

“…It is well known that G i ( x , t , u x , u x x ,⋯), i ≥ 1 is a symmetry if and only if it leaves the partial differential equation (PDE) invariant for the replacement u → u + ϵ G i within order ϵ . The generalized symmetries for the Burgers equation are given as …”

“…A hereditary symmetry Φ( u ) is an operator‐valued function of u having the property that it generates a hierarchy of evolution equations for which Φ( u ) is a recursion operator . The hereditary symmetry Φ( u ) is a recursion operator of the following hierarchy of evolution equations …”