Problem: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $2u_{tt} - 3u_{xx} - 6(u^2)_{xx} - 5u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 13$. Please place your final solution in a $\boxed{}$ LaTeX Environment.
Solution: $$ \boxed{ u(x,t) = 13\,\mathrm{sech}^2\left(\sqrt{\frac{13}{5}}\left(x-\sqrt{\frac{55}{2}}t\right)\right) } $$