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

Description

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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