Exact solutions of the high-dimensional extended generalized Broer-Kaup equations with nonlocal symmetry

Abstract To analyze the generalized Broer-Kaup (GBK) system, we have employed the methodology suggested by Lou et al , which involves constructing high-dimensional equations from conservation laws associated with low-dimensional partial differential equations. For the first time, we have derived the (2+1)-dimensional integrable GBK equations by leveraging the conservation law of the (1+1)-dimensional GBK equations. Through the imposition of constraints within the (2+1)-dimensional GBK equations, we have obtained a novel (1+1)-dimensional GBK equations. Subsequently, we pioneered the use of a nonlocal approach to analyze this new equations. Specifically, we constructed a closed system incorporating nonlocal symmetries. This was achieved by introducing the Lax pairs associated with potential functions and GBK equations, utilizing conservation laws. By applying the nonlocal symmetries to this closed system, we were able to deduce the generating element. The exact solution of the equation is achieved by combining finite symmetry transformations with a symmetry reduction technique that involves approximations. The dynamic behavior of the equations is studied by means of figures of the exact solutions.