PHB says that Wizards should use 'fine inks' for 50 gp for every levels of the spell he want to copy into his spellbook.

The problem is that there are no any 'fine inks' in the PHB goods list, just common 'inks'.

The question is: should a Wizard buy a lot of bottles of these common inks per 50 gp each? Or could he just use any inks (such as from the scholar pack) and drop 50 gp in the forest?

What should a Wizard sitting with common inks and a lot of gold in his bag the middle of the forest do to copy a spell from one spellbook to another? Is it possible?

This is an Abstraction your DM is expected to handle

"Fine Inks" is not a proper noun in 5th Edition D&D. You don't see a statblock for an item "Fine Inks" because it's just a colloquial term: "Inks that are of relatively high quality [hence why they cost 50gp, which eclipses the annual salary of a regular peasant by an order of magnitude]".

So if you manage to find a spell scroll or some other printed spell that you intend to copy into your spellbook, you need only say to your DM "I want to copy this spell into my spellbook", they'll say "alright, go to the market and buy the ink/components you need, do you have enough gold pieces?", and depending on what stage of the campaign you're in, the DM might try to throw in a plot hook during this process, or kidnap your Cleric, or some other normal D&D campaign things. Same as mostly anything else that happens when you're not physically navigating your way through a dungeon.

I could perhaps see some DMs get stingy on this process depending on where you're at (maybe the town you're in is especially roughshod and unlikely to have access to the quality of ink you need?) but other than that, there's really not much else to it.

Also, a reminder: the 50gp worth of materials is not solely the inks:

For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.

—Spellbook, Player's handbook, pg. 114

So while there's definitely some amount of ink involved in copying the spell, the costs are also implied to be various other material components you might need to properly master the spell/verify its behavior. So in the same breath as above, your DM would probably ask you to visit a Reagent shop to pick up the various components you need.

The wizard just needs the gold pieces

The book states (emphasis mine)

For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.

A wizard doesn't actually need to have any components or inks on hand to copy a spell .... other than the spellbook(s); the gold is a standin.

As myself and other have said many times: D&D is a poor reality simulator and its better off not to view it under too fine a lens, lest the frayed (and fantastic) threads beneath be visible.

Personally, I've always imagined that the wizard somehow melts the gold into an ink itself, but that's purely my own interpretation.

It depends on how much detail and bookkeeping you want

The cost of copying spells is specified only in value and what they represent. The relevant excerpt from the Player's Handbook (p. 114):

For each level of the spell, the process takes 2 hours and
costs 50 gp. The cost represents material components you
expend as you experiment with the spell to master it, as well
as the fine inks you need to record it.

The simple solution is to say your gold is simply spent upon you copying the spell. One could say that the Wizard is assumed to have bought the relevant ingredients and inks when last they had opportunity. This in most cases gives the smoothest game-play as minimal time has to be dedicated to bookkeeping.

If your table prefers more realism (or accounting) you will need to buy it. At this point DM fiat will come in heavily, but saying you buy X gp worth of 'spell-learning-components and inks' that you can use as you find spells is not unreasonable.

If you want an awful lot of bookkeeping you could be required to track down all of the different stuff and possibly mix the ink yourself, however such an approach is gonna have your DM determining what exactly all of that stuff is and most of us aren't going to do that. If they suddenly do, it's probably being used as a plot hook...

You need to buy special inks

The equipment list and table in the PHB are not exhaustive. If you cannot find something there, it does not mean that it does not exist or that it cannot be purchased.

The fine inks in the wizard feature are a special item necessary for copying spells. You have to purchase them somewhere. If you do not have it (or the other necessary components) on hand, you cannot copy spells into your book.