Coverage
The universe of the site is the set of autonomous polynomial equations E(w, w′, …) = 0 of order K ≤ 6 and degree at most 5 that occur in physics, mechanics, chemistry or biology, mostly as travelling-wave or stationary reductions of evolution equations. Coverage is measured on the catalogue, which was harvested from the literature; it is not a statement about all equations of a given order and degree.
A cell of the grid counts the catalogue equations of that order and degree and shows the level of the answer for each of them. The levels, and the two ways an entry can be without a report, are defined on the method page; the legend below uses the same names.
Orders 1–6, degrees 2–5
Which equations the literature treats
The counting in full
By the order of the travelling-wave equation
The catalogue as a whole
By field
By weight
Weight 5: treated in hundreds of papers on exact solutions; weight 1: rarely.
What the theorems say, type by type
The grid above counts equations of the catalogue. This part is about all equations of a given shape. Take an equation solved for the top derivative, w^(K) = Φ, with Φ a polynomial. At a pole of order p a monomial of degree n with J derivatives has weight np + J, and w^(K) has weight p + K. The equation is of type (K, p) if every monomial of Φ has weight at most p + K and some nonlinear monomial has weight exactly p + K. Those monomials form the core; all other monomials are lower terms. Sub-types by the shape of the core: a pure power wⁿ; J1, the monomial wⁿ⁻¹w′ alone; a slice, monomials of one degree with at least two derivatives (for example w w″ and w′²); mixed, monomials of two or more degrees.
Three levels of statement are told apart. All: true for every equation of the sub-type (core coefficients nonzero, any lower terms, including none). Generic: what the engine finds for a member with all lower terms present and random rational coefficients; it then holds off a proper subset of the coefficient space. None: no theorem of the list gives the class of the solutions even for the generic member. In the table, “W” stands for “every meromorphic solution is rational, simply periodic or elliptic”, “≤ m” for “at most m poles in a period”, and “no poles” for “no meromorphic solution has a pole”.