A nonlinear partial differential equation is an expression depending on derivatives of $u$ $$f(x,t,u,u_x,u_t,\cdots)=0,$$ where the derivatives of $u$ can be obtained from the Taylor series of $u$.
There is another expansion of $u$, that is the Fourier expansion, and the Fourier transform can be used in linear partial differential equations.
Now, my question is:
Is there any nonlinear equation depending on Fourier coefficients?
These nonlinear equations are not derived from Fourier transform of linear equations, but from the real physics setting.