Problem: Find a localized self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\partial_{xx} u + \tanh(u \partial_x u) \sech(u \partial_y u) + \sin^2(\partial_{xy} u) - e^{xy} = 0$, $u(0, y) = \cosh(y)$, $\partial_x(0, y) = \sinh(y)$ with $u(0, 0) = 1$ as the maximum value. Localization means $u(x,t)$ and its derivatives vanish at $t= \pm \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.
Solution: $\boxed{u(x, y) = sech(\sqrt{\frac{1}{2}} (x-\frac{y}{2}))}$