no. 6
Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations
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.

DOI : 10.1016/j.anihpc.2013.08.007
Classification : 35Q55, 35Q53
Mots clés : Nonlinear Schrödinger equation, Gross–Pitaevskii equation, Landau–Lifshitz equation, (Generalized) Korteweg–de Vries equation, (Generalized) Kadomtsev–Petviashvili equation, Weakly transverse Boussinesq system 
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     title = {Error bounds for the {(KdV)/(KP-I)} and {(gKdV)/(gKP-I)} asymptotic regime for nonlinear {Schr\"odinger} type equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1175--1230},
     publisher = {Elsevier},
     volume = {31},
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Chiron, D. Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1175-1230. doi : 10.1016/j.anihpc.2013.08.007. http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.007/

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