N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Abstract

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

Extended Calogero-Bogoyavlenskii-Schiff equation and its dynamical behaviors

“…Furthermore, Manakov et al discovered in [9] that the interactions of lump waves do not result in a pattern of phase changes. Regarding that, many powerful methods for finding the lump solutions of NPDEs have been developed over the past decades, including the long-wave limit approach [7,10], the nonlinear superposition formulae [11], the inverse scattering transformation [12,13], the invariance and Lie symmetry analysis [14,15], the Bäklund transformation [16,17], the bilinear neural network method [18][19][20][21][22][23][24], the Darboux transformation [25,26] and the Hirota bilinear method [27][28][29][30][31], Symbolic computation method [32][33][34][35] and other different methods [36][37][38][39][40][41][42][43]. Among the approaches stated above, taking a 'long wave' limit of the corresponding N-soliton solutions plays an important role in the investigation of M-lump solutions for nonlinear partial differential equations.…”