Variational approach to (۴+۱)-dimensional Boiti–Leon–Manna–Pempinelli equation

The (۴+۱)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is a typical high-order nonlinear integrable partial differential equation (PDE), which plays a crucial role in describing multi-dimensional nonlinear wave phenomena in plasma physics, fluid mechanics, and nonlinear optics. However, its high dimensionality (four spatial variables + one time variable) and strong nonlinear coupling pose significant challenges to constructing a variational formulation and solving soliton solutions. To address this issue, this work focuses on the variational method for the (۴+۱)-dimensional BLMP equation and proposes a construction strategy for an approximate variational formulation based on the semi-inverse method. Through two-step variable transformations (order-reduction transformation and auxiliary potential function introduction), the high-order and nonlinear terms of the original equation are simplified, and the approximate form of the Lagrangian density F is derived. Consequently, an approximate variational formulation of the (۴+۱)-dimensional BLMP equation is obtained, and consistency verification confirms that the extremum condition of the functional is exactly equivalent to the solution of the original equation. Notably, the approximate form of F not only balances computational efficiency and physical accuracy but also provides guidance for the improvement of the original equation from an energy perspective. A prominent open problem arising from this work—the exact determination of F from the variational derivative constraint equations—invites mathematical enthusiasts and researchers in nonlinear PDEs to explore innovative solutions, which will advance the general theory of variational principles for high-dimensional nonlinear integrable systems. The research results offer an effective theoretical tool for solving the (۴+۱)-dimensional BLMP equation and analyzing its dynamic characteristics, with broad application potential in simulating multi-dimensional nonlinear wave phenomena.

Keywords:

(۴+۱)-dimensional Boiti–Leon–Manna–Pempinelli equation , Semi-inverse method , Variational formulation , Nonlinear integrable partial differential equation , Soliton solution , Open problem