Localized wave solutions and mixed interaction structures in the AB system

Abstract

For AB system, in the first place we calculate the modulational instability for the possible reason of the formation of localized wave. Secondly, based on the generalized (n,N−n)-fold Darboux transformation, we construct order-n degenerate soliton solutions and their interaction solutions sitting on zero background of this system. Specifically, we discuss the three cases of degenerate soliton with interaction solutions obtained above in detail: (a) For order-1 degenerate soliton solutions, we show that they can be decomposed into two order-1 solitons with variable “phase shift”. (b) For higher-order degenerate soliton solutions, we achieve the bright-dark transformation of these solutions by changing the positive and negative of eigenvalues λi. Then a number of the corresponding structural maps are clearly displayed. (c) For interaction solutions with soliton and degenerate soliton solutions, we study the collision under various structures, and these collision characteristics are significantly different from those before. At last, we also sum up various mathematical features of degenerate soliton and mixed interaction solutions.