Recursion operator for the fully-nonlinear fifth-order equation u_t = 1 / u_{5x}^{2/3}

Determine a recursion operator for the fully-nonlinear fifth-order evolution partial differential equation u_t = 1 / u_{5x}^{2/3}. Prior analysis indicates that no standard local recursion operator of order six or less exists for this equation, and the operator is expected to be nonlocal.

Background

The paper studies symmetry-integrable, fully-nonlinear evolution equations and maps them to known semilinear integrable equations via multipotentialisations and (generalised) hodograph transformations. While the third-order equations admit standard second-order recursion operators, the authors examine a particular fifth-order fully-nonlinear equation u_t = 1 / u_{5x}^{2/3}.

In Remark 2, the authors show that this fifth-order equation does not admit a recursion operator of a commonly used local form up to sixth order, suggesting that any recursion operator for this equation must be nonlocal. Despite mapping the equation to a semilinear form with known recursion structures, the recursion operator for the original fully-nonlinear equation remains unknown.

The open problem is explicitly stated in the concluding remarks, emphasizing the need to identify a (likely nonlocal) recursion operator for the fifth-order equation to fully characterize its symmetry-integrable hierarchy.