Problem: Find a self-similarity solution (soliton behavior) for $\partial_{tt} u - \partial_{xx} u + 2u ((\partial_t u)^2 - (\partial_x u)^2) = 2u^5 - u$ that travels at velocity $v = 1/\sqrt{2}$, subject to the boundary condition of $u(0, 0) = 1$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants.
Solution: $$\boxed{u(x, t) = 2(e^{\sqrt{2}(x - (1/\sqrt{2})t)} + e^{-\sqrt{2}(x - (1/\sqrt{2})t)})^{-1}}$$