Wave envelopes with second-order spatiotemporal dispersion. I. Bright Kerr solitons and cnoidal waves

We propose a simple scalar model for describing pulse phenomena beyond the conventional slowly varying envelope approximation. The generic governing equation has a cubic nonlinearity, and we focus here mainly on contexts involving anomalous group-velocity dispersion. Pulse propagation turns out to be a problem firmly rooted in frames-of-reference considerations. The transformation properties of the model and its space-time structure are explored in detail. Two distinct representations of exact analytical solitons and their associated conservation laws (in both integral and algebraic forms) are presented, and a range of predictions is made. We also report cnoidal waves of the governing nonlinear equation. Crucially, conventional pulse theory is shown to emerge as a limit of the more general formulation. Extensive simulations examine the role of the new solitons as robust attractors.

Figure 1

Schematic illustrating forward (FWD) and backward (BWD) pulses in the laboratory frame. For physically meaningful solutions, the FWD (BWD) soliton [upper (lower) signs in Eq. (13a)] must always span the first and third (second and fourth) quadrants of the $(\tau ,\zeta )$ plane. This condition, which is captured by $W>0$, ensures that the peak of the pulse is always moving forward in time. Dotted lines denote the trajectories $\tau \mp W\zeta =0$.

Figure 2

Plot of parameter $W$ [given by Eq. (12b) with $\alpha =1.0$] as a function of frequency $\Omega$ for solitons with ${\rho }_0=1.0$ and for two (opposite-sign) values of the spatial dispersion parameter $\kappa$. Inset: space-time plane $(\tau ,\zeta )$. Forward-propagating solitons with $\Omega =0$ move along the line $\tau -V_0\zeta =0$ at uniform speed $1/V_0$. Solutions with $\Omega <0$ ($\Omega >0$) travel at faster (slower) speeds.

Figure 3

Schematic for a forward-propagating pulse illustrating the cases of (a) $\mathrm{sgn}\left(s\kappa \right)=+1$ and (b) $\mathrm{sgn}\left(s\kappa \right)=-1$. The “0” subscripted coordinates $({\tau }_0,{\zeta }_0)$ define the rest frame (i.e., the frame in which the pulse is stationary and, hence, travels along the ${\zeta }_0$ axis); these coordinates can be obtained from Eqs. (6a) and (6b) by selecting $W$ as the velocity parameter. Note that the pulse width must be measured at a fixed longitudinal position (so that $\Delta \zeta =\Delta {\zeta }_0=0$).

Figure 4

Self-reshaping of a unit-amplitude pulse (${\rho }_0=1.0$) from initial condition (32) where $\kappa =+{10}^{-3}$ and $\alpha =1.0$. The peak amplitude decreases, and the pulse width tends to increase in such a way that the area increases. Horizontal bars denote the asymptotic soliton parameters as predicted by inverse-scattering techniques [14].

Figure 5

Self-reshaping of a unit-amplitude pulse (${\rho }_0=1.0$) from initial condition (32) where $\kappa =-{10}^{-3}$ and $\alpha =1.0$. The peak amplitude increases, and the pulse width tends to decrease in such a way that the area decreases. Horizontal bars denote the asymptotic soliton parameters as predicted by inverse-scattering techniques [14].