RATIONAL AND SEMI-RATIONAL SOLUTIONS TO THE NONLOCAL MEL'NIKOV EQUATION VIA DETERMINANTS

Based on a combination of the bilinear method and the KP hierarchy reduction method, explicit rational and semi-rational solutions, given in terms of determinants, to the nonlocal Mel'nikov equation are investigated. We first start from the tau functions of the single component KP hierarchy, and construct general periodic solutions to the nonlocal Mel'nikov equation. By taking a long-wave limit of the periodic solutions, general rational and semi-rational solutions to the nonlocal Mel'nikov equation are constructed. The obtained rational solutions describe lumps on a constant background. The semi-rational solutions have three different dynamical behaviours: lumps of any order on a periodic line waves background, the mixture of lumps of any order and breathers of any order on a constant background, and the mixture of lumps of any order and breathers of any order on a periodic line waves background.