I am looking for suggestions on a good introductory book to nonlinear PDEs and the ideas of phase transitions or pattern formation.
My background in PDEs is mostly self-taught. So I worked through Haberman's intro book on analytical PDEs, but I found a lot more use from Leveque's book on numerical solutions to PDEs--both the original finite difference book and the subsequent book on hyperbolic conservation laws and finite volume methods. I also have some exposure to the ideas from computational fluid dynamics as well, though not too much. Finally, I did read through Strogatz's book on nonlinear ODEs, so I understand how bifurcations work in ODEs.
I have been reading some papers that take agent based model and convert them to PDEs using continuum mechanics. The ideas are pretty understandable--basically they perform the opposite of the usual pde discretization. In a PDE discretization you take a continuous object and make it discrete by assigning some solution points on a grid or mesh. When converting an ABM to a PDE, you start with the mesh and knot points and take the limit in the spatial and time domains to recover the PDE. In the case of the referenced paper, the PDE is a Reaction-Diffusion model that looks similar to models of chemotaxis.
The part that I have trouble with is when the papers start to discuss different parameter regimes and order parameters, etc.--in a PDE context. By taking the ABM and converting it to a PDE, the authors can then examine those same systems using PDE tools that may not work in the discrete context. I have read the Schroeder book on "Thermal Physics," so I have a sense of phase transitions and order parameters in statistical mechnanics, but I don't have a sense of how these tools are applied to PDE problems. In other words, how do mathematicians usually find the bifurcation points of nonlinear PDEs, and how do they look at phase transition in PDEs. Is this usually done numerically, though trial and error? Or are there nicer methods like studing Lyapunov stability analysis, or something like Von Neumann analysis to find regions where the solution changes its stability properties, etc.
Any references would be appreciated.