Problem: For the nonlinear PDE $$u_t + 6u\cdot u_x + u_{xxx} = 0$$ with $x\in [-L,L] \ \text{(periodic)}, \ t>0$, seek a traveling‐wave solution. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Box your final answer.
Solution: $$\boxed{u(x,t) = \frac{1}{2}\cdot\text{sech}^2\left(\frac{1}{2}\cdot(x - t)\right)}$$