Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation

Two types of analytical undular bore solutions, of the initial value problem for the modified Korteweg-de Vries (mKdV), are found. The first, an undular bore composed of cnoidal waves, is qualitatively similar to the bore found for the KdV equation, with solitons occurring at the leading edge and small amplitude linear waves occurring at the trailing edge. The second, a newly identified type of undular bore, consists of finite amplitude sinusiodal waves, which have a rational form. At the leading edge is the mKdV algebraic soliton, while, again, small amplitude linear waves occur at the trailing edge. The initial-boundary value (IBV) problem for the mKdV equation is also examined. The solutions of the initial value problem are used to construct approximate analytical solutions of the IBV problem. An alternative analytical solution for the IBV problem, based on the assumption of an uniform train of solitons, is also developed. The parameter regimes, in which the different types of solution occur, for both the initial value and IBV problem are identified and excellent comparisons are obtained between the numerical and approximate solutions, for both problems.