$\overline{\mathrm{\partial }}$-dressing method for the three-coupled fourth-order nonlinear Schrödinger system

This article mainly investigated the three-coupled fourth-order nonlinear Schrödinger (NLS) systems, which characterized the alpha-helical proteins with interspine coupling at the fourth-order interaction. By the $\overline{\mathrm{\partial }}$-dressing method, we studied the three-coupled fourth-order NLS systems. We derived a new spectral problem starting from the $\overline{\mathrm{\partial }}$-problem. Then, we obtained the three-coupled fourth-order NLS hierarchy associated with the spectral problem using a recursive operator. Finally, we analyzed the $N$-soliton solutions of the three-coupled fourth-order NLS systems by determining the spectral transform matrix in the $\overline{\mathrm{\partial }}$-problem. We also provided detailed descriptions of the $1$-, $2$-, and $3$-soliton solutions, along with dynamic features and graphical illustration.