  We present the discovery of a class of exact spatially localized as well as
periodic wave solutions within the framework of the modified Korteweg-de Vries
equation. This class comprises breather and interacting soliton solutions as
well as interacting periodic wave solutions. The functional forms of these
solutions include a joint parameter which can take both positive and negative
values of unity. It is found that the existence of those closed form solutions
depend strongly on whether the cubic nonlinearity parameter should be
considered positive or negative. The derived wave structures show interesting
properties that may find practical applications.
