The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e i ω t φω(x) for a nonlinear Schrödinger equation with an inhomogeneous nonlinearity V (x)|u| p - 1 u, 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 infinity, where 0 < b < 2, we show that e i ω t φ ω(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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics