# Nonlinear, stationary electrostatic ion cyclotron waves: Exact solutions for solitons, periodic waves, and wedge shaped waveforms

## Abstract

The theory of fully nonlinear stationary electrostatic ion cyclotron waves is further developed. The existence of two fundamental constants of motion; namely, momentum flux density parallel to the background magnetic field and energy density, facilitates the reduction of the wave structure equation to a first order differential equation. For subsonic waves propagating sufficiently obliquely to the magnetic field, soliton solutions can be constructed. Importantly, analytic expressions for the amplitude of the soliton show that it increases with decreasing wave Mach number and with increasing obliquity to the magnetic field. In the subsonic, quasi-parallel case, periodic waves exist whose compressive and rarefactive amplitudes are asymmetric about the 'initial' point. A critical 'driver' field exists that gives rise to a soliton-like structure which corresponds to infinite wavelength. If the wave speed is supersonic, periodic waves may also be constructed. The aforementioned asymmetry in the waveform arises from the flow being driven towards the local sonic point in the compressive phase and away from it in the rarefactive phase. As the initial driver field approaches the critical value, the end point of the compressive phase becomes sonic and the waveform develops a wedge shape. This feature and the amplitudes of the compressive andmore »

- Authors:

- Department of Mathematics, Statistics and Physics, Durban University of Technology, Steve Biko Campus, Durban 4001 (South Africa)
- (South Africa)
- Materials Research Division, iThemba LABS, P.O.Box 722, Somerset West, 7129, South Africa and School of Chemistry and Physics, University of KwaZulu-Natal, Private Bag: X54001, Durban 4001 (South Africa)

- Publication Date:

- OSTI Identifier:
- 22068880

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 19; Journal Issue: 11; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1070-664X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; AMPLITUDES; ASYMMETRY; CORIOLIS FORCE; DIFFERENTIAL EQUATIONS; ENERGY DENSITY; EXACT SOLUTIONS; FLUX DENSITY; FUNDAMENTAL CONSTANTS; ION PLASMA WAVES; MACH NUMBER; MAGNETIC FIELDS; NONLINEAR PROBLEMS; PERIODICITY; SOLITONS; SUBSONIC FLOW; SUPERSONIC FLOW; WATER WAVES; WAVE FORMS; WAVELENGTHS

### Citation Formats

```
McKenzie, J. F., School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag: X54001, Durban 4001, Doyle, T. B., and Rajah, S. S.
```*Nonlinear, stationary electrostatic ion cyclotron waves: Exact solutions for solitons, periodic waves, and wedge shaped waveforms*. United States: N. p., 2012.
Web. doi:10.1063/1.4769031.

```
McKenzie, J. F., School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag: X54001, Durban 4001, Doyle, T. B., & Rajah, S. S.
```*Nonlinear, stationary electrostatic ion cyclotron waves: Exact solutions for solitons, periodic waves, and wedge shaped waveforms*. United States. doi:10.1063/1.4769031.

```
McKenzie, J. F., School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag: X54001, Durban 4001, Doyle, T. B., and Rajah, S. S. Thu .
"Nonlinear, stationary electrostatic ion cyclotron waves: Exact solutions for solitons, periodic waves, and wedge shaped waveforms". United States. doi:10.1063/1.4769031.
```

```
@article{osti_22068880,
```

title = {Nonlinear, stationary electrostatic ion cyclotron waves: Exact solutions for solitons, periodic waves, and wedge shaped waveforms},

author = {McKenzie, J. F. and School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag: X54001, Durban 4001 and Doyle, T. B. and Rajah, S. S.},

abstractNote = {The theory of fully nonlinear stationary electrostatic ion cyclotron waves is further developed. The existence of two fundamental constants of motion; namely, momentum flux density parallel to the background magnetic field and energy density, facilitates the reduction of the wave structure equation to a first order differential equation. For subsonic waves propagating sufficiently obliquely to the magnetic field, soliton solutions can be constructed. Importantly, analytic expressions for the amplitude of the soliton show that it increases with decreasing wave Mach number and with increasing obliquity to the magnetic field. In the subsonic, quasi-parallel case, periodic waves exist whose compressive and rarefactive amplitudes are asymmetric about the 'initial' point. A critical 'driver' field exists that gives rise to a soliton-like structure which corresponds to infinite wavelength. If the wave speed is supersonic, periodic waves may also be constructed. The aforementioned asymmetry in the waveform arises from the flow being driven towards the local sonic point in the compressive phase and away from it in the rarefactive phase. As the initial driver field approaches the critical value, the end point of the compressive phase becomes sonic and the waveform develops a wedge shape. This feature and the amplitudes of the compressive and rarefactive portions of the periodic waves are illustrated through new analytic expressions that follow from the equilibrium points of a wave structure equation which includes a driver field. These expressions are illustrated with figures that illuminate the nature of the solitons. The presently described wedge-shaped waveforms also occur in water waves, for similar 'transonic' reasons, when a Coriolis force is included.},

doi = {10.1063/1.4769031},

journal = {Physics of Plasmas},

issn = {1070-664X},

number = 11,

volume = 19,

place = {United States},

year = {2012},

month = {11}

}