Lithium-ion batteries are the customary energy storage for fixed-wing airplanes, but despite being better than any other type of battery in this regard, they're still not very energy dense compared to hydrocarbon fuels used in aviation. Careful experimentation with the size of the battery, on glider-type airframes, may be able to maximize flying time to several hours. One way past this is to use a very low amount of energy storage and pack the plane with solar cells. This can only be very successful on low speed, very low wing loading, high glide ratio aircraft with large, slow-rpm props.
A 48 hour run, which basically indicates an indefinite flying pattern, has been achieved in aerospace R&D labs, at extreme cost and complexity, high altitudes with experimental airspace exemptions, and severe problems with fragility. Still, it does suggest that the concept of a daytime solar flyer could be practically applied to a real, usable product.
Two brands seem to be mentioned a lot in terms of previous experimentation with solar panels: Sunpower's A300 high-efficiency cells which have been used in several famous professionally-designed planes, and Powerfilm's thin film cells - including an ultra-lightweight version without UV-protection encapsulant which is marketed specifically towards RC/UAV usage.