Justification of the NLS approximation for a quasilinear water wave model

We show how for a quasilinear water wave model the NLS approximation can be justified. The model presents several new difficulties due to the quadratic terms which have to be eliminated by a normal-form transformation. Due to the quasilinearity of the problem there is some loss of regularity associated with the normal-form transformation and there is a nontrivial resonance present in the problem. The loss of regularity is dealt with by using a Cauchy–Kowalevskaya-like method to treat the initial value problem and the nontrivial resonance is dealt with via a rescaling argument.