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.

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. Whether to allow this is entirely in the hands of your DM.

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. This gives minimal bookkeeping while still maintaining some realism with regards to equipment and expenses.

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...

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 All Depends:

Remember that unless you are playing in a tournament these sorts of rules are very much a guideline. It also just doesn't make sense that all spells would cost 50gp to transcribe - perhaps on average. What about very simple cantrips, or very high-level spells with exotic components? I imagine a Unicorn's horn, or dragon scale is worth a heck of a lot more than 50gp - that might get you a very small portion of powdered unicorn horn.

In 'real-world' ritual magic the components for the spell are often mixed into the ink solution used to transcribe the rite.

For example some spells require odd or somewhat grotesque mixtures - perhaps the practitioner's or victim's blood or other bodily fluids, and perhaps significant elements such as powdered sulfur, or essential oils of some sort or other.

These are usually mixed together (the fundamental part of the rite) with or as the ink and then used to write the incantation to whatever conveys its power - written to a scroll to be burned, or painted on a door as a ward, etc.

The components for a spell can be quite varied; so either use the indicated items from the D&D spell, or if desired the DM can define some variants of standard items - fine herbs, gems, etc. which are listed.

The process of transcribing spells to a spell-book in D&D could conceivably follow this pattern - ritually preparing and casting the spell against the medium (the ink-potion) as a narrative action and then writing it to the spell-book as the immediate action.

The 'physics' of Magic are kind of loosely defined so that DMs can fit it into their campaign in fun effective way without over burdening players with minutia at a campaign or story scale. Where does your world's magic come from? Is it ubiquitous, or does everyone have some latent talent?

Implement as the DM Pleases, It's Never Going to Make Much Sense

The Gold is the Important Part; the Materials are Abstract

The key line is that "the process takes 2 hours and costs 50 gp." It then goes on to what seems to be a much fluffier section about what this cost "represents". While arguments based on what is not said are always problematic, if actually acquiring the materials in the normal game sense was considered important it really seems that the more natural way of phrasing this rule would be something along the lines of "the process takes to hours and expends 50 gp worth of materials. These materials are...". As written, the player actually having particular barely explained inks and unnamed materials, seems to be a low priority at most.

If you want to make actual tangible materials and inks a strict necessity, that's great; it makes more sense (it's what I do). But this is hardly required or necessary, and really no different than requiring a player to come up with a background or event that supports them being able to suddenly multiclass into sorcerer, because their abruptly acquired sorcererness represents some sort of ancestral magic. This might be a logic/lore necessity, but not an actual intended mechanical constraint.

Implementation is Heavily at the DM's Discretion, as is the Whole Ability

Giving the DM particularly broad powers in this instance to construe implementation of the rule at his or her discretion is consistent with the use of the term "represents" rather than any clear language about it "expending X things" as there is for when a spell gobbles up a diamond or what have you. Saying "the wizard must first acquire X things" would also be a natural way to emphasize that yes indeed you NEED to go shopping. It does not read in such a way.

But DM discretion is also in the spirit of the spell copying ability in general, given that how much this ability can be used is entirely dependant on how plentiful the DM makes magic scrolls and spellbooks. Choosing to require actual purchase of these things is just another tool in the DM toolbox so far as doling out magic to the party's wizard goes.

As a matter of personal preference I'd very much rather the gold have to be spent on something rather than just disappear into the aether, but mechanically it seems to have been left to DMs to feel this way or otherwise, and it makes sense that it should be.

The Rule is a Conceit for the Sake of Mechanics and Balance

To me, of all constrictions on class abilities this entire need to have gold at all for spell copying is one of the most verisimilitude breaking ones, fine inks or otherwise. It is obviously designed for mechanics at the cost of in universe sense. Inks and materials are a thin justification; a pretense explained with as little detail as they could manage.

Notice that it doesn't say you must acquire or have the items (presumably then you might find them, craft them, etc), but rather that you must spend the gold. This gold correlates exactly to spell level for each and every spell. The price in the aggregate of these "fine inks" and the "material components you expend as you experiment with the spell" always follow the same cost formula despite there presumably being very different material components for different spells of the same level, as is the general lore of magic in the game. Also, normally one can cover most material components for actually casting these spells with a component pouch or arcane focus. Unless there are some strange, uniform spell-mastering-process components this really makes no sense.

The intent seems to be simply to force wizards to have to make decisions about when they copy which spells, and generally to somewhat limit their power versus other spellcasters. The lore/fluff of it is pretty stupid and illogical however you implement it, so getting hung up on inks is just a sign you should take a deep breath and step away.

Many Mechanical Conceits are Illogical

Once you think about it, realism-wise, inexplicably copying the spell without a shopping expedition makes no less sense than wizards and some other spellcasters suddenly learning particular spells at level up, or all classes suddenly gaining all manner of abilities, skills, etc. upon level up. At most tables there is no strict rule that you have to practice these things before abruptly getting good at them, but it is, of course, one of the core mechanics of the game.

At the end of the day it is a game built around mechanical conceits. It is also at the DM's discretion how to limit them. Do you let characters level up immediately upon acquiring the XP or make them wait for a rest? Do you describe the injuries characters do and receive or just talk about HP, whatever that is? How much of an explanation do you demand for what is actually being done to accomplish an investigation check or the help action?

Personally I like to put in all sorts of practices and limitations to add as much verisimilitude to this elf game as possible, for the sake of immersiveness and just feeling like the world has some sort of rules. But the designers are not mandating this, and they're not mandating you go on an ink shopping expedition.