# Peakompactons: Peaked compact nonlinear waves

## Abstract

This paper is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of nonstandard solitary waves termed peakompactons. We present that these peaked compactly supported waves arise as solutions to nonlinear evolution equations from a hierarchy of nonlinearly dispersive Korteweg–de Vries-type models. Peakompactons, like the now-well-known compactons and unlike the soliton solutions of the Korteweg–de Vries equation, have finite support, i.e., they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, i.e., a higher spatial derivative suffers a jump discontinuity at the wave’s crest. Here, we construct such solutions exactly by reducing the governing partial differential equation to a nonlinear ordinary differential equation and employing a phase-plane analysis. Lastly, a simple, but reliable, finite-difference scheme is also designed and tested for the simulation of collisions of peakompactons. In addition to the peakompacton class of solutions, the general physical features of the so-called K ^{#}(n,m) hierarchy of nonlinearly dispersive Korteweg–de Vries-type models are discussed as well.

- Authors:

- Purdue Univ., West Lafayette, IN (United States)
- Univ. of North Carolina, Chapel Hill, NC (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1356135

- Report Number(s):
- LA-UR-16-27517

Journal ID: ISSN 0217-9792

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- International Journal of Modern Physics B

- Additional Journal Information:
- Journal Volume: 31; Journal Issue: 10; Journal ID: ISSN 0217-9792

- Publisher:
- World Scientific

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS AND COMPUTING; Mathematics

### Citation Formats

```
Christov, Ivan C., Kress, Tyler, and Saxena, Avadh.
```*Peakompactons: Peaked compact nonlinear waves*. United States: N. p., 2017.
Web. doi:10.1142/S0217979217420085.

```
Christov, Ivan C., Kress, Tyler, & Saxena, Avadh.
```*Peakompactons: Peaked compact nonlinear waves*. United States. doi:10.1142/S0217979217420085.

```
Christov, Ivan C., Kress, Tyler, and Saxena, Avadh. Thu .
"Peakompactons: Peaked compact nonlinear waves". United States.
doi:10.1142/S0217979217420085. https://www.osti.gov/servlets/purl/1356135.
```

```
@article{osti_1356135,
```

title = {Peakompactons: Peaked compact nonlinear waves},

author = {Christov, Ivan C. and Kress, Tyler and Saxena, Avadh},

abstractNote = {This paper is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of nonstandard solitary waves termed peakompactons. We present that these peaked compactly supported waves arise as solutions to nonlinear evolution equations from a hierarchy of nonlinearly dispersive Korteweg–de Vries-type models. Peakompactons, like the now-well-known compactons and unlike the soliton solutions of the Korteweg–de Vries equation, have finite support, i.e., they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, i.e., a higher spatial derivative suffers a jump discontinuity at the wave’s crest. Here, we construct such solutions exactly by reducing the governing partial differential equation to a nonlinear ordinary differential equation and employing a phase-plane analysis. Lastly, a simple, but reliable, finite-difference scheme is also designed and tested for the simulation of collisions of peakompactons. In addition to the peakompacton class of solutions, the general physical features of the so-called K#(n,m) hierarchy of nonlinearly dispersive Korteweg–de Vries-type models are discussed as well.},

doi = {10.1142/S0217979217420085},

journal = {International Journal of Modern Physics B},

number = 10,

volume = 31,

place = {United States},

year = {Thu Apr 20 00:00:00 EDT 2017},

month = {Thu Apr 20 00:00:00 EDT 2017}

}