The Korteweg-de Vries equation describes the unidirectional propagation of long waves in a large class of nonlinear dispersive media. The author of the present paper uses a defocusing modified Korteweg-de Vries equation, which admits a given shock wave solution (or kink solution), up to an arbitrary phase shift by a trigonometric expression. This is a canonical dispersive equation which arises in the study of a number of physical phenomena, particularly in the areas of fluid plasma mechanics with applications that include the two-layer fluid model and a plasma with non-Maxwellian electron distribution. The normalized modified Korteweg-de Vries equation is included in an initial-value problem for which the initial data have a discontinuous step. To obtain the complete large asymptotic structure of the solution to this problem, the method of matched asymptotic coordinate expansions is used. Also, the author includes a brief discussion of the structure of the large-time solution to the above mentioned equation when the initial data are given by the general discontinuous step.