A formula for symmetry recursion operators from non-variational symmetries of partial differential equations

Abstract

An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a general formula that produces a pre-symplectic operator from a non-gradient adjoint-symmetry. These formulas are illustrated by several examples of linear PDEs and integrable nonlinear PDEs. Additionally, a classification of quasilinear second-order PDEs admitting a multiplicative symmetry recursion operator through the first formula is presented.

Appendix: index notation

We adapt the index notation used in Ref. [2].

Partial derivatives of u with respect to $x^i$ are denoted $u_i = \partial _{x^i} u$ and $u_{i\ldots j} = \partial _{x^i}\ldots \partial _{x^j}u$. Summation over a repeated single index i is assumed unless otherwise noted.

Higher derivatives are denoted by multi-indices which are defined by $u_J:= u_{j_1\ldots j_m}$, where $J=\{j_1,\ldots ,j_m\}$ is an unordered set that represents the differentiation indices and $|J|= m$ is the differential order. Summation over a repeated multi-index will be denoted by $\sum$.

If $J=\{j_1,\ldots ,j_m\}$ and $K=\{k_1,\ldots ,k_{m'}\}$ are multi-indices, then their union is defined by $J,K := \{j_1,\ldots ,j_m,k_1,\ldots ,k_{m'}\}$ and $|J,K| := m+m'$. If $K\subset J$, then J/K denotes set of indices remaining in J after all indices in K are removed. Note the differential order is $|J/K| = |J|-|K|$.

In this notation, total derivatives with respect to $x^i$ are given by $D_J := D_{j_1}\cdots D_{j_m}$ where $D_j = D_{x^j} = \partial _{x^j} + \sum _{K} u_{j,K}\partial _{u_K}$.

For expressing the product rule $D_J(fg)$ in a short notation, it is useful to introduce the binomial coefficient ${\textstyle \left( {\begin{array}{c}J\\ K\end{array}}\right) } := \frac{(\# J)!}{(\# K)!(\# J/K)!}$. Here, $\# J\!:= (\#1,\ldots ,\# n)$ where $\# i$ denotes the multiplicity (number of occurrences) of each integer $i=1,\ldots ,n$ in the set J; and $(\# J)! := \prod _{1\le i\le n} (\#i)!$ where ! is the standard factorial. A useful identity is ${\textstyle \left( {\begin{array}{c}J\\ K\end{array}}\right) } = {\textstyle \left( {\begin{array}{c}J\\ J/K\end{array}}\right) }$.

Then, $D_J(fg) = \sum _{K\subseteq J}{\textstyle \left( {\begin{array}{c}J\\ K\end{array}}\right) } D_{J/K} f D_K g$ is the product rule.