What the site does and does not do

The visitor enters an autonomous polynomial ordinary differential equation, or picks a named evolution equation and gets its travelling-wave reduction. For that equation the site reports which solutions are meromorphic in the whole complex plane, as far as this can be decided, and says what each statement rests on.

• Who made this
• The question
• Rational, simply periodic, elliptic
• The number d
• Local analysis
• The three lists
• The verdict
• The labels
• The independent check
• The theorems
• Two versions
• Limits

Who made this

The results on this site were computed and written by AI models: the engine, the proofs labelled agent-proof, the catalogue and these pages. A statement labelled agent-proof has been read by a second AI model and has not been checked by a person. Statements labelled published or preprint are quoted from the works cited with them. A certificate is an exact computation that anyone can rerun, for instance on the page Check a solution.

The question

The unknown is a function w(z) of a complex variable. The equation is

E(w, w′, …, w^(K)) = 0,

where E is a polynomial in w and its derivatives whose coefficients are rational numbers or parameters. The variable z does not occur: the equation is autonomous. The order is K; the degree is the total degree of E in w, w′, …. The equation is solved for the highest derivative if E = c w^(K) − Φ with a nonzero constant c.

A solution is meromorphic in ℂ if it is analytic in the whole plane except for poles. Entire functions (no poles at all) and constants are included. Solutions with branch points or essential singularities at finite points, and solutions that exist only on a part of the plane, are not the subject of the site.

Every verdict is therefore a statement about solutions that are meromorphic in the whole complex plane. Travelling waves with other singularities in the complex plane — for example fronts that exist for a range of speeds, peakons, compactons — are outside its scope: “complete” means that all meromorphic solutions are listed, not that all travelling waves are.

For an evolution equation in u(x,t) the travelling-wave reduction u = w(z), z = x − ct is applied first and, where possible, integrated; the constants of integration become parameters. The result page shows the reduction that was applied.

Rational, simply periodic, elliptic

Three kinds of meromorphic functions have closed forms that the site can find and check:

• rational functions of z;
• simply periodic functions that are rational functions of one exponential s = exp(νz); hyperbolic and trigonometric tangents are of this kind;
• elliptic functions, that is, doubly periodic meromorphic functions. They are written with the Weierstrass function ℘(z; g₂, g₃) and its derivative, which satisfy ℘′² = 4℘³ − g₂℘ − g₃.

In the literature the union of these kinds, or its transcendental part, is often called the class W. The site says “rational, simply periodic or elliptic”. In formulas that can be copied as text, wp(z, g2, g3) stands for ℘ and wpp(z, g2, g3) for ℘′.

The number d

For a solution w, the number d is the transcendence degree over ℂ of the field generated by w and all its derivatives: the largest number of algebraically independent functions among w, w′, w″, …

• d = 0: constants.
• d = 1: nonconstant rational functions, rational functions of one exponential, elliptic functions.
• d = 2: for example exp(2z) ℘(exp z).
• d ≥ 3: for example Painlevé transcendents in suitable variables.

The site treats d = 1 completely where a theorem allows it, reports solutions with d = 2 where they are known, and for d ≥ 3 says only “ruled out” (with the theorem) or “not ruled out”.

Local analysis

At a pole of order p at z₀ a solution has a Laurent series

w = Σ c_k (z − z₀)^(k−p)

The weight of a monomial ∏ (w^(j))^e_j at pole order p is Σ e_j (p + j): the order of its pole. The monomials of largest weight form the dominant part; a pole is possible only where at least two of them balance, and this fixes p and a₀.

The Fuchs indices are the roots k of the indicial polynomial of the dominant part at (p, a₀); −1 is always one of them. At a positive integer Fuchs index the recurrence for the coefficients gives a compatibility condition; if it holds, the coefficient c_k is free. The equation has the finiteness property if only finitely many Laurent series with a pole at a given point satisfy it.

The result page gives this table for every equation (section “Local analysis”). By itself it decides nothing about global solutions; it is the input of the theorems and of the searches.

The three lists

Found. Solutions meromorphic in the whole plane, in closed form. Each one carries an exact check: the formula is substituted into the equation and the result is reduced to zero by exact computation (for elliptic solutions modulo ℘′² = 4℘³ − g₂℘ − g₃). Anyone can rerun the check on the page Check a solution. A solution may hold only under conditions on the parameters; the conditions are listed with it. A family marked special parameter values exists only where the parameters satisfy an equality; in a long list these families are collected in one part below the families for generic values. A family marked direct search was found by a search on parameter values that the decision procedure did not analyse; the list on those values is not claimed to be complete. A formula whose check did not run is kept apart under “Candidates (not verified)” and is not counted.

Ruled out. Kinds of meromorphic solutions that cannot exist. Each item names its basis: a theorem of the list below, together with a record of how every hypothesis was verified for this equation, or an exact computation that can be rerun.

Not decided. What remains open for this equation, with the partial information that exists. A partial answer is shown next to what is not decided; it is not suppressed and it is not presented as more than it is.

The sentence “these are all meromorphic solutions” appears only when a theorem of the list below gives it. With symbolic parameters the answer describes generic values and lists the parameter conditions that change it, as far as the engine can tell.

The verdict

Every report starts with a level. Where the parameters of an equation matter, the level takes the special parameter loci into account: “complete for generic parameters” says that the list is complete off those loci, and the report states the weakest level reached on them.

On the pages Catalogue and Coverage an entry without a report is counted as “no report yet”, or as “time limit reached” when its analysis was stopped at the time limit.

The labels

Trust is expressed by six labels and by nothing else. Each item of a report carries one of them; the badge shows its meaning when it is touched or pointed at. A verdict carries the weakest label among the statements it needs (agent-proof is weaker than preprint, preprint weaker than published, published weaker than certificate), and each of those statements is shown with its own label.

Everything that is reported is computed in exact arithmetic: rational numbers, algebraic numbers, symbols. Floating-point numbers are used to find candidates and to draw pictures, never to certify.

A found solution can carry a second mark, “independent check”. The certificate of the engine is the exact computation: the formula is substituted and the result is reduced to zero. The independent check is a redundancy test by different code outside the engine, which substitutes the same formula again, exactly where it can and otherwise numerically to 30 digits at random points of the parameter locus; it guards against an error of the program and is not a second proof. Its totals over all reports are on the page Checks.

Two versions

The same pages exist in two versions. The light version is a static site: the reports of the catalogue are computed in advance, and an equation that is typed in is analysed in the browser by the engine (Python and SymPy, run by Pyodide; the first use downloads 10–15 MB). The heavy version serves the same pages from a local server that runs the full engine with longer time limits and more algebra software. Every report states the mode that produced it; a light report can contain fewer items than a heavy one for the same equation, and then it says so in “Not decided”.

Limits

• Only autonomous polynomial equations in one unknown function are read. An equation that becomes polynomial after a standard substitution is listed in the catalogue with that substitution.
• The catalogue covers orders up to six and degrees up to five, with a few named equations of order seven or eight. Universal families of orders five and six with all coefficients symbolic are outside the scope.
• The question is about solutions meromorphic in the whole plane. A solution that is real and smooth on the real axis but has other singularities than poles in the complex plane is not covered, and the site says nothing about it.
• “Not decided” means: not decided by this site for this equation.
• A result labelled search states its bound and claims nothing beyond it.
• A statement labelled agent-proof has not been checked by a person. A statement labelled preprint comes from a text that has not been refereed.
• The engine in the browser works under a time limit; when the limit is reached the report keeps what was obtained and lists the rest as not decided.
• The catalogue is a sample of the literature, not a bibliography. Where a solution or an equation occurs in a paper, the paper is cited, and nothing more is said about it.
• Plots are drawn with floating-point numbers. They illustrate a formula and certify nothing.