Normal form and solitons

We present a review of the normal form theory for weakly dispersive nonlinear wave equations where the leading order phenomena can be described by the KdV equation. This is an infinite-dimensional extension of the well-known Poincaré-Dulac normal form theory for ordinary differential equations. In particular, the normal form theory shows that the perturbed equations given by the KdV equation with higher order corrections are asymptotically integrable up to the first-order correction, and the first-order corrections can be transformed into a symmetry of the KdV equation called the fifth-order KdV equation. We then give the explicit conditions for the asymptotic integrability up to the third-order corrections. As an important example, we consider the Gardner-Miura transformation for the modified KdV equation and show that the inverse of the transformation is a normal form transformation. We also provide a detailed analysis of the interaction problem of solitary waves as an important application of the normal form theory. Several explicit examples are discussed based on the normal form theory, and the results are compared with their numerical simulations. Those examples include the ion acoustic wave equation, the shallow water wave equation and the Hirota bilinear equation having a seventh-order linear dispersion.