On the Invariant Subspace Method for Nonlinear Second-Order Evolution Equations

We investigate invariant finite-dimensional linear spaces for a class of nonlinear evolution equations with second-order time dependence of the type ${\partial }_{tt}u=-{\partial }_x\left(A(x,u)u\right)+{\partial }_{xx}\left(B(x,u)u\right)+f\left(u\right),$ where the coefficients may depend nonlinearly on the solution. Using direct algebraic verification and invariant subspace reductions, we derive several exact solutions and reduce the governing equations to finite-dimensional systems of ordinary differential equations. Various examples are presented to illustrate the applicability of the method to nonlinear transport, diffusion, and reaction models.