PHP has some persistent version of well-known functions such as mysql_pconnect (deprecated) or pfsockopen
From what I've read, I understand that such functions cache the resource produced by the first call and reused when needed.
Long story short , how does all the "p" functions really works in background ? The official doc about all this is close to nil
The resource in question isn't really cached, not in the true meaning of cache that is. The connections are just kept open until the script terminates or until you call the appropriate function that closes it (so don't call e.g. mysql_close() if you want to use a persistent connection).
Now, the real question is - what does "until the script is terminated" mean. This may vary or may not be possible, depending on the PHP SAPI that's currently used, but let's take Apache with support for HTTP/1.1 + mod_php as an example - it's the most popular and easiest to explain ...
HTTP/1.1 is required, because it has Keep-Alive - something rather strange for a stateless protocol. When you access a page, Apache runs your PHP script under a thread or a new system process.
And here's the catch - your script does execute in it's entirety, but Apache does NOT terminate the thread or process in question, allowing it to maintain persistency for resources that are left open. How exactly it's done I'm not sure I can explain, but with Keep-Alive your next HTTP request will be attached by Apache to the same thread or process id, allowing possibility to reuse that resource.
The second part is more of just a basic sanity check - you can only attach to a persistent connection if you're using the same "options" and/or credentials that you've used to create it. So if you call mysql_connect() with a different username - it will create a new connection and won't reuse the old one (it's signature doesn't match). But given that supposedly the same script is re-executed, this is only a "fomality" in most cases.
I hope that makes sense to you, it took me a few hours of experimenting in order to fully understand how it works. You can read more about it on php.net - the article is named "Persistent Database Connections", but that's only because originally persistent connections were only available for databases - the described behavior is true for all other equivalents.
Edit:
Forgot to answer your more specific questions:
But for exemple with pfsockopen what happen to the socket when it's cached ?
Nothing - it's just kept alive, as already explained above.
Is it still open and connected ?
Yes
If so , what happen to the potential data received during the "sleep" time ?
It waits for you to read it.
What is the TTL of such cached connection ?
That in particular I can't answer - it may depend on the options that you pass to the p* function, your php.ini settings and/or what you're connecting to. It may also be a hard-coded value and to be different for all of them.