Problem: Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $$u_t + 0.1 (u_x)^2 = u_{xxxx} - 16 u; \quad u(0, t) = 1, u(x, 0) = 0 $$ The solution is sought for $x \ge 0$ and $t \ge 0$. Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \to \infty$. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \boxed{} LaTeX Environment. If there are any free constants, approximate them numerically. There should be no variable (non-evaluated) constants or free constraints, however. Do not box more than one equation! The approximation should include the zeroth-order term and the first-order correction term accounting for the nonlinearity.
Solution: $\boxed{u(x, t) \sim \frac{599}{600}e^{-2x} + \frac{1}{600}e^{-4x}}$