Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations

Chiron, D.

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1175-1230.

We consider the (KdV)/(KP-I) asymptotic regime for the nonlinear Schrödinger equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg–de Vries equation (in dimension 1) and to the Kadomtsev–Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/(KP-I) equation in the spirit of the work of Lannes and Saut, and then prove a comparison result with quantitative error estimates. For either suitable nonlinearities for (NLS) either a Landau–Lifshitz type equation, we derive a (mKdV)/(mKP-I) equation involving cubic nonlinearity. We then give a partial result justifying this asymptotic limit.

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