The idea that nature is described by a nonlinear system of equations was the idea that Einstein had in the 1920s, and motivated his search for a unified field theory. It doesn't work, and it's philosophically less worthwhile than current theories anyway, so even if it did work, it wouldn't be simpler than string theory, or as elegant.
The idea that you can describe what's going on with local equations is false, as is demonstrated conclusively by Bell's inequality violations. The Bell inequality tells you that you can send electrons to far-away locations with spins that can be measured in 3 directions, A,B,C. The spin of the two electrons in each direction are 100% correlated (it's actually anti-correlated, but same difference for the argument), so if you measure the spin in direction A, and one electron is up the other is 100% certain to be up. Same for direction B and C, the two electrons always report the same spin in any of the three directions.
The spin in directions A and B are 99% correlated, meaning if you measure A on one of the electrons is up, then B on the other electron is up 99% of the time, and B is down 1% of the time. The spin in directions B and C are 99% correlated, so if you measure B is down on one electron, C is up on the other electron 1% of the time.
From the 100% correlation of the electrons, you conclude that the nonlocal field state (hidden variable) on one electron has the property that
A and B are 99% the same, 1% different
B and C are 99% the same, 1% different
From this you deduce that
A and C must be at least 98% the same
meaning that whatever field configuration is happening to make A, the field configuration for C can only give different results 2% of the time, the sum of those times when it gives different answers than B plus the times B gives different answers than A.
This bound is called Bell's inequality, and it is violated by quantum mechanics. A and C are different 96% of the time.
This means any type of local-in-space description, linear, nonlinear, complicated, simple, whatever, will never ever work to describe nature. Your description is either nonlocal in the sense of faster-than-light communication, or nonlocal in the sense of having a global notion of state which is entangled nonlocally by measurements. This is why nobody looks for nonlinear field equations to describe nature anymore. It can't possibly work.
But the main ideas of Einstein's nonlinear field theories have survived to inspire developments in later physics.
- The pions are excitations of a sigma-model, which is a type of nonlinear field theory. They are small oscillations of the quark condensate in the vacuum.
- The proton can be thought of as the topological soliton of the sigma model. In quantum mechanics, it can still be a fermion even though the sigma model has no fundamental fermionic variables.
- The field equations of 11-dimensional supergravity, which are a central part of string theory, generalize General Relativity in pretty much the only nontrivial ways known--- they give the biggest extension of spacetime symmetry possible, and they include a new field, constrained by the supersymmetry.
So these ideas are not a dead end, but they cannot work without quantum mechanics by themselves. If you want to understand quantum behavior emerging from some sort of nonlinear dynamics underneath, this dynamics can't be local.