Problem: Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $$\frac{\partial u}{\partial t} + 3u^2 \frac{\partial u}{\partial x} = 0.3 \frac{\partial^2 u}{\partial x^2} - 1.5 u; \quad u(0, t) = 1 \quad \text{for } t > 0.$$ The solution is sought for $x \ge 0$ and $t \ge 0$. The initial condition is $$u(x, 0) = 0 \quad \text{for } x > 0.$$ Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \rightarrow \infty_-$. 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: $$\boxed{u(x, t \rightarrow \infty) \approx \left(1 + \frac{\sqrt{5}}{4}\right) e^{-\sqrt{5}x} - \frac{\sqrt{5}}{4} e^{-3\sqrt{5}x}}$$