Primitive solutions of the Korteweg–de Vries equation

We survey recent results connected with constructing a new family of solutions of the Korteweg–de Vries equation, which we call primitive solutions. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg–de Vries equation as the number of solitons tends to infinity. A primitive solution is determined nonuniquely by a pair of positive functions on an interval on the imaginary axis and a function on the real axis determining the reflection coefficient. We show that elliptic one-gap solutions and, more generally, periodic finite-gap solutions are special cases of reflectionless primitive solutions.