Solving non-linear differential equations

Question

When a nonlinear differential equation cannot be solved exactly, an approximate way is to consider the unknown function as a combination of its equilibrium value plus a small deviation from equilibrium value and then putting it back into the equation and neglecting the higher order terms. In many textbooks of physics this method is used. What is the underlying theory for this method ?

Suppose, I have a non-linear differential equation in an unknown variable Y. If the equation cannot be solved exactly , i have seen that people consider Y=Y0 + Y1 and put it back into the equation. They then neglect the terms like Y0*Y0 , Y1*Y1, Y1*grad(Y1) etc. as they consider them to be HIGHER ORDER. I understand that they are small quantities and hence can be neglected. But for me this whole scheme appears to be coming from nowhere. Is there any rigorous mathematical theory behind this ?

Answer 1

This is only the first step in studying a non-linear equation: finding its stationary points and studying their stability. Therefore using small deviations is justified, as we are interested here in the behavior near these stationary points.

Since I was asked to suggest the books, I will give here a few recommendation. As I mentioned in the comments, I read them in Russian, so I cannot guarantee the availability and/or the quality of their English translations. I am also pretty sure that there are excellent and more up-to-date books available.

• Oscillations and Waves in nonlinear systems by Rabinovich and Trubetskov is a standard university level textbook, written for physicists rather than mathematicians - something that may be crucial here.
• This book is written in English in the same spirit (by one of the authors), but significantly shortened.
• "Oscillations theory" by Andronov, Vitt and Khaikin is a classic written in Stalin era, so it saw some of its authors disappearing and reappearing on the cover over the course of the multiple editions. I am not aware whether it exists in translations. Oscillations theory is a Russian term for non-linear physics.
• George Zaslavsky has written multiple books on non-linear theory on different level - he was a big specialist in chaos. I think that Nonlinear physics is the introductory course.