How can the following equations for meandering rivers be theoretically obtained?
I'm gonna hazard a guess here: They can't. While I don't have any evidence for this claim, it seems likely that there are too many small-scale non-linear processes feeding in to the overall generation process to ever be able to sensibly derive any kind of analytical solution based on physical first-principles. I think that's what you're asking.

So, basically, these equations are an empirical model. The parameters don't have any relation to the physical processes that generated the meanders - they are just descriptions of the resultant shape. The generation process for a model like this is something along the lines of:


Inspect the data qualitatively
Come up with some potential models that could approximate what you're seeing. In this instance, sinusoidal and sequentially linked circles look like they might work.
For each model, apply and verify:

Attempt to fit the model to some of the data.
Apply the model to some of the rest of the data, and check it's performance (also, pay attention to parsimony).

Choose the model with the best performance as your model.
With your model, perform some analysis of the model fit (residual analysis, etc). If it looks like there are patterns in the residuals that could be reasonably modelled, go back to 2. and come up with some model variants.


As far as I can tell from the referenced paper, this is roughly the process that the authors used. The values of each of the parameters are basically empirically obtained, to allow the model to best fit the majority of the data.