A mathematical preprint outlines a conditional route for deciding when a class of nonlinear partial differential equations (PDEs) in one-dimensional space-time can have solutions expressed in elementary functions of the spatial coordinate. Its proposed test, called a nested partition, groups multi-index terms with matching coefficients and monomials so the spatial operator can be written as an algebraic combination of nested derivatives.
This is a formal symbolic analysis, not an empirical study: it works with functions, differential operators and constructed equations rather than a sampled population or dataset. The paper states that its nested-derivative expansion is established by induction.
From structure to candidate solutions
The proposed check begins with the operator's structure. If the required nested partition can be found, the paper presents solving an appropriate nested equation as a conditional route to elementary solutions of the spatial operator equation. It notes that the correspondence equations needed for the grouping are not always straightforward to solve.
For homogeneous nested-derivative equations, candidate solutions are built from resolving functions determined recursively. In the nonlinear homogeneous case, the construction first sets P(u)=V and solves the associated linear equation, then recovers u from an algebraic equation.
The paper also gives a sufficient integration-by-parts test for elementary antiderivatives: when suitable elementary auxiliary functions satisfy the stated derivative conditions, the target integral is elementary. But the existence of those auxiliary functions is not automatic.
Forcing terms add another condition
For nonhomogeneous equations with a forcing term, the stated particular-solution construction uses recursively defined functions and antiderivatives, including a function G whose spatial derivative is the forcing term g. The paper gives analogous criteria for deciding whether the resulting particular solutions are elementary, while assuming the needed auxiliary functions exist.
Examples remain illustrative
The paper includes an invariant-space route for the case in which the temporal operator is linear. In the single-basis case, a solution can be treated as an eigenfunction of the spatial operator. One example reports a possible solution family whose time factor is the one-parameter Mittag-Leffler function.
A separate constructed example treats an associated cubic polynomial as solvable by radicals and uses its branches to generate possible solutions. An appendix defines a real root-of-unity generalization and identifies ordinary cosine as a special case.
Those examples illustrate constructions, not a complete catalogue of solutions. The supplied analysis does not establish completeness or uniqueness for the displayed families, and it does not systematically examine global domains, singularities, branch choices, or boundary and initial conditions.
The result remains conditional
The framework's scope is conditional. It applies only when an operator has the required nested partition and when the relevant elementary-function or radical conditions hold; it does not show that every nonlinear PDE in the stated class has an elementary solution or that every spatial operator admits such a partition. Nor does it establish an effective algorithm for arbitrary operators or provide numerical, empirical or physical-system validation.
Further checking is needed for the recursive coefficients, the nested-partition criterion and the worked examples. Open questions also include whether the method extends beyond polynomial nonlinearities and algebraic equations solvable by radicals.
The document is an arXiv preprint, version 2 dated 21 Aug 2026. Its supplied end matter contains no funding statement.
Paper data and sources
Original title: The nested derivative criterion for determining elementary solutions to certain non-linear PDEs in one-dimensional space-time
Authors: Franceso Maltese
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text