Dynamics of solitons and breathers on a periodic waves background in the nonlocal Mel'nikov equation

Dynamics of general line solitons and breathers on a periodic line waves (PLWs) background in the nonlocal Mel'nikov (MK) equation are investigated via the KP hierarchy reduction method. By constraining different parametric conditions for a general type of tau functions of the KP hierarchy, two families of mixed solutions to the nonlocal MK equation are derived. The first family of mixed solutions illustrates general line solitons on a PLWs background. The simplest case of such mixed solutions shows the two-line solitons on a PLWs background, and the two-line solitons possess five different patterns: a mixture of one-dark-soliton and one-antidark-soliton, two-antidark-soliton, two-dark-soliton, degenerated two-dark-soliton, and degenerated two-anti-dark-soliton. The high-order mixed solutions display superposition of several individual simplest solutions. The second family of mixed solutions demonstrates general breathers on a PLWs background or on a nonzero constant background. The breathers are periodic in time and do not move in the (x, y)-plane as time propagates.