5 research outputs found
On the Whitham equations for the defocusing nonlinear Schrodinger equation with step initial data
The behavior of solutions of the finite-genus Whitham equations for the weak
dispersion limit of the defocusing nonlinear Schrodinger equation is
investigated analytically and numerically for piecewise-constant initial data.
In particular, the dynamics of constant-amplitude initial conditions with one
or more frequency jumps (i.e., piecewise linear phase) are considered. It is
shown analytically and numerically that, for finite times, regions of
arbitrarily high genus can be produced; asymptotically with time, however, the
solution can be divided into expanding regions which are either of genus-zero,
genus-one or genus-two type, their precise arrangement depending on the
specifics of the initial datum given. This behavior should be compared to that
of the Korteweg-deVries equation, where the solution is devided into the
regions which are either genus-zero or genus-one asymptotically. Finally, the
potential application of these results to the generation of short optical
pulses is discussed: the method proposed takes advantage of nonlinear
compression via appropriate frequency modulation, and allows control of both
the pulse amplitude and its width, as well as the distance along the fiber at
which the pulse is produced.Comment: 44 pages, 21 figure
Ămergence de flaticons dans les fibres optiques
ConfĂ©rence pouvant ĂȘtre vue sur http://youtu.be/p9OnhcHQ3MwNational audienceNous Ă©tudions expĂ©rimentalement la propagation non-linĂ©aire d'une onde continue menant Ă l'Ă©mergence d'impulsions au sommet plat et sans dĂ©rive de frĂ©quence. Ces impulsions, appelĂ©es flaticons, subissent une Ă©volution auto-similaire de leur partie centrale et prĂ©sentent des oscillations temporelles marquĂ©es dans leurs flancs
Refraction of dispersive shock waves
We study a dispersive counterpart of the classical gas dynamics problem of
the interaction of a shock wave with a counter-propagating simple rarefaction
wave often referred to as the shock wave refraction. The refraction of a
one-dimensional dispersive shock wave (DSW) due to its head-on collision with
the centred rarefaction wave (RW) is considered in the framework of defocusing
nonlinear Schr\"odinger (NLS) equation. For the integrable cubic nonlinearity
case we present a full asymptotic description of the DSW refraction by
constructing appropriate exact solutions of the Whitham modulation equations in
Riemann invariants. For the NLS equation with saturable nonlinearity, whose
modulation system does not possess Riemann invariants, we take advantage of the
recently developed method for the DSW description in non-integrable dispersive
systems to obtain main physical parameters of the DSW refraction. The key
features of the DSW-RW interaction predicted by our modulation theory analysis
are confirmed by direct numerical solutions of the full dispersive problem.Comment: 45 pages, 23 figures, minor revisio
Refraction of dispersive shock waves
We study a dispersive counterpart of the classical gas dynamics problem of the interaction of a shock wave with a counter-propagating simple rarefaction wave, often referred to as the shock wave refraction. The refraction of a one-dimensional dispersive shock wave (DSW) due to its head-on collision with the centred rarefaction wave (RW) is considered in the framework of the defocusing nonlinear Schrödinger (NLS) equation. For the integrable cubic nonlinearity case we present a full asymptotic description of the DSW refraction by constructing appropriate exact solutions of the Whitham modulation equations in Riemann invariants. For the NLS equation with saturable nonlinearity, whose modulation system does not possess Riemann invariants, we take advantage of the recently developed method for the DSW description in non-integrable dispersive systems to obtain main physical parameters of the DSW refraction. The key features of the DSW-RW interaction predicted by our modulation theory analysis are confirmed by direct numerical solutions of the full dispersive problem