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120 8 Domains and Spatial Solitons

8.2.2 Subcritical Hopf Bifurcation

We consider two possible modifications of (8.1) that lead to subcritical solutions: (a) a modification of the local nonlinear terms, and (b) a modification of the nonlocal terms, containing spatial derivatives.

(a) The Quintic Complex Swift–Hohenberg Equation. In this case, a nonlinear term of fifth order is considered:

∂A∂t = pA − bA |A|2 − A |A|4 + i a 2 − ∆ A − g a 2 − ∆ 2 A . (8.3)

The traveling-wave solution behaves as depicted in Fig. 8.3.

Fig. 8.3. Dependence of the intensity on the criticality parameter p and on the squared wavenumber k2, in the case of a subcritical Hopf bifurcation generated by a quintic nonlinear term

The subcritical character of the bifurcation follows from the left-hand graph in Fig. 8.3 (there is a coexistence of solutions for p < 0). Di erently from the supercritical case, the traveling-wave branch is disconnected from the trivial-solution branch (see the right-hand graph in Fig. 8.3). As a consequence, the trivial solution is stable for any wavenumber, reflecting an absolute bistability (bistability is not absolute if the upper branch connects with the trivial branch, as will be discussed in the next example).

A physical system showing this behavior is a laser with a saturable absorber, whose corresponding order parameter equation is

The e ect of this type of nonlinearity will be analyzed in detail in the following chapter. Note that the fifth-order nonlinearity corresponds to the first terms in the Taylor expansion of the first two terms in (8.4).

In this example, the gain must be larger than the losses for stable solitons. This condition can be visualized by plotting the first and second terms in (8.4), as shown in Fig. 8.4.

loss

gain

(b) Spatial Nonlinear Resonance. Usually the resonant wavenumber, corresponding to the solution selected by the system, is constant and given by k02 = −∆/a. In some cases, however, the resonant wavenumber is intensitydependent, a phenomenon known as nonlinear resonance. The complex Swift– Hohenberg equation then takes the form

The properties of the traveling-wave solution of (8.5) are illustrated in Fig. 8.5: the nonlinear resonance results in a tilt of the resonance curve (a linear resonance corresponds to a symmetric parabola).

Note that, unlike the case of the quintic complex Swift–Hohenberg equation, the instability now is not absolute: there always exist a wavenumber (in fact, the band of wavenumbers between the two intersections of the resonance

A 2

Fig. 8.5. Dependence of the intensity on the criticality parameter p and on the squared wavenumber k2, in the case of a subcritical Hopf bifurcation generated by a nonlinear resonance term

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curve with the axis) where the trivial solution is unstable. In order to have absolute bistability, this instability region must be removed.

One way of doing this is by shifting the curve to the left, since negative values of k02 have no physical meaning. We then deal with a “cut” nonlinear resonance (Fig. 8.6). The bistability is now absolute, since the traveling-wave branch is completely disconnected from the trivial solution.

An example of a physical system showing a nonlinear resonance is a laser (or, in general, a nonlinear resonator) with intracavity focusing. This case will be treated in detail in Chap. 10.

Fig. 8.6. The e ect of shifting of the nonlinear resonance curve: (a) the whole curve, including unphysical solutions; (b) Cut nonlinear resonance, showing absolute bistability

8.3 Amplitude and Phase Domains

A spatial soliton can be regarded as a limiting case of a more general solution, a domain. A domain is a region of space of arbitrary size or shape (in a system with one or two spatial degrees of freedom, respectively) where a field of given amplitude and phase is separated from regions with a di erent field amplitude and phase by domain boundaries or domain walls. The domain walls correspond to the connections between two stable solutions.

The classification of bifurcations given in Sect. 8.1 (see Fig. 8.1) can be used to classify the di erent types of domains. When the fields inside and outside the domain di er only in their phase (thus having the same intensity in all of the space except at the walls), we refer to phase domains. Phase domains are related to a pitchfork bifurcation, as shown in Fig. 8.1c. Subcriticality is not required for phase domains. A detailed discussion of phase patterns is left to Chap. 11.

When, however, the fields di er in their amplitude, we refer to amplitude domains. In this case, a subcritical bifurcation is always needed.

Domains can be generated by means of a hard-excitation mechanism: a su ciently strong spatially localized perturbation of the lower state can bring a portion of the system into the other state, even though a weak perturbation will not. Once a domain has been formed, it shows a dynamic behavior. The

domain walls expand or contract, depending on the parameters, eventually bringing the entire system into one of the two homogeneous states.

If the system parameters are not far away from a modulational stability boundary, the switching fronts connecting two homogeneous states show damped spatial oscillations on either side of the front.

Although a complete characterization of solitons requires, in general, a numerical solution of the model, some important predictions can be made by analyzing (i) the character of the bifurcation, as was done in Sect. 8.1, and (ii) the stability of the homogeneous solutions, to be discussed in the next section.

8.4 Amplitude and Phase Spatial Solitons

Usually, amplitude and phase solitons are referred to as bright solitons and dark ring solitons, respectively. This terminology comes from their intensity distribution: amplitude solitons connect solutions with di erent amplitudes (and also di erent intensities), and they appear as bright spots surrounded by a background of lower intensity (usually zero) (see Fig. 8.7a). On the other hand, phase solitons connect solutions of opposite phase (but of the same intensity), and they appear as dark rings (in 2D) or lines (in 1D) on a background of finite intensity (see Fig. 8.7b). As already stated, amplitude solitons always require subcriticality, but this is not necessary for phase solitons.

Fig. 8.7. Amplitude (left) and phase (right) solitons that are solutions of the equations for a degenerate optical parametric oscillator in di erent parameter regions. For the plot at the left, E = 1.2, and for the plot at the right, E = 2.0

Depending on the stability of the homogeneous solutions, amplitude spatial solitons can be interpreted in two ways. First, if a homogeneous solution corresponding to one of two bistability branches is modulationally unstable, then a soliton can occur as a homoclinic connection between a stable homogeneous state and a modulated (stripe or hexagon) state, as shown by Fauve and Thual in [3]. A soliton in this interpretation is a single, isolated band of

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a stripe pattern in the 1D case, or a single isolated spot of a hexagonal pattern in the 2D case. The background solution (the solution far away from the soliton) corresponds to the stable solution branch in this interpretation. For bright solitons, the upper branch is usually modulationally unstable, and the radiation corresponding to the stable lower branch serves as a background. In the case of dark solitons, the unstable (modulated) solution is the lower branch [1].

In the case when both the upper and the lower branches are modulationally stable, solitons (or domains in general) can be interpreted as homoclinic connections between the two homogeneous states, as shown by Rosanov [4]. A spatial domain corresponding to one solution branch can contract to a minimum size and not contract further, owing to the interaction between domain boundaries. The bright soliton in this interpretation is a spatial domain of minimum size corresponding to the upper bistability branch, whereas the radiation corresponding to the lower branch acts as the background. For dark solitons, the opposite is true. The mechanisms of the interaction between domain walls that leads to stable solitons are under investigation, but it is more or less clear that a nonmonotonic spatial decay of the domain boundary plays a significant role in the stability of solitons. The stronger the spatial oscillations of a nonmonotonically decaying domain boundary are, the larger is the stability range of a spatial soliton of the Rosanov type. The e ect of modulations will be treated in detail in Chap. 12.

Both type of solitons (amplitude and phase) can coexist in the same optical system if it is characterized by a subcritical pitchfork bifurcation. In this case the subcriticality is responsible for bright solitons, and the pitchfork bifurcation for phase solitons. The example shown in Fig. 8.7 corresponds to intensity distributions found by numerical integration of the equation for a DOPO in two transverse dimensions, with the same initial conditions but di erent parameters.

References

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2.G. Ioos and D.D. Joseph, Elementary Stability and Bifurcation Theory (Springer, New York, 1990). 117

3.S. Fauve and O. Thual, Solitary waves generated by subcritical instabilities in dissipative systems, Phys. Rev. Lett. 64, 282 (1990). 123

4.N.N. Rosanov, Transverse Patterns in Wide-Aperture Nonlinear Optical Systems, Progress in Optics, vol. 35, ed. by E. Wolf (North-Holland, Amsterdam, 1996). 124