How Can I Fix Branch Cut Issues in a Hamiltonian System with NDSolve?

[Grandpa]

So I'm trying to solve the following Hamiltonian system using Mathematica.

solution = NDSolve[{x'[t] == 2p[t], x[0] == 2,
p'[t] == I*(2 + 1/2)(I*x[t])^(1 + 1/2), p[0] == 1-2*I}, {x, p}, {t,0,10}, MaxSteps -> Infinity][[1]];

I'm letting E=1, so at all points t, it should be that
(p[t]/.solution)^2-(I*x[t]/.solution)^(2+1/2)=1.

That is the case until the function crosses a branch cut on the complex x-plane that runs from 0 to i*(Infinity). Once the function crosses the branch cut, the system no longer preserves E and the value changes to something around 1.3.

How can I go about fixing this problem? I've already tried working with the precision and accuracy goals, and that didn't work. I also tried the SymplecticPartitionedRungeKutta method, and that didn't work either. I'm not sure what else to consider. Any help would be appreciated.

I was thinking about having the program just continue in the same direction when x[t] gets really near the branch cut, but I'm not sure what to use to program that.

Thanks,
Alex

 

[jvicens]

it is important to carefully analyze the problem and understand the underlying physics before attempting to solve it computationally. In this case, it seems that the Hamiltonian system you are trying to solve has a singularity at the branch cut on the complex x-plane. This means that the system does not have a well-defined solution at that point, and any numerical method will struggle to accurately solve it.

One approach to address this issue is to modify the Hamiltonian system itself. This can be done by introducing a regularization parameter, which essentially smooths out the singularity and allows for a well-defined solution to be obtained. This approach is commonly used in the numerical solution of singular Hamiltonian systems.

Another option is to use a different numerical method that is specifically designed to handle singularities. One such method is the "symplectic integrator," which is known to preserve the Hamiltonian structure of the system and can handle singularities more accurately than standard methods.

It is also worth considering the physical implications of the branch cut in your system. Is it a physical singularity, or just a mathematical artifact? Depending on the answer, it may be possible to reformulate the problem in a different way that avoids the singularity altogether.

In summary, there are a few different approaches you can try to address the issue of the branch cut in your Hamiltonian system. It may also be helpful to consult with other researchers in the field or seek guidance from experts in numerical methods to find the best solution for your specific problem. Good luck!