are nonlinear partial differential equations the hardest form

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i never really thought about it before but PDEs does seem like a really "disjoint" field of study, more like a blanket term for a bunch of crazy-ass mathematical magic tricks

>>8678834
it really does seem that way. is there just no way to look at it as an underlying theory like how functions and their derivatives/integrals have been slapped together under the fundamental theorem of calculus? do we need a fundamental theorem of ᚠᚢᚦᚨᚱᚲ that will change the way we live forever?

>>8678825
lookup numerical analysis, its usually the last class in an engineering degree and it pretty much says "heres a computer that will solve all those de/pde youve been memorizing loool" it teaches how to 'find' the equation you are looking for solution to solve (the computer solves it with numbers plugged in, since there is no "general" solution ever )

>>8679774
aren't those just approximations like newtons method? i mean perelman used NPDEs to solve the poincaire conjecture so it can't be just as easy as plugging it in, otherwise why didn't someone do that and make a million bucks?

>>8679788
Yeah it is shit like Newton's method or 4th order Runge-Kutta.
Problem is that they don't always converge.
Even simple shit like Euler flow equations can shit their pants with non-trivial geometry of walls.

>>8679795
oh wow really? I didn't think that would actually be the case i was just spitting hot air. would understanding scalar fields help solve them?

>>8678825
When it comes to nonlinear PDEs, people do in fact spend their entire career studying one equation (with appropriate generalizations). It can certainly compose an entire thesis.

They are solved using a wide variety of techniques. I'd have to say that bifurcation theory comes up quite often, and can be a very powerful tool for showing nontrivial solutions to a wide variety of nonlinear PDE problems.

This is my field. Nonlinear PDEs are great. I like seeing all the crazy shit that can happen, and I like reaching for a wide variety of tools (including functional analysis, bifurcation theory) to try to tackle them.

>>8679830
oh i've seen bifurcation theory pop up before, i have a few books on it. what would you reccomend as essential reading for someone who wants to delve into the world of NPDEs? Spectral Theory? inverse scattering method? Toda Lattices? it seems so massive of a field that doesn't really give you a guide on how to approach it.

PDEs = god tier math

>>8679842
I like Kielhöfer's bifurcation theory book. As for spectral theory; I don't really have any recommendations - I just had it from previous background. I'm not familiar with the other techniques you listed. As you said, it's a huge field, and there are so many tools out there to try. I wouldn't say there is an over-arching guide.

Another "technique" is a very brute-force perturbation series method that's used a lot. I think Stokes used it himself for his water waves, and it can be made rigorous. It might be interesting to read up on these techniques. I don't know where it can be found in books (I'm sure it can - just search "perturbation theory") - but it can be learned by reading others' papers and seeing the techniques in action.

>>8678825
Source of image tho ?

>>8678825

Yes. I studied linear PDE, it was hard as fuck, and now the non linear...

http://mathoverflow.net/questions/15292/why-cant-there-be-a-general-theory-of-nonlinear-pde

>>8680338
>To elaborate on Steve Huntsman's comment, I remember reading the following on Terence Tao's blog: there exist PDE that can simulate Newtonian mechanics, and using such a PDE and the correct initial conditions it is possible, in principle, to simulate an arbitrary analog Turing machine. So a general-purpose algorithm to determine even the qualitative behavior of an arbitrary PDE cannot exist because such an algorithm could be used to solve the halting problem.
kek

>>8680393
Please no there are thousands of pdes surely they cam be reduced right?

>>8678825
>is there an inherent powergrap to understanding just even one equation that it would be considered worthy of a phd thesis and an entire sub-field modelling nature?
Yes. You really shouldn't think of PDE as a single field. It has a variety of domains dealing with widely different objects and using widely different techniques

SOURCE OF IMAGE ? ? ? ?

>>8678841
There is underlying theory. Have you studied any analysis, modern algebra or linear algebra?

Just use numerical methods to solve them and make tons of money like the engineers do.

So what do I need to study starting now to solve the 7 Millennium Problems?

Fuck telling me it's impossible

>>8681843
algebraic geomtry