I study Bethe ansatz, although my background is mathematics not physics. Can somebody explain to me what is plane waves? I have seen in many papers this expression that "The idea of the Bethe ansatz is that eigenfunctions are a linear combination of plane waves", or superposition of plane waves.
My understanding is for each particle a complex number which is called plane wave has been defined! then an eigenfunction is a combination of these complex numbers over all possible permutations. But why is such a complex number for each particle defined, and why is it called a plane wave?