How should white push for advantage in the Nimzowitsh attack of the Petrov's defense? This line of the Petrov defense can often lead to opposite side castling and the general rule for that is attacks happening on both sides of the board. However, I noticed that it's very difficult to attack black in the Petrov defense. 

My question is basically this: In what way should I play to get an advantage against this line of the Petrov defense, whether it should be in a dynamic or positional manner?

[FEN ""]

1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. Nc3, Nxc3 6. dxc3 Be7
I noticed that it's very difficult to attack black in the Petrov
  defense.


That is why it is popular at the elite levels: It is super-solid. The character of the game is going to depend on black, and the choice of where to castle. If black castles short, since white castles long, it is going to be an all-out pawn storm on opposite wings, and the person, who calculates, and attacks, better will win. in practice, white has fared better there, scoring about 70% per the Mega 2020 database compared to about 60% in the other lines where both sides castle long. It is a much more dangerous choice.

When both sides castle long, the game definitely becomes more positional since both sides have relatively safe kings. It will come down to things like h4, and Ng5 by white, and if that Ng5 can either be traded for the Be6, gaining the two bishops, or if white can force black to capture the Ng5 with the Be7(Bf6), again, gaining the two bishops. White has doubled pawns, but they are not a big deal, and in fact, they give white more maneuvering room since he has an extra semi-open file to use. If white can post a piece, maybe the Bf1-c4-d5 on d5, if may force a concession c6, weakening the black queenside. After c6, d5 is often forced, and then white can hit at that center with c4 if timed right.

If this type of semi-positional game occurs, it is not quite the same as the positional games that occur in d-pawn openings. The positions are much more fluid than those with blocked centers, so there is much more calculating, and by that, I mean exact calculation, and positional evaluation. And, black is more cramped.

Lastly, of course there is the open e-file. no matter what, it is likely to lead to the trade of the rooks, because neither side can afford to let the other dominate the only fully open file, and not contest it. In positions that have only one open file, if you can fully control it, it is often a decisive factor. In the distant past, when I played e4, I "made a living" playing the Exchange French, and often, before my opponents knew it, I owned the e-file, and the game.

In the end, white's slight space advantage is probably why the 60% winning percentage, but it requires patience, and there is no "do this and get an advantage" rule. I would try to read through a bunch of white wins in the line, and see how the games played out, and try to emulate the ones you liked the most.

 [FEN ""]

 1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. Nc3 Nxc3 6. dxc3 Be7 7. Be3 Nc6 8. Qd2 Be6 9. O-O-O Qd7 (9... O-O 10. h4 Ne5 11. Ng5 Qd7 12. Kb1 {With an all-out opposite sides attack.}) 10. Kb1 Bf6 11. h4 O-O-O 12. Ng5 Bf5 13. Bc4 Ne5 14. Bd5 {And the game has become semi-positional.}