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SIAM J. NUMER. ANAL. c 2006 Society for Industrial and Applied Mathematics Vol. 43, No. 6, pp. 22722293
 

Summary: SIAM J. NUMER. ANAL. c 2006 Society for Industrial and Applied Mathematics
Vol. 43, No. 6, pp. 2272­2293
ARTIFICIAL BOUNDARY CONDITIONS FOR ONE-DIMENSIONAL
CUBIC NONLINEAR SCHR¨ODINGER EQUATIONS
XAVIER ANTOINE, CHRISTOPHE BESSE, AND ST´EPHANE DESCOMBES
Abstract. This paper addresses the construction of nonlinear integro-differential artificial
boundary conditions for one-dimensional nonlinear cubic Schr¨odinger equations. Several ways of
designing such conditions are provided and a theoretical classification of their accuracy is given.
Semidiscrete time schemes based on the method developed by Dur´an and Sanz-Serna [IMA J. Nu-
mer. Anal. 20 (2000), pp. 235­261] are derived for these unusual boundary conditions. Stability
results are stated and several numerical tests are performed to analyze the capacity of the proposed
approach.
Key words. nonlinear cubic Schr¨odinger equation, artificial boundary conditions, pseudodiffer-
ential operators, stable semidiscrete schemes, solitons interaction
AMS subject classifications. 35Q55, 35Q51, 47G30, 26A33, 65M12
DOI. 10.1137/040606983
1. Introduction. In many physical and technological domains of interest, the
numerical solution to a one-dimensional cubic nonlinear Schr¨odinger (NLS) equation
of the form
itu + 2

  

Source: Antoine, Xavier - Institut de Mathématiques Élie Cartan, Université Henri Poincaré - Nancy 1

 

Collections: Mathematics