I'm trying to solve the following non-linear diffusion equation: $$ \frac{\partial}{\partial t} u(x,t)= \frac{\partial^{2}}{\partial x^{2}}u(x,t)^{3}$$ $$ -1\leq x \leq1, t \geq 0 $$ with the boundary conditions: $$u(1,t)=-u(-1,t)=1$$ and the initial conditions: $$u(x,0)=x$$ I'm trying to implement a Crank-Nicolson scheme but I'm a little stuck. Here's what I have so far: $$\frac{u_{i}^{n+1} - u_{i}^{n}}{\Delta t} = \frac{(u^{3})_{i-1}^{n+1} - 2(u^{3})_{i}^{n+1} + (u^{3})_{i+1}^{n+1} + (u^{3})_{i-1}^{n} - 2(u^{3})_{i}^{n} + (u^{3})_{i+1}^{n}}{2\Delta x^2}$$ I'm not sure how to evaluate $$(u^3)_i^{n+1}$$ How should I proceed?

I'm trying to solve the following non-linear diffusion equation: $$ \frac{\partial}{\partial t} u(x,t)= \frac{\partial^{2}}{\partial x^{2}}u(x,t)^{3}$$ $$ -1\leq x \leq1, t \geq 0 $$ with the boundary conditions: $$u(1,t)=-u(-1,t)=1$$ and the initial conditions: $$u(x,0)=x$$ I'm trying to implement a Crank-Nicolson scheme but I'm a little stuck. Here's what I have so far: $$\frac{u_{i}^{n+1} - u_{i}^{n}}{\Delta t} = \frac{(u^{3})_{i-1}^{n+1} - 2(u^{3})_{i}^{n+1} + (u^{3})_{i+1}^{n+1} + (u^{3})_{i-1}^{n} - 2(u^{3})_{i}^{n} + (u^{3})_{i+1}^{n}}{2\Delta x^2}$$ I'm not sure how to evaluate $$(u^3)_i^{n+1}$$ How should I proceed?

Edit: Forgot to mention that the analytical solution is: $$u(x,\infty) = x^{\frac{1}{3}}$$