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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.

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