Was any Russian design feature incorporated in modern US fighters jets or vice-versa?
Three things come to mind:
- Geometric shaping for stealth is based on the theoretical work of Petr Ufimtsev, a Russian mathematician who explained how to calculate the reflection of radio waves from the 1960s on.
- The sight-controlled missile targeting which appeared in the MiG-29 in the 1980s. In combination with the wide off-boresight capabilities of the aircraft's radar and the missiles, it allowed to get a lock on aircraft flying actually behind it. When US F-16 met ex-GDR MiGs in exercises in the early 1990s, they were shocked to find how much superior the Russian design was and soon copied it.
- The rotating collars which were used on the Yak-141 to deflect the engine exhaust straight down for vertical take-off have been used in the VTOL version of the F-35. Lockheed-Martin co-operated with Yakowlev in the early Nineties to learn how the Russian airplane works.
From a historical perspective you can add the Russian (actually, Georgian) aircraft designer Alexander Kartveli, who headed engineering at Republic, and the influence of Georgia-born Alexander de Serversky (actually of Russian descent) who propagated strategic bombing to win future wars, even though this concerns bombers, not fighters.
Variable wing geometry ("swing wings") is a candidate. The U.S. Bell X-5 was the first aircraft to fly with a system for changing the sweep angle of the wings in flight; it was built based on a captured German aircraft whose wings could be adjusted on the ground. The Russians got wind of the X-5 program and copied the idea.
The first production fighter on either side to fly with swept wings was the U.S. F-111 Aardvark fighter-bomber in 1964, however the Soviets were only 3 years behind with the first flight of their variable-geometry fighter, the Soviet MiG-23 interceptor, then the F-14 interceptor and MiG-27 ground-attack variant followed in 1970, with the B-1 penetration bomber and European Panavia Tornado multirole fighter in 1974 being the last swing-wing aircraft to enter production.
The Russians also tried to copy the U.S. XB-70 Valkyrie. For its day, this was a real threat to Russian air defenses, as it could theoretically fly above the faster low-level SAMs and outrun the high-altitude SAM systems the Soviets had at the time. Countering like-for-like to maintain MAD, the Soviets developed the Sukhoi T-4, which has an external design almost exactly matching the XB-70. The U.S. ended up cancelling the B-70 project as Soviet SAM advances and the MiG-25 interceptor made speed alone insufficient to fly over the USSR with impunity (the SR-71 also never overflew Soviet territory, to the best of civilian knowledge). Later on, the Soviets, in similar fashion to the Valkyrie project, would also copy the basic idea of the Space Shuttle to produce the Energia-Buran launch system.
I remember hearing reports in the late Soviet era that Soviet MiG-29s and Su-27s had surpassed American radar technology and the Americans were struggling to catch up. The technological leap, IIRC, was passive electronically-scanned radar (PESA), which both sides had been using for ground and ship installations but the Soviets were the first to get the system small and self-contained enough for aircraft use. The Americans' fighters of the day (F-14, F-15, F-16, F-18) still used mechanically-scanned arrays with a moving antenna which slowed the scanning rate, especially in "deep" scanning modes (the tradeoff at the time was higher resolution; the F-15E's radar is capable of producing a map of the target zone from dozens of miles out, allowing the WSO to point the aircraft's targeting systems with extreme accuracy). The U.S. more or less leapfrogged the use of PESA radars in aircraft (only the B-1B and B-2 ever used them) and instead worked with the Japanese on AESA, which was first seen on Japan's F-2 variant of the F-16, and is used on later F/A-18 Super Hornet variants and the F-22 in the U.S. arsenal.
Lastly on my list, after the collapse of the Soviet Union, former Warsaw Pact territories including East Germany began sharing the technology the Soviets had left behind, and in some cases even bringing Soviet technology into NATO armed forces. Among the new information gained, the capabilities of the AA-11 Archer short-range IR missile system truly scared NATO top brass. The missile was more agile and had higher off-boresight capabilities than anything NATO was fielding at the time (most countries were at the time using the U.S.'s AIM-9M Sidewinder variant, which was good but demonstrably inferior especially in targeting). This was in part thanks to the helmet-mounted cueing system Peter Kampf mentioned, allowing a pilot to get a seeker lock on a target above or to the sides of his aircraft just by looking at it, instead of having to point the entire plane. The U.S. and NATO responded with the Joint Helmet Mounted Cueing System, the AIM-9X, and several other missile development programs including the MICA and ASRAAM, all borrowing from lessons learned by the AA-11.
In his book 747, Joe Sutter (who led the 747 design team and came up with the idea of a widebody design rather than multiple decks) describes a meeting hosted by the US State Dept with some leading Soviet aircraft designers, for an exchange of ideas on non military aircraft design. Yes, they did do this, even in the depths of the cold war.
One question the Soviet designers asked was why US commercial jets mounted the engines in pods beneath the wings, rather than in the wings, which is where British and Soviet jet transports of the 1950's and 1960's had their engines mounted.
Sutter gave two big reasons for the engine pod mounting:
Easier service - the housings could simply be removed, giving full access to the engine. Also, changing an engine became much simpler. Servicing or replacing a jet engine that's buried in the middle of the wing is far more difficult.
Limiting damage from an exploding engine. The early turbojets would occasionally go bang. With the engine mounted in a pod beneath the wing, this limited the damage to the wing in the event of an engine malfunction.
Some time after that meeting, the Soviets brought out the IL76 transport... with engines mounted in pods beneath the wings.