Frobenius integrable decompositions for ninth-order partial differential equations of specific polynomial type

Keywords: Frobenius integrable decompositions Partial differential equations Differential polynomials Bäcklund transformations Ordinary differential equations Nonlinearity a b s t r a c t Frobenius integrable decompositions are presented for a kind of ninth-order partial differential equations of specific polynomial type. Two classes of such partial differential equations possessing Frobenius integrable decompositions are connected with two rational Bäcklund transformations of dependent variables. The presented partial differential equations are of constant coefficients, and the corresponding Frobenius integrable ordinary differential equations possess higher-order nonlinearity. The proposed method can be also easily extended to the study of partial differential equations with variable coefficients. While exploring soliton phenomena, one found a large number of partial differential equations (PDEs) exhibiting soliton phenomena in many science subjects such as fluid physics, solid physics, elementary particle physics, biological physics, superconductor physics, etc. It is a quite fascinating research topic how to solve such PDEs, particularly obtain interesting solutions including solitons, and the topic also attracts much attention of mathematicians, physicists and dynamicists. Over the past several decades, there have been numerous efficient methods for constructing exact solutions to nonlinear PDEs, one after the other [1–19]. Though the existing solution methods are diverse and different, appropriate reductions (e.g., similarity reductions, symmetry constraints, travelling wave reductions) are often employed to reduce given PDEs to simpler PDEs (normally linear) and/or integrable ordinary differential equations (ODEs). By employing linear differential equations, a novel kind of exact solutions – complexitons – was presented successfully, indeed [3,4,6,7]. Recently, Ma et al. [20] presented Frobenius integrable decompositions (FIDs) for two classes of nonlinear evolution equations (NEEs) with logarithmic derivative Bäcklund transformations in soliton theory. The discussed NEEs are transformed into systems of Frobenius integrable ODEs with cubic nonlinearity. You et al. [21] obtained two classes of PDEs with variable coefficients possessing FIDs, including the KdV and the potential KdV equation, the Boussinesq equation, and the generalized BBM equation.