I'm solving the fractional nonlinear (time-independent) Schroedinger equation of the form

$$\frac{1}{2}u-\frac{1}{2}\frac{\partial ^{\alpha}u}{\partial x^{\alpha}}-u^3=0.$$

The fractional derivative is defined as a matrix acting on a discretized version of $u$, i.e. $u$ is a column vector. $\frac{\partial ^{\alpha}}{\partial x^{\alpha}}$ is a dense matrix. I'm having trouble finding an algorithm that converges to known solutions, for example, when $\alpha=2$ we should get $u(x)=sech(x).$ It needs to satisfy given initial conditions, $u(0)$ and $u'(0)$.

I view this as a root-finding problem and have tried Newton's method/secant method for systems, but the initial conditions are not satisfied because at each iteration the solution of $u$ is modified (in all elements of $u$). Working in Matlab I've also tried using Matlab library's root-finding algorithms but there is no way of enforcing these initial conditions either.

It seems I'm taking away from the problem by taking this trial and error approach. Is there a better way of going about it? I appreciate any assistance.



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