# Instability of coherent states of a real scalar field

## Abstract

We investigate stability of both localized time-periodic coherent states (pulsons) and uniformly distributed coherent states (oscillating condensate) of a real scalar field satisfying the Klein-Gordon equation with a logarithmic nonlinearity. The linear analysis of time-dependent parts of perturbations leads to the Hill equation with a singular coefficient. To evaluate the characteristic exponent we extend the Lindemann-Stieltjes method, usually applied to the Mathieu and Lame equations, to the case that the periodic coefficient in the general Hill equation is an unbounded function of time. As a result, we derive the formula for the characteristic exponent and calculate the stability-instability chart. Then we analyze the spatial structure of the perturbations. Using these results we show that the pulsons of any amplitudes, remaining well-localized objects, lose their coherence with time. This means that, strictly speaking, all pulsons of the model considered are unstable. Nevertheless, for the nodeless pulsons the rate of the coherence breaking in narrow ranges of amplitudes is found to be very small, so that such pulsons can be long-lived. Further, we use the obtained stability-instability chart to examine the Affleck-Dine-type condensate. We conclude the oscillating condensate can decay into an ensemble of the nodeless pulsons.

- Authors:

- Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation of the Russian Academy of Sciences (IZMIRAN), Troitsk, Moscow Region, 142190 (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 20768725

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 2; Other Information: DOI: 10.1063/1.2167918; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AMPLITUDES; ANNIHILATION OPERATORS; EIGENSTATES; HILL EQUATION; INSTABILITY; KLEIN-GORDON EQUATION; NONLINEAR PROBLEMS; PERIODICITY; PERTURBATION THEORY; SCALAR FIELDS; SOLITONS; STABILITY; TIME DEPENDENCE

### Citation Formats

```
Koutvitsky, Vladimir A., and Maslov, Eugene M..
```*Instability of coherent states of a real scalar field*. United States: N. p., 2006.
Web. doi:10.1063/1.2167918.

```
Koutvitsky, Vladimir A., & Maslov, Eugene M..
```*Instability of coherent states of a real scalar field*. United States. doi:10.1063/1.2167918.

```
Koutvitsky, Vladimir A., and Maslov, Eugene M.. Wed .
"Instability of coherent states of a real scalar field". United States.
doi:10.1063/1.2167918.
```

```
@article{osti_20768725,
```

title = {Instability of coherent states of a real scalar field},

author = {Koutvitsky, Vladimir A. and Maslov, Eugene M.},

abstractNote = {We investigate stability of both localized time-periodic coherent states (pulsons) and uniformly distributed coherent states (oscillating condensate) of a real scalar field satisfying the Klein-Gordon equation with a logarithmic nonlinearity. The linear analysis of time-dependent parts of perturbations leads to the Hill equation with a singular coefficient. To evaluate the characteristic exponent we extend the Lindemann-Stieltjes method, usually applied to the Mathieu and Lame equations, to the case that the periodic coefficient in the general Hill equation is an unbounded function of time. As a result, we derive the formula for the characteristic exponent and calculate the stability-instability chart. Then we analyze the spatial structure of the perturbations. Using these results we show that the pulsons of any amplitudes, remaining well-localized objects, lose their coherence with time. This means that, strictly speaking, all pulsons of the model considered are unstable. Nevertheless, for the nodeless pulsons the rate of the coherence breaking in narrow ranges of amplitudes is found to be very small, so that such pulsons can be long-lived. Further, we use the obtained stability-instability chart to examine the Affleck-Dine-type condensate. We conclude the oscillating condensate can decay into an ensemble of the nodeless pulsons.},

doi = {10.1063/1.2167918},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 47,

place = {United States},

year = {Wed Feb 15 00:00:00 EST 2006},

month = {Wed Feb 15 00:00:00 EST 2006}

}