### Abstract

In this paper we consider the problem of constructing solutions of several well known nonlinear partial differential equations (p.d.e.s) in phase space (i.e,. the Fourier transform domain). We seek solutions representing travelling focussed pulses. As such, based on a technique used to construct such solutions (so called Localized Wave solutions) of linear p.d.e.s, we look for phase space solutions constisting of a generalized funtion whose support is a particular line or surface, together with a suitable weighting function. The support of the phase space solutions must be such that it regenerates itself after the appropriate nonlinear operation. In one spatial dimension we construct the usual well known soliton solutions of several equations. For the case of higher spatial dimensions we construct a travelling "slab" pulse solution of the nonlinear Schrödinger equation. We also discuss some issues involved with the extra freedom one has for the phase space support, leading perhaps to more exotic spacetime domain solutions.

Original language | English (US) |
---|---|

Pages (from-to) | 115-123 |

Number of pages | 9 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 78 |

Issue number | 1-2 |

DOIs | |

State | Published - Nov 1 1994 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*78*(1-2), 115-123. https://doi.org/10.1016/0167-2789(94)00121-9

**Soluitions of nonlinear partial differential equations in phase space.** / Donnelly, R.; Ziolkowski, Richard W.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 78, no. 1-2, pp. 115-123. https://doi.org/10.1016/0167-2789(94)00121-9

}

TY - JOUR

T1 - Soluitions of nonlinear partial differential equations in phase space

AU - Donnelly, R.

AU - Ziolkowski, Richard W

PY - 1994/11/1

Y1 - 1994/11/1

N2 - In this paper we consider the problem of constructing solutions of several well known nonlinear partial differential equations (p.d.e.s) in phase space (i.e,. the Fourier transform domain). We seek solutions representing travelling focussed pulses. As such, based on a technique used to construct such solutions (so called Localized Wave solutions) of linear p.d.e.s, we look for phase space solutions constisting of a generalized funtion whose support is a particular line or surface, together with a suitable weighting function. The support of the phase space solutions must be such that it regenerates itself after the appropriate nonlinear operation. In one spatial dimension we construct the usual well known soliton solutions of several equations. For the case of higher spatial dimensions we construct a travelling "slab" pulse solution of the nonlinear Schrödinger equation. We also discuss some issues involved with the extra freedom one has for the phase space support, leading perhaps to more exotic spacetime domain solutions.

AB - In this paper we consider the problem of constructing solutions of several well known nonlinear partial differential equations (p.d.e.s) in phase space (i.e,. the Fourier transform domain). We seek solutions representing travelling focussed pulses. As such, based on a technique used to construct such solutions (so called Localized Wave solutions) of linear p.d.e.s, we look for phase space solutions constisting of a generalized funtion whose support is a particular line or surface, together with a suitable weighting function. The support of the phase space solutions must be such that it regenerates itself after the appropriate nonlinear operation. In one spatial dimension we construct the usual well known soliton solutions of several equations. For the case of higher spatial dimensions we construct a travelling "slab" pulse solution of the nonlinear Schrödinger equation. We also discuss some issues involved with the extra freedom one has for the phase space support, leading perhaps to more exotic spacetime domain solutions.

UR - http://www.scopus.com/inward/record.url?scp=43949153544&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43949153544&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(94)00121-9

DO - 10.1016/0167-2789(94)00121-9

M3 - Article

VL - 78

SP - 115

EP - 123

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -