Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces

In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated. The full scope of obtaining finite-gap solutions of the KdV equation is presented. It is proven that the multiply periodic ${\mathrm{\wp }}_{1,1}$-function on the Jacobian variety of a hyperelliptic curve of arbitrary genus serves as the finite-gap solution, the genus coincides with the number of gaps. The subspace of the Jacobian variety where ${\mathrm{\wp }}_{1,1}$, as well as other $\mathrm{\wp }$-functions, are bounded and real-valued is found in any genus. This result covers every finite phase space of the KdV hierarchy, and can be extended to other completely integrable equations. A method of effective computation of this type of solutions is suggested, and illustrated in genera $2$ and $3$.