open access

https://doi.org/10.14943/83851

https://hdl.handle.net/2115/69505

Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities

英語

2004

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

700

1-18

The effect of inhomogenity of nonlinear medium is discussed concerning the stability of standing waves eiωtφω(x) for a nonlinear Schrödinger equation with an inhomogeneous nonlinearity V (x)|u|p-1u, where V(x) is proportional to the electron density. Here, ω > 0 and φω(x) is a ground state of the stationary problem. When V(x) behaves like |x|-b at in nity, where 0 < b < 2, we show that eiωtφω(x) is stable for p < 1 + (4 - 2b)=n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x|-b. Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.

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