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We consider the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases. We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result [35] beyond the regime of small solitary waves and extend the results of [19,29] from orbital to asymptotic stability for a suitable family of solitary waves. [ABSTRACT FROM AUTHOR]

We consider the initial-value problem for the one-dimensional cubic nonlinear Schrödinger equation with a repulsive delta potential. We prove that small initial data in a weighted Sobolev space lead to global solutions that decay in |$L^{\infty }$| and exhibit modified scattering. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential (δ-NLS). We shall prove that for a given small asymptotic profileuap, there exists a solutionuto (δ-NLS) which converges touapinL2(ℝ) ast → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the initial value problem of a specific system of cubic nonlinear Schrödinger equations. Our aim of this research is to specify the asymptotic profile of the solution in L ∞ as t → ∞. It is then revealed that the solution decays slower than a linear solution does. Further, the difference of the decay rate is a polynomial order. This deceleration of the decay is due to an amplification effect by the nonlinearity. This nonlinear amplification phenomena was previously known for several specific systems, however the deceleration of the decay in these results was by a logarithmic order. As far as we know, the system studied in this paper is the first model in that the deceleration in a polynomial order is justified. [ABSTRACT FROM AUTHOR]