Invariant Subspaces and Exact Solutions for Nonlinear Variable-Coefficient Beam-Type Equations

This paper investigates a class of generalized nonlinear variable-coefficient beam-type evolution equations motivated by the modeling of nonhomogeneous elastic structures with nonlinear effects. The invariant subspace method is employed to construct exact solutions by identifying appropriate finite-dimensional invariant subspaces associated with the nonlinear operators. This approach reduces the original partial differential equations to systems of nonlinear ordinary differential equations. Several representative nonlinear beam-type equations are studied, leading to polynomial, trigonometric, hyperbolic, and negative-power invariant subspaces together with their corresponding exact solutions. The results demonstrate the effectiveness of the invariant subspace method for constructing explicit exact solutions of nonlinear variable-coefficient fourth-order beam-type equations and highlight its applicability to a broad class of nonlinear beam models.