We have done precision numerical simulations of water molecules as they flow through carbon nanotubes that are so narrow that the water molecules are lined up into a single file. Two possible arrangements of water molecules seem possible–one in which water molecules nestle between each adjacent pair of carbon atoms of the nanotube, and another in which the water molecules ignore the spacing imposed by the nanotube and space themselves according to the forces from neighboring water molecules. We found that the most interesting occurrences were neither of these two spacings. Under some conditions, most water molecules could be found located in between carbon atoms, but isolated groups of water molecules would be bunched up without obvious relation to the position of the carbon atoms. These bunches of water molecules were highly unstable in the sense that if given a small push, they would rapidly speed down the length of the nanotube. These highly mobile arrangements propagate as solitons–a type of nonlinear wave that has extremely unique properties. For example, solitons can propagate even at extremely low temperatures. So, water in nanotubes will continue to flow due to the propagation of solitons even when the temperature is much below the freezing temperature. Furthermore, while we usual think of, say, copper as a solid at room temperature, when put inside a nanotube, a single file of copper atoms will flow through the nanotube via the propagation of solitons. And, the amount of material that leaves the end of the nanotube, is precisely the excess number of molecules in the soliton. An interesting case is when the soliton carries only one excess molecule. Then, each soliton on reaching the end of the nanotube releases a single molecule. So, soliton propagation in a nanotube becomes a means to produce a flux through the nanotube can be controlled molecule by molecule.