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 ∂ttu=−∂x(A(x,u)u)+∂xx(B(x,u)u)+f(u), 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.