Consider the differential equation
$$P(f '(x)) = Q(f(x))$$
Where $P(x),Q(x)$ are polynomials.
Examples are $f'(x) = 1 + f(x)^2$ where we get a tan solution and $f'(x)^2 = 4 f(x)^3 - g_2 f(x) - g_3$ where we get a Weierstrass elliptic function solution. One of them is periodic , the other double periodic.
In general, When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic ?
I looked at some famous elliptic functions and most of them are defined with 2 or 3 functions like the Dixon elliptic functions with $cm'(x) = - sm^2(x),sm'(x) = cm^2(x)$ what is related to the Fermat curve $x^3 + y^3 = 1$ and the Eisenstein integers. Or the lemniscate elliptic functions with $sl'(x) = (1+ sl^2(x)) cl(x) , cl'(x) = -(1 + cl^2(x)) sl(x)$. However the lemniscate elliptic function $sl$ also satisfies a selfreference one : $(sl'(x))^2 = 1 - sl(x)^4$ thereby satifying the type of differential equation I was looking for.
It is basically just that solving
$$P(x) + Q(y) = 1$$
for $x$ or $y$ results in at least one function that satisfies :
$$P(f '(x)) = Q(f(x))$$
with the right initial conditions.
Usually what I find is that the degrees of $P$ and $Q$ are between $2$ and $4$.
So, what is going on ?
Does every pair of polynomials $P,Q$ with degrees between $2$ and $4$ give double periodic functions ? Are degrees above $4$ possible to get double periodic functions ?
And when do we get periodic functions that are not double periodic ?
Does the Fermat curve
$$x^5 + y^5 = 1 $$
or
$$x^7 + y^7 = 1$$
and their related differential equations give us double periodic functions ?
I want to point out that a function of a periodic function is also periodic and the same applies to the double periodic case.
Also we get the trivial case for a polynomial $M(x)$ :
$$ M(P(f'(x))) = M(Q(f(x)))$$
which has as its solutions the same function $f$ as if $M$ was the identity function.
What basicly is an answer to my question of bounded degree, but I am looking for more insightful and general results ofcourse.
Some ideas I had were plugging in a fourier series with variable coefficients. But I was dealing with infinitely many variables and not sure if my fourier series was even valid ; was it still analytic and did it still agree with the function it was describing ?? Another ideas was an analogue for fourier series, a series expansion for double periodic functions. But I got stuck there too. Not sure if that was going in the right direction or not. Even if that works, I want to prove it does.
I tried some (complex analysis and geometric function theory ) theorems but they had problems with the poles and analytic continuation around those. The taylor radius was too small and fourier requires $L^2$ spaces anyways. I might be able to solve a specific case but I want the general idea.
I am ofcourse slightly aware of some basic results such as relating the period(if it exists) with some coefficients ( such as $g_2,g_3$ in the Weierstrass case ) and rewriting the equations as an integral. Or writing functions in terms of eachother. Or some infinite sums. But that does not give me the insight I seek.
I am not an expert at the addition formula's but I also understand that an addition formula implies periodic or double periodic. But again that does not give me what I seek.
How to look at this ?