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The short pulse equation and associated constraints (906).
| Content Provider | CiteSeerX |
|---|---|
| Author | Theodoros P. Horikis, A. |
| Abstract | The short pulse equation (SPE) is considered as an initial-boundary value problem. It is found that the solutions of the SPE must satisfy an integral relation otherwise the temporal derivative exhibits discontinuities. This integral relation is not necessary for a solution to exist. An infinite number of such constraints can be dynamically generated by the evolution equation. Key words: short pulse equation, initial data constraints. PACS: 02.30.Ik, 02.30.Jr The standard model for describing propagation of a pulse-shaped complex field envelope in nonlinear dispersive media is the nonlinear Schrödinger (NLS) equation. In the context of nonlinear optics, the main assumption made when deriving the NLS equation from Maxwell’s equations is that the pulse-width is large as compared to the period of the carrier frequency. When this assumption is no longer valid, i.e., for pulse duration of the order of a few cycles of the carrier, the evolution of such “short pulses ” is better described by the so-called short-pulse equation (SPE) [1]. The SPE can be expressed in the following dimensionless form, uxt = u + 1 6 (u3)xx (1) where subscripts denote partial derivatives. The SPE forms an initial-boundary problem when accompanied by the initial data u(0, x) = u0, and sufficiently fast decaying boundary conditions u(t, ±∞) = 0. Much like the NLS equation, the SPE is integrable [2] and exhibits soliton solutions in the form of loopsolitons [3]. However, when it is formed as an evolution equation certain conditions must apply otherwise, as shown below, the temporal derivative exhibits discontinuities. Despite the fact that the equation is integrable via the inverse scattering transform [4], there are certain subtleties that need to be clarified. Integration of Eq. (1) introduces the operation ∂ −1 ∫ x x u(t, x) = u(t, x ′ ) dx ′ |
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| Access Restriction | Open |
| Subject Keyword | Short Pulse Equation Integral Relation Nls Equation Temporal Derivative Exhibit Discontinuity Partial Derivative Carrier Frequency Maxwell Equation Initial Data Constraint Infinite Number Pulse-shaped Complex Field Envelope Short Pulse Nonlinear Dispersive Medium Nonlinear Optic Soliton Solution Initial Data Evolution Equation Standard Model Main Assumption Pulse Duration So-called Short-pulse Equation Evolution Equation Certain Condition Initial-boundary Value Problem Initial-boundary Problem Certain Subtlety Boundary Condition Following Dimensionless Form Nonlinear Schr Dinger |
| Content Type | Text |
