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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.
Keywords: Nonlinear Schrödinger equation, Gross–Pitaevskii equation, Landau–Lifshitz equation, (Generalized) Korteweg–de Vries equation, (Generalized) Kadomtsev–Petviashvili equation, Weakly transverse Boussinesq system
@article{AIHPC_2014__31_6_1175_0,
author = {Chiron, D.},
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},
number = {6},
year = {2014},
doi = {10.1016/j.anihpc.2013.08.007},
mrnumber = {3280065},
zbl = {1307.35274},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.08.007/}
}
TY - JOUR AU - Chiron, D. TI - Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 1175 EP - 1230 VL - 31 IS - 6 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.08.007/ DO - 10.1016/j.anihpc.2013.08.007 LA - en ID - AIHPC_2014__31_6_1175_0 ER -
%0 Journal Article %A Chiron, D. %T Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 1175-1230 %V 31 %N 6 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.08.007/ %R 10.1016/j.anihpc.2013.08.007 %G en %F AIHPC_2014__31_6_1175_0
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
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