What is the character arc of Captain Jack Sparrow?

Subverting expectations

I am unable to understand how he changes at the beginning to end in each of the stories.

That is because Jack, contrary to common narrative tropes, does not change.

Captain Jack Sparrow specifically defies the expectation that characters must grow between the start and conclusion of their story arc. Jack Sparrow is the narrative equivalent of what in chemistry is called a catalyst:

Catalysis is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst. Catalysts are not consumed in the catalyzed reaction but can act repeatedly.

In narrative terms, Jack definitely sparks events in the plot, but he himself does not change during the plot. His narrative role is purely as an agent in a plot that doesn't really affect him. Not significantly, at least, nor does the plot change any other pirates (more on that later).

He remains a Pirate, acts on selfish desires, wants to pillage and plunder. In the beginning, he has no ship but in the end he gets a ship, but that is not the change or learning.

When you focus on the trilogy instead of the first movie, you can see that Jack ends up in the same place at the end than where he was in the beginning: having lost the Pearl to Barbossa, drunk, on a dingy, on a badly planned mission that no one really understands.

Jack figuratively follows his own compass in the same way that he literally follows his magic compass: without explanation.

Swings and roundabouts

When you observe the beginning of the first movie and the endings of the three movies, another pattern emerges: you win some, you lose some, but in the end it's all a zero-sum-game.

• Beginning of CotBP: Jack has been mutinied and abandoned
• End of CotBP: Jack is captain of the Black Pearl
• End of DMC: While this is only revealed in the beginning of the third movie, the end of Dead Man's Chest puts Jack in the position of being the only crew member (and captain) on board of the Pearl, stuck in Davy Jones' locker. He does, however, eventually avoid Davy's intended punishment, just like he did back when he made the deal with Davy.
• End of AWE: Jack has been mutinied and abandoned

Jack won some, then lost some, and ended up in the same place, to do it all over again. He already went through this cycle twice.

This sort of cyclical/repeating events happen often in the trilogy:

• Curse of the Black Pearl begins and At World's End ends with "Yo-Ho A Pirate's Life For Me" being sung
• Barbossa twice leads the same mutiny against the same captain (Jack) on the same ship
• Just like with the first mutiny, the crew ends up feeling guilty over leading a mutiny against Jack. The first time this happened was before the first movie, and is what led to Bootstrap Bill's death as he could not keep that guilt inside.
• Pirates will set aside their feelings/morality when shown proof of treasure. The first mutiny against Jack happened specifically because Jack did not give concrete proof of treasure and Barbossa did (which led them to Cortes' treasure, the source of the curse in the first movie).
• While the two EITC guards end up becoming pirates (the two pirates at the back in this scene instead or EITC soldiers, they are essentially still ship crew members following a captain's orders.
• Gibbs remains Jack's loyal right hand. This was the case in the beginning of Curse of the Black Pearl and at the end of At World's End. Notice the similarities with how Jack finds Gibbs and wakes him.
• The sword Will Turner makes in the beginning of Curse of the Black Pearl travels through many hands during the trilogy, but Davy Jones ends up taking it from Norrington, and Will inherits it from Davy when he inherits his position. Will ends up with the same sword he made in the beginning of the story.
• Calypso did not display any softer character nor intention to change after being released from her imprisonment.

Over all, while some changes do happen in the plot, the predominant motif in these movies is that pirates will be pirates, and were so before the trilogy, and will be so after the trilogy.

More to the point, all pirates revert to doing what they were doing before the trilogy: womanizing, chasing treasure, backstabbing each other. None of them came away with any life lessons or changed behavior. The only people whose arc actually changed them are non-pirates.

Jack Sparrow is the leading example of this motif.

TL:DR Jack Sparrow is an escapist/trickster character (in a world full of tricksters aka "Pirates") and has a yet-to-be-completed redemption arc.

In modern literature, the trickster survives as a character archetype, not necessarily supernatural or divine, sometimes no more than a stock character.

Often, the trickster is distinct in a story by his acting as a sort of catalyst; his antics are the cause of other characters' discomfiture, but he himself is left untouched.

While the first film, Curse of the Black Pearl, doesn't go too deep into Jack's morality, and essentially everything ends well for the two innocent characters that are thrown into Jack's reality (Will Turner III and Elizabeth Swann), the next two films (Dead Men's Chest, At World's End) set up a major redemption arc for Jack Sparrow.

In these two films, Jack proves to be an escapist character that begins to corrupt the relationship between Will and Elizabeth and is seeking immortality, while also trying to escape a debt he owes Davy Jones.

At the end of Dead Man's Chest, after tempted Elizabeth to become more of a pirate, she does a very "piratey" (semi-immoral) thing by leaving Jack shackled to The Black Pearl, so they can escape, since Davy Jones' kraken is only really after Jack. She tempts him to stay behind to let the others escape and encourages him to be "a good man."

Jack does this a bit reluctantly trying to escape, but since there is nowhere to go by the time he frees himself from the chains, he bravely plunges into the mouth of the kraken.

In At Worlds End many of the main characters need to go to Davy Jones Locker, where Jack now resides, in order to stop Davy Jones by assembling the Brethren of the Court. Jack has one of the "nine pieces of eight" needed to make this happen.

But Davy Jones' Locker is not just a convenient plot device, it is a place that acts as a sort of hell or purgatory, as the Tia Dalma explains:

TIA DALMA: Jack Sparrow is taken, body and soul, to a place not of death, but punishment. The worst fate a person can bring upon himself...stretching on forever. That's what awaits at Davy Jones' Locker.

When we first see Jack, his ship The Black Pear is stuck on land (sand) and he has to deal with commanding multiple versions of himself, demoting his usually comical antics to mere caricatures. We know from what Jack told Elizabeth in Curse of the Black Pearl that this would be "hell" for Jack, because of his feelings about what a ship means to him,

JACK SPARROW: Wherever we want to go, we go. That's what a ship is, you know. It's not just a keel and a hull and sails; that's what a ship needs. Not what a ship is. What the Black Pearl really is, is freedom.

Ultimately, being stuck with himself is also a metaphor for a person who has not yet comes to terms with whom they really are or what they have done to other people...

This then helps push the tragic part of the bittersweet ending when Jack can't decide to stab Davy Jones' heart and become the Captain of the Flying Dutchman, giving him immortality, but knowing if he doesn't do the job of ferrying the souls of the dead to the other side (spiritual moral vocation), he will become part sea creature like Davy Jones and his crew. Will is then stabbed by Davy Jones, but Jack stabs Davy Jones' heart with Will's hands, allowing Will to live by taking Jones place...

But the caveat is that Will can only step on land once every ten years, meaning he would not spend most of his life with Elizabeth as intended.

So the next film On Stranger Tides seems like it is removed from the first trilogy, but in actuality it begins to go deeper into Jack's past by introducing a scornful old flame of Jack's out for revenge for what he allegedly did to her, along with giving viewers a new love story between a clergy boy named Philip and a mermaid named Serena, as Jack again seeks immortality through the Fountain of Youth.

When one looks at the situation with a clergy boy (Philip) and his romance with a mermaid (Serena), it's easy to see that Philip is ideal candidate to replace Will on the Flying Dutchman, so that he can return home to Elizabeth and his son Henry. If Jack would eventually help make this a reality, he would be closer to redemption...

But for whatever reason (perhaps some of the cast was busy or refused to return?) this is not the direction that Pirates of the Caribbean: Dead Man Tells No Tells takes.

Instead it appears they found another work around from Jack's childhood with The Trident of Poseidon first introduced in the young Jack Sparrow adventure novels.

And while the Will and Elizabeth romance is resolved through the Trident undoing curses, the film seems to intentionally drag out Sparrows' redemption by both allowing him to hit an all time low, becoming exceedingly closer to those caricatures of himself presented in Davy Jones Locker (while also again diving further into his past with Salazar & Morgan, and because one of the trailers for the film included a line stating, "The final adventure begins", suggesting this was beginning of a final series of films. Davy Jones also seems to return in the post credit scene!

It implies that things aren't really resolved. One could argue that Jack wasn't really the hero of the film at all, as Barbarossa, Carina, and Henry are more of the pro-active catalysts setting out to resolve things (and interestingly Barbossa gets a to fulfill his redemption arc instead) with Jack unwillingly getting dragged along this time, with the caveat being that his destruction is betraying the compass, which *might also be about betraying himself. This all points to a major turning point for the character.

Originally Pirates of the Caribbean five and six were suppose to film back to back, but plans changed. Then Johnny Depp was fired in part due to the domestic abuse accusations against his former spouse, along with reports that this situation effected his performance and behavior on set. At one point the actor seriously injured his hand and was flown back to the US for immediate surgery. There are conflicting reports about why this happened, including the most recent testimony from Johnny Depp himself.

Allegedly Disney is waiting for the final verdict of Depp's libel case against The Sun before considering bringing him back for Pirates of the Caribbean six.

However, there are also reports that this would be Jack Sparrow's last film transitioning into a spin-off in which he will be passing the buck onto a new character, the recently re-imaged Redd character featured within the amusement park ride.

So it remains to be seen if the character will finally comes to terms with himself and find resolve or a meaningful redemption...

As an aside, earlier films almost optioned Tim Power's' novel On Stranger Tides, which was later picked up for the fourth film.

In 2003, the first Pirates of the Caribbean movie, The Curse of the Black Pearl, was released. Author Tim Powers went and saw it, though wondered if his book was used. Then, before the second movie (Dead Man's Chest) was released, Disney had approached Powers' agent and said they wanted to option On Stranger Tides in case they do a fourth Pirates movie, despite that there had only been the one movie at that point. But Tim Powers was hugely optimistic, and he and his wife Serena saw Dead Man's Chest the first day it came out just to add their "two votes" to making it a big success. Having saw the film, Powers saw another possible element or two from his book; the idea of a character up in the rigging of a mast standing on a cross spar holding two poles to work a puppet down on the deck below, Powers then wondered if that would've been there if he hadn't written his book. But by that time they had got the option so he figured, "carry on." And then finally they wouldn't actually buy the book until they were virtually filming the week before, as it was standard with movies that they don't outright purchase the property until they're setting up the cameras...

As someone who has read the book several times, there are a lot commonalities between all of the POTC films and this book and not just the fourth installment, with one major difference being that the main character "Jack Shandy", is more dramatically straight character and non-comedic, which is what Disney was also originally going for, but Depp convinced them otherwise.

The novel itself in part deals with a story about a young man who is essentially press-ganged into a pirates' life and he has to come to terms with the corruption of his evil uncle, while another plot features a scholar/professor who lost his wife and is seeking magic in order to bring his wife's conscious back, but into his daughter's body! Jack falls in love with her and finally does the right thing by trying to save her, thus a redemption arc is also present.

While none of these plots are confirmed and may serve only as conjecture, it does make sense given what the whole story of Jack Sparrow has been so far and POTC themes of unrequited or disrupted love stories being at the heart of it's story-telling, that the story may eventually end in similar way, where he finally does something truly good on his own, because of love...