Derivation, nonlocal symmetry analysis, and exact solutions of modified Broer-Kaup equations

Abstract

Throughout this work, the conservation laws of the (1+1)-dimensional Broer-Kaup (BK) equations are constructed using the multiplier method. These conservation laws are then used to derive higher dimensional BK equations. By introducing constraint conditions, the higher dimensional equation is reduced to a (1+1)-dimensional modified Broer-Kaup (mBK) equations. Subsequently, the mBK equations are researched through the nonlocal symmetry method. A new closed system, which is nonlocally symmetric, is constructed using the Lax pair and the introduction of a potential function. By applying finite symmetry transformations and symmetry reductions to the closed system, the exact solutions of the mBK equations are obtained. By selecting different parameters, a set of knot solutions and dark soliton solutions are derived, and their dynamical behavior is analyzed.