Is it impossible to make a general theory of nonlinear partial differential equations? Isee them everywhere, from plasma confinement tonjosephson junctions, and im curious if you can derive those named equations from a Universal theory. I see most of the millenium prize probelems dealing with them too, so im curious if there is a profound nature behind understanding what a solution is in a general way.
Not many differential equations are explicitly solvable, OP. Often times you have to settle for a description of qualitative behavior as opposed to an explicit solution. Any "general theory of nonlinear partial differential equations" would have to be so general that it is in practice virtually unusable.
I've seen a rigorous proof for the unsolvability of 5th order polynomials, I have not seen a proof for non-linear PDE classes. We have exact analytical solutions to many non-linear PDEs, a statement like "not many differential equations are explicitly solvable" is extremely premature.
it's not premature seeing as PDEs have been of great consideration for nearly as long as ODEs--and the former theory still has a great deal of trouble in finding exact solutions. His statement is from experience, there simply isn't a general approach to ODEs. There is great numerical approach to linear ODEs, but these still usually do not have a nice closed form expression (again, in practice). Non-linear ODEs can be tricky as you don't have a large toolset to approach them. PDEs increase the difficulty in that your solution space gets more complex. The nature of PDEs is finicky, which is not a precise statement but I mean to describe the relative ease in which even simple looking PDEs can not only lack a closed form solution, but lack any solution entirely.
Rather than complain on an anonymous imageboard, you should ask yourself 'what do I mean by a general PDE?' and work from there to see what troubles you might encounter. A great deal of mathematics from Lie Algebras to Differential Geometry and the more structured spaces studied in Functional Analysis are motivated by alternative perspectives on solving PDEs.