Could there ever be a fully solar powered airliner capable of transatlantic flights?
Short answer: No

Long answer: No. Look at the solar impulse project, this is some of the best solar-electric technology that's out there, and it's barely able to carry one person, very very slowly. If you took an airliner, covered it in the best solar cells available and connected it to the best electric engines available it probably wouldn't even be able to taxi. 

Some science here: The maximum energy density of the sun on the Earth is about 1300 watts/meters squared. About 30% is lost in the atmosphere so the best you'll get on the surface of the earth is about 900 W/m2. The 737, for example, has 102 square meters of wing area, that would equate to about 90000 watts (90KW) of sun energy hitting the wing. The best power conversion we have of solar to electricity we have is about 10%, so you would get at most 9KW of power from a 737 solar array. As we see from this answer it takes 90 MW (megawatts) of power to get a 747 into the air, assuming it takes about 1/3 or that (30MW or 30000KW) to get a 737 into the air, then that array will only generate 0.3% of the electricity needed, and that's the best possible case in the strongest sun. If it's cloudy or at night you are completely out of luck.  

Any solution based on current battery technology is addressed here, and it's very much a no at this point. 

Theoretically it is not possible that a solar powered passenger airplane could be as good as a jet-a1 powered passenger jet, this is because the energy from the sun is not sufficient even if you had 100% conversion to electricity. If battery technology were to become orders of magnitude better to the point its energy density compared to liquid fuels then it very well could, this is a long way off, the best batteries that are currently lab concepts are about 1/5 of the density of liquid fuel. Once batteries get there then solar power could charge them for the flight. It's actually just as possible that solar power could be used to make liquid fuel in a renewable way from CO2 and other elements in the environment. Bio-fuels are solar powered, but we can't make much of a dent in fuel use with them without taking food off of people's plates, and personally I'd rather eat!