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, Volume 31 (2014) no. 6, p. 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 : https://doi.org/10.1016/j.anihpc.2013.08.007
Classification:  35Q55,  35Q53
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},
publisher = {Elsevier},
volume = {31},
number = {6},
year = {2014},
pages = {1175-1230},
doi = {10.1016/j.anihpc.2013.08.007},
zbl = {1307.35274},
mrnumber = {3280065},
language = {en},
url = {http://www.numdam.org/item/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, Volume 31 (2014) no. 6, pp. 1175-1230. doi : 10.1016/j.anihpc.2013.08.007. http://www.numdam.org/item/AIHPC_2014__31_6_1175_0/

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