150 11 Phase Domains and Phase Solitons

11.3 Dynamics of Domain Boundaries

Before investigating the dynamics of phase domains in 2D, let us analyze the case of stationary, straight boundaries, which is actually the 1D case. Straight boundaries between su ciently large domains are stationary, since two phases (two stationary solutions) corresponding to neighboring domains are always in equilibrium here.1 Domain boundaries can move either because of curvature e ects (as investigated in Sect. 11.4), or because of their mutual interaction (Sect. 11.5).

No analytic expression exists for a straight domain boundary in the case of the RSH equation. However, such boundaries can be found analytically in the case of the real Ginzburg–Landau (RGL) equation,

∂A	= A − A3 + 2A .	(11.4)
∂t

Equation (11.4) has a kink-form solution in 1D, corresponding to a straight domain boundary in 2D,

A(x) = ± tanh	√2		,	(11.5)
	x

which represents a solution directed along the y axis.

The RSH equation in the limit of large negative detuning actually transforms into the RGL equation (11.4). Therefore, let us assume that the domain boundary solution of the RSH equation possess a form similar to the kink

solution of the RGL equation.						This suggest the use of the following ansatz:
A(x) =	1 − ∆2 tanh			x0		,	(11.6)
					x

√

where 1 − ∆2 is the modulus of the homogeneous solution of the RSH equation and x0 is the (unknown) half-width of the domain boundary.

11.3.1 Variational Approach

The RSH equation is a variational equation, and thus it can be also written in the gradient form ∂A/∂t = −δF/δA , with a potential F (A) given by [7]

∞	−	A2		A4			2 + ∆ A		2	dx dy .
F =			+		+						(11.7)
		2		4			2
−∞

1This is not always true in DOPOs, since in a certain parameter range the Ising– Bloch transition can be present, leading to a drift of the walls at constant velocity. However, this transition requires a complex order parameter, and thus cannot be obtained from the Swift-Hohenberg equation.

11.3 Dynamics of Domain Boundaries

151

We use a variational approach in order to (i) determine the half-width x0 of the straight domain boundary that minimizes the potential (11.7) in 1D, and (ii) to analyze the motion of curved domain boundaries in 2D.

After the ansatz (11.6) is substituted into (11.7), the integration results in an infinite value for the potential. This is due to the contribution of the homogeneous background A0. Therefore we need to calibrate the potential (11.7) by subtracting this constant contribution [8], which in the 1D case yields

	−∞ −		2			4
F =	∞	A2	− A02	+	A4	− A04	+

	2	/∂x	2			2	−	2	dx .
	∂		2+ ∆ A					∆22A0
									(11.8)

Integration of (11.8) with the ansatz (11.6) now gives a finite value for the potential,

F =

1 − ∆2

5x4

1

−

∆2

+ 8

−

20∆x2

,

(11.9)

15x03

0

0

which depends on the unknown parameter x0 and on the detuning ∆. The value of the half-width can be found by minimizing the potential (11.9), and is given by

2

√

−

√

x2 =

6 − ∆2

5∆

.

(11.10)

0

√

1

−

∆2

5

Substituting (11.10) into (11.9), we find the corresponding potential,

2

(3/2)

∆2

2

√

√

.

F1D =

3

2√5

2

(11.11)

√6

√5∆ (3/2)

8

1 − ∆

2 + 3∆ −

6 − ∆ 5∆

−

−

The dependences (11.10) and (11.11) are plotted in Fig. 11.2. As expected, the dark line broadens monotonically with decreasing detuning. The calibrated potential (11.8) is positive over almost the whole detuning range. This is plausible, since the dark line is a defect in a homogeneous pattern, and thus increases the potential energy of the system. However, for large values of the detuning the potential becomes negative, which has profound consequences for the dynamics of domains in 2D. Indeed, if for a small detuning the presence of a kink in 1D increases the variational potential, then in 2D a domain wall should tend to be as short as possible, and the domains should contract. On the other hand, if for a large detuning the presence of a kink in 1D decreases the variational potential, then in 2D a domain wall should, correspondingly, tend to be as long as possible, and thus the domains should expand. This dynamic behavior of 2D domains is investigated in the next section.

152 11 Phase Domains and Phase Solitons

3

x0

2

1

F1D

0
-1	-0.5	0	0.5	Fig. 11.2. The half-width x0 of the dark line
				1

∆tional (11.11), as a function of the detuning ∆

11.3.2Two-Dimensional Domainsgiven by (11.10), and the calibrated 1D func-

To study analytically the motion of a domain boundary in 2D, we assume a ring-shaped form. This represents, equivalently, a circular domain centered at the origin of a polar coordinate system. Such a dark ring can be described by the ansatz

					x0			x0
		−
A(r) =	1		∆2 tanh		r − r0		tanh	r + r0	,	(11.12)

where r0 is the radius of the ring, and the half-width x0 is given by (11.10). Owing to the cylindrical symmetry of the ansatz (11.12), the variational potential can be written as

F = 2π

0

− 2 +

4

+ 2

∂r2

+ r ∂r + ∆

A

2

r dr , (11.13)

∞

A2

A4

1

∂2

1

∂

which has to be calibrated in the same way as in the 1D case, to avoid the background contribution.

Analytical integration of (11.13) using the ansatz (11.12) is not possible in the 2D case; however, two other complementary approaches can be used. One possibility is to evaluate approximately the integral in some limiting cases. The other possibility is to calculate the integral (11.13) numerically.

When the radius of the ring is large enough (when r0/x0 1), the order parameter A(r) is nearly an odd function with respect to the ring radius r0. Making the change of variables x = r − r0, assuming the integrated function f(x) to be even, and extending the integration limits to the whole space, we obtain

∞		∞
F2D = 2π	f(x)(x + r0) dx = 2πr0		f(x) dx = 2πr0F1D . (11.14)

−∞

−∞

11.3 Dynamics of Domain Boundaries

153

This result is not exact, because the integration limits in reality do not extend to infinity, and also because the integrated function is not even, but has a small odd part. The errors occurring because of these two assumptions give corrections to (11.14) of orders O(exp (−r0/x0)) and O(1/r0), respectively. Therefore, for an asymptotically large radius of the ring r0, and for a width of the dark line x0 = O(1), we obtain the potential

F2D	= 2πr0F1D	+ O exp −x0			+ O r0	.	(11.15)
			r0		1

Thus the potential in 2D is roughly equal to the potential in 1D (11.11) multiplied by the length of the weakly curved (circular) dark line. In this approximation, the sign of F1D determines the evolution of the ring. As can be seen from Fig. 11.2, around zero detuning F1D is positive, and thus the longer the domain boundary is, the larger the 2D potential is. As the solution tends to minimize the potential, the domains contract. For a detuning larger than some ∆c, the 1D potential is negative, and the domains expand. The

particular case of stationary rings occurs at ∆c = 2/7 ≈ 0.535.

The result of numerical integration of the RSH equation (11.4) shows contraction or expansion of the domains, depending on the detuning. In Fig. 11.3, an example of domain contraction for a small value of the detuning is given.

Figure 11.4, in contrast, shows domain expansion for a large detuning value. The asymptotic pattern in this case is a labyrinth structure. The numerical results indicate that rings of large radius are marginally stable at a detuning ∆c = 0.45 ± 0.05, which di ers from the analytically evaluated

Fig. 11.3. Evolution of phase domains obtained using the RSH equation for a small signal detuning, ∆ = 0.25. The integration was performed in a box of size 70 units (the integration grid contained 128×128 points), with periodic boundaries. Time increases from left to right. In the upper row the field intensity is plotted, and the lower row shows the field phase. The pictures were obtained at times t = 0 (the initial distribution), t = 20, t = 100 and t = 250. The last domains disappear at t = 370

154 11 Phase Domains and Phase Solitons

Fig. 11.4. Evolution of phase domains obtained using the RSH equation for a large signal detuning, ∆ = 0.65. Other parameters as in Fig. 11.3. The initial picture at time t = 0 was prepared by integrating the RSH equation with a detuning ∆ = 0.25 (as in the previous figure), and further calculations were then performed with the new detuning value. The other plots were obtained at times t = 20, t = 150 and t = 1000

equilibrium detuning value given above. This di erence occurs because of the inexact form of the ansatz. The domain boundaries are not exactly of hyperbolic-tangent form (with monotonically decaying tails), but show an oscillatory decay of the tails. A more accurate analysis that takes account of these spatial oscillations and leads to a better correspondence with the numerical values is performed in Sect. 11.5.

In this way, by varying the detuning ∆, one can manipulate the domain dynamics, forcing the domains to expand or to shrink. Expanding domains keep their topological properties during the evolution for moderate values of the detuning: the number of domains in a finite labyrinth pattern is equal to that in the initial pattern, as can be seen in Fig. 11.4.

The situation is di erent for larger values of the detuning, in particular

when ∆ > 2/3. In this case, as follows from the linear stability analysis, the homogeneous solution is modulationally unstable. Domain growth and labyrinth formation are now accompanied by the appearance of new domains (nucleation), as shown in Fig. 11.5. Another peculiarity of the large-detuning case is that the dark lines in the labyrinths can break and reconnect, which also leads to topological changes of the domains.

Fig. 11.5. Evolution of phase domains obtained using the RSH equation a for large signal detuning, ∆ = 0.85. Other parameters as in Fig. 11.3. The plots were obtained at times t = 0, 20, 150 and 1000