# Is there any nonlinear equations depending on Fourier coefficients?

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.