Novel bright-dark mixed N -soliton for the ( 3+1 )-component Mel’nikov system and its multi-component generalization

Based on the KP-hierarchy reductionmethod, we construct the novel bright-dark mixed N -soliton for the ( 3+1 )-component Mel’nikov system including 3-component short waves (SWs) and one-component long wave (LW) for all possible combinations of nonlinearity coefficients. It is verified that dark or bright solitons can exist in the SW components, but only bright solitons appear in the LW component. According to different combinations of solitons in three different SW components, the bright-dark mixed N -soliton for the ( 3+1 )-component Mel’nikov system is mainly discussed into two types that two-bright-one-dark and one-bright-two-dark solitons. Finally, the ( 3+1 )-component Mel’nikov system can be directly extended to ( M+1 )-component ( M≥ 3 ) case comprised of M -component SWs and one-component LW, and the bright-dark mixed N -soliton for the multi-component generalization is also given in Gram determinant form.