# Small solutions of nonlinear Schr\"odinger equations near first excited states

@article{Nakanishi2010SmallSO,
title={Small solutions of nonlinear Schr\"odinger equations near first excited states},
author={Kenji Nakanishi and Tuoc Van Phan and Tai-Peng Tsai},
journal={arXiv: Analysis of PDEs},
year={2010}
}
• Published 20 August 2010
• Mathematics
• arXiv: Analysis of PDEs

## Figures from this paper

### Global dynamics below excited solitons for the nonlinear Schr\"odinger equation with a potential

Consider the nonlinear Schr\"odinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the

### Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations

• Mathematics
SIAM J. Math. Anal.
• 2011
Certain finite-codimension regions of the phase space are constructed consisting of solutions converging to these excited states at time infinity ("stable directions").

### On instability of some approximate periodic solutions for the full nonlinear Schr

• Mathematics
• 2011
Using the Fermi Golden Rule analysis developed in several results by the first author, we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a

### On Instability for the Quintic Nonlinear Schrödinger Equation of Some Approximate Periodic Solutions S

Using the Fermi Golden Rule analysis developed in [CM], we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a quintic nonlinear Schrödinger

### On instability for the quintic nonlinear Schrodinger equation of some approximate periodic solutions

• Mathematics
• 2012
Using the Fermi Golden Rule analysis developed in (CM), we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a quintic nonlinear Schrodinger

### A Resonance Problem in Relaxation of Ground States of Nonlinear Schrodinger Equations

In this paper we consider a resonance problem, in a generic regime, in the con- sideration of relaxation of ground states of semilinear Schrodinger equations. Different from previous results, our

### LOCAL DYNAMICS NEAR UNSTABLE BRANCHES OF NLS SOLITONS

• Mathematics, Physics
• 2013
Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions,

### Adiabatic theorem for the Gross–Pitaevskii equation

• Mathematics
• 2015
ABSTRACT We prove an adiabatic theorem for the nonautonomous semilinear Gross–Pitaevskii equation. More precisely, we assume that the external potential decays suitably at infinity and the linear

### On Selection of Standing Wave at Small Energy in the 1D Cubic Schrödinger Equation with a Trapping Potential

• Mathematics
Communications in Mathematical Physics
• 2022
Combining virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk, Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear induced dissipation on discrete modes, and in

## References

SHOWING 1-10 OF 35 REFERENCES

### Relaxation of excited states in nonlinear Schrödinger equations

• Mathematics
• 2001
We consider a nonlinear Schrodinger equation in $\R^3$ with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance

### STABLE DIRECTIONS FOR EXCITED STATES OF NONLINEAR SCHRÖDINGER EQUATIONS

• Mathematics
• 2001
ABSTRACT We consider nonlinear Schrödinger equations in . Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct

### Classification of Asymptotic Profiles for Nonlinear Schrodinger Equations with Small Initial Data

• Mathematics
• 2002
We consider a nonlinear Schrodinger equation with a bounded lo- cal potential in R 3 . The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance

### Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations

• Mathematics
SIAM J. Math. Anal.
• 2011
Certain finite-codimension regions of the phase space are constructed consisting of solutions converging to these excited states at time infinity ("stable directions").

### On Asymptotic Stability in Energy Space of Ground States for Nonlinear Schrödinger Equations

• Mathematics
• 2007
AbstractWe consider nonlinear Schrödinger equations $$iu_t +\Delta u +\beta (|u|^2)u=0\, ,\, \text{for} (t,x)\in \mathbb{R}\times \mathbb{R}^d,$$ where d ≥ 3 and β is smooth. We prove that symmetric

### Asymptotic dynamics of nonlinear Schrödinger equations: Resonance‐dominated and dispersion‐dominated solutions

• Mathematics
• 2002
We consider a linear Schrödinger equation with a nonlinear perturbation in ℝ3. Assume that the linear Hamiltonian has exactly two bound states and its eigen‐values satisfy some resonance condition.

### Selection of the ground state for nonlinear schrödinger equations

• Mathematics
• 2004
We prove for a class of nonlinear Schrodinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic

### Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

• Physics, Mathematics
• 1999
Abstract. We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a

### Dynamics of Nonlinear Schrodinger / Gross-Pitaevskii Equations; Mass Transfer in Systems with Solitons and Degenerate Neutral Modes

• Physics
• 2008
Nonlinear Schrodinger / Gross-Pitaevskii equations play a central role in the understanding of nonlinear optical and macroscopic quantum systems. The large time dynamics of such systems is governed

### Asymptotic stability of small solitons for 2D Nonlinear Schr

We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schr\"{o}dinger equations  iu_t+\Delta u=Vu\pm |u|^{p-1}u \quad\text{for