Are there any fully analytically solvable nonlinear oscillators? I'm trying to find a simple one-dimensional problem, in which a particle would oscillate with some energy, and the period of oscillation would depend on particle energy (unlike in harmonic oscillator). I could take any potential, different from parabolic, and be done, but all such potentials appear to not lead to explicit closed form solutions.
I've tried various $x^\alpha$ potentials, tried radial Kepler effective potential $K/r^2-Q/r$, some others, but all they appear to give $t(x)$ instead of explicit $x(t)$ as closed forms.
Another way might be to use potential $\propto|x|$, and consider it piecewise, but the solution quickly becomes quite ugly and requires using such functions as $\min$, $\max$ etc., cluttering the final expression a lot.
So my question is: are there any shapes of potential, for which equation of motion would be solvable in simple explicit closed form $x(t)$, and at the same time period would depend on energy?
 A: Maybe the pendulum would be an interesting example. The equation of motion of an ideal pendulum of length $\ell$ in a uniform gravity field is
$${d^2\theta\over dt^2}+{g\over \ell} \sin\theta=0$$
It is well known that in the small angle approximation this reduces to a harmonic oscillator, but the precise solution involves elliptic integrals, and can be expressed in the inverses of such integrals, which are called elliptic functions. This class of functions was an object of intense study in the 19th century, and several special forms got specific names. In particular, the pendulum can be solved in terms of a Jacobi elliptic function.
A: The answer to this question depends an awful lot on what you mean by "closed form" solutions.
Consider the standard harmonic oscillator where the solution is
$$x(t) = A \cos(\omega t + \phi) \, .$$
In what sense, if any, is this a "closed form"?
That $\cos$ function is defined in terms of a power series
$$\cos(x) \equiv \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} \, .$$
Lots of differential equations can be solved with power series methods, and in cases where the solved equation is important those power series solutions tend to get Fancy Names$^{\text{TM}}$ like "Bessel functions", "Jocobi elliptic functions" (see doetoe's answer), etc.
Those other special functions aren't really any less elementary than $\sin$ and $\cos$: they have recurrence relations, completeness/orthogonality relations, and reasonably simple transformation under derivatives, etc.
You can also always use harmonic balance to write the (homogeneous) solution of an oscillator in terms of a Fourier series.
Aside:
The reason $\sin$ and $\cos$ are preferred is that they form a closed basis under time translation:
$$\cos(t + \delta t) = A \cos(t) - B \sin(t)$$
where $A = \cos(\delta t)$ and $B = \sin(\delta t)$.
This makes them nice in systems where the equations of motion themselves are time translation invariant because all time derivatives turn $\cos$ and $\sin$ into each other.
