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Semiclassical Nonlinear Schrödinger equations with potential and focusing initial data

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Carles, R. and Miller, L. Osaka J. Math. 41 (2004), 693–725 SEMICLASSICAL NONLINEAR SCHR ¨ODINGER EQUATIONS WITH POTENTIAL AND FOCUSING INITIAL DATA R ´EMI CARLES and LUC MILLER (Received January 20, 2003) 1. Introduction We study the semi-classical limit ε → 0 of solutions uε : ( ) ∈ R × R → C of the equation ε∂ uε + 1 2 ε2 uε = ( )uε + λ|uε|2σuε where λ > 0 (the nonlinearity is repulsive), with concentrating initial data uε(0 ) = ( − 0 ε ) ( ·η0/ε) Similar problems were studied for attractive nonlinearities (λ < 0), by Bronski and Jerrard ([1]), and Keraani ([15]). In that case, if the power is 2-subcritical (σ < 2/ ) and is the ground state solution of an associated scalar elliptic equation, then when is smooth with ∈ 2 ∞, the following asymptotics holds in := ∞loc(R; 2(R )), 1 ε /2 ∥∥∥∥uε( )− ( − ( ) + ε ε( )ε ) ( ·η( )/ε)+ θε( ) ∥∥∥∥ = (√ε ) 1 ε /2 ∥∥∥∥ε∇ (uε( )− ( − ( ) + ε ε( )ε ) ( ·η( )/ε)+ θε( ) )∥∥∥∥ = (√ε )(1.1) where θε( ) ∈ [0 2π[, ε : R → R is locally uniformly bounded and ( ( ) η( )) are the integral curves associated to the classical Hamiltonian (1.2) ( τ η) = τ + 1 2 |η|2 + ( ) with initial data ( 0 η0). 2000 Mathematics Subject Classification : 35B40, 35Q55, 81Q20, 35P25. This work was partially supported by the ACI grant “ ´Equation des ondes : oscillations, dispersion et controˆle”. These results were improved while the first author was in the University of Osaka, invited by N. Hayashi, to whom he wishes to express his gratitude. 694 R. CARLES AND L. MILLER In this paper, we address the case of a defocusing nonlinearity (λ > 0), when the potential is a polynomial of degree at most two. In the case λ > 0, a different qualitative behaviour is expected. Intuitively, disper- sive effects prevent the solution from keeping a concentrating aspect as in (1.1), for it is well known (see e.g. [5]) that the solutions to the nonlinear Schro¨dinger equation (1.3) ∂ ψ + 1 2 ψ = |ψ|2σψ have the same dispersive properties as the s

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