On a class of solutions of KdV

Practically every book on the Inverse Scattering Transform method for solving the Cauchy problem for KdV and other integrable systems refers to this method as nonlinear Fourier transform. If this is indeed so, the method should lead to a nonlinear analogue of the Fourier expansion formula u(t,x)=∫-∞+∞u^(k)eikx-ω(k)tdk. In this paper a special class of solutions of KdV whose role is similar to that of eikx-ω(k)t is discussed. The theory of these solutions, referred to here as harmonic breathers, is developed and it is shown that these solutions may be used to construct more general solutions of KdV similarly to how the functions eikx-ω(t) are used to perform the same task in the theory of Fourier transform. A nonlinear superposition formula for general solutions of KdV similar to the Fourier expansion formula is conjectured.