Piccard and Borschberg are looking forward to a future that includes more solar power.
Branson, owner of Virgin Airways, joined Piccard and Jones in Geneva for a champagne toast.
"It's a little bit like being in a dream, " Piccard said as he stepped on the tarmac.
"I've never had a champagne shower before, " Piccard laughed after Branson sprayed him with a well-shaken bottle.
"I think we share the same spirit, " Piccard said of Lindbergh, the first pilot to solo across the Atlantic.
Piccard and Jones had hoped to touch down alongside the pyramids at Giza.
Don Walsh and Jacques Piccard both risked their lives to descend that deep, and only the former is still alive.
Jones and Piccard both said they felt they were guided by an "almost invisible hand" as they circumnavigated the globe.
"Personally, my best memory is the precise moment between the takeoff and the landing, " said Piccard, 41, a Swiss psychiatrist.
Bertrand Piccard and Brian Jones flew nonstop around the world in a balloon, landing safely Sunday in the Egyptian desert.
Piccard hopes Solar Impulse will start a similar cycle that will lead to unimaginable new dimensions in the development of technology.
With Piccard and Jones aboard, the Breitling Orbiter 3 landed about 640 kilometers (400 miles) southwest of Cairo on Sunday.
Bertrand Piccard, the plane's other creator, and Borschberg are planning an around-the-world flight in an improved version of the plane in 2015.
Piccard's wife, Michelle, and three daughters, were aboard the plane from Cairo.
Today, Solar Impulse's pilots, Bertrand Piccard and Andre Borschberg, came to NASA's Ames research center to announce their plan to fly across America.
However, in 2010, Swiss balloonist Bertrand Piccard plans to launch Solar Impulse, a manned plane in which he will attempt to circumnavigate the globe.
Global excitement from that 1927 flight sparked the beginning of a "new cycle, " Piccard said, which led to the international airline industry we enjoy today.
Mr Jones, 51, from Erlestoke, Wiltshire, and Swiss psychiatrist Mr Piccard, 41, can safely say they have conquered one of the last great aviation challenges.
Borschberg and Bertrand Piccard, Solar Impulse co-founder and chairman, said the plane should be ready for the cross-country journey on May 1, depending on the weather.
"All the big pioneers of the 20th century have tried to fly coast to coast across America, " said co-pilot and one of the plane's founders, Bertrand Piccard.
"All the big pioneers of the 20th century have tried to fly coast to coast across America, " Piccard, one of the plane's founders, said before the flight.
In 1960, Piccard's father, Jacques, descended seven miles beneath the Pacific Ocean in another pressurized module to set a deep-dive record that has been matched only twice.
WSJ: A Solar-Powered Plane Travels Light | WSJ. Magazine May 2013
Piccard and Borschberg will take turns to pilot the plane.
FORBES: Video: Solar-Powered Plane Takes Off For A U.S. Maiden Voyage
With each stop on their itinerary, Piccard and Borschberg bring with them a message: Use technology that saves energy and support government-mandated targets for creating electricity from renewable sources.
Swiss pilot Bertrand Piccard and British co-pilot Brian Jones landed about 20 km from the nearest road after their epic flight which made aviation history, smashing records as it went.
Video posted on the expedition's website showed a smiling pilot Bertrand Piccard shortly after landing, as he waved to well-wishers and held up a flag emblazoned with the Solar Impulse name.
"We want to show that with clean technologies, a passionate team and a far-reaching pioneering vision, one can achieve the impossible, " Dr Piccard said at the announcement of the mission in March.
Mr Piccard's Solar Impulse has a wingspan of just over 63 metres, similar to that of a giant Airbus A340 airliner, but its carbon-fibre fuselage has only enough room for one pilot.
The HB-SIA craft was piloted by Bertrand Piccard, a co-founder of the effort, who is perhaps best known for being the first person to circumnavigate the globe in a hot-air balloon, in 1999.
In the day's opening race, America's Daron Rahlves was involved in a spectacular crash when the 36-year-old, one of the United States' most decorated downhillers, collided with France's Ted Piccard and landed hard on his back.
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