Soluitions of nonlinear partial differential equations in phase space

Research output: Contribution to journalArticle

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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 languageEnglish (US)
Pages (from-to)115-123
Number of pages9
JournalPhysica D: Nonlinear Phenomena
Volume78
Issue number1-2
DOIs
StatePublished - Nov 1 1994

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Nonlinear Partial Differential Equations
partial differential equations
Partial differential equations
Phase Space
Weighting Function
Linear partial differential equation
Soliton Solution
Fourier transform
weighting functions
Nonlinear Equations
Space-time
pulses
Solitons
Nonlinear equations
nonlinear equations
Line
Fourier transforms
slabs
solitary waves

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

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

In: Physica D: Nonlinear Phenomena, Vol. 78, No. 1-2, 01.11.1994, p. 115-123.

Research output: Contribution to journalArticle

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