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Pulled fronts in the Cahn–Hilliard equation
1 April 2002
Physics Letters A 295 (2002) 267–272 www.elsevier.com/locate/pla
Pulled fronts in the Cahn–Hilliard equation B.A. Malomed a , D.J. Frantzeskakis b , H.E. Nistazakis b , A.N. Yannacopoulos b , P.G. Kevrekidis c,∗ a Department of Inderdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel b Department of Physics, University of Athens, Panepistimiopolis, GR 157 84, Athens, Greece c Department of Mathematics and Statistics, University of Massachusetts, Lederle Graduate Research Tower,
Amherst, MA 01003-4515, USA Received 19 September 2001; accepted 10 December 2001 Communicated by A.R. Bishop
Abstract A possibility of pulling a front by a moving inhomogeneity is considered in the context of the Cahn–Hilliard equation, which is a generic model of nonequilibrium phase-separation processes. The critical (maximum) velocity of the inhomogeneity, at which it is still able to steadily drag the front, is found in an analytical approximation, using both perturbation theory and a quasiparticle description of the front. A case of steep inhomogeneities is studied in detail by means of direct simulations, showing that the analytical prediction for the critical velocity is in very good agreement with numerical results for small and moderate values of the inhomogeneity’s strength. If the driving velocity exceeds the critical value, the kink is eventually destroyed. If the perturbation is strong, the simulations show that the actual critical velocity is larger than a formally extended analytical value, i.e., the kink turns out to be more robust than it is expected from the perturbative results. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction A well-established model for the study of phase transitions in binary alloys is the Cahn–Hilliard (CH) equation [1,2], ∂ 2 ∂ 2u ∂u =− 2 (1) + f (u) , ∂t ∂x ∂x 2 where u is the relative concentration of the two species in the alloy, and f (u) is a polynomial nonlinearity usually taken in the form f (u) = bu − u3 , where b is a constant coefficient. In fact, the CH equation is * Corresponding author.
E-mail address: [email protected] (P.G. Kevrekidis).
a fairly universal one, as it describes nonequilibrium processes in a number of different physical systems, see papers [3] and references therein. The CH equation is well known (see, e.g., Ref. [4]) to possess a Lyapunov functional F , such that the equation can be written in the gradient form, ∂ 2 δF ∂u = 2 , (2) ∂t ∂x δu where δ/δu is the Frechet (variational) derivative of the functional with respect to the function u. For Eq. (1), the Lyapunov functional is ∞ 2 1 ∂u − F (u) dx, F= 2 ∂x −∞
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268
B.A. Malomed et al. / Physics Letters A 295 (2002) 267–272
where F (u) ≡ f (u) du. The value of the Lyapunov functional corresponding to any nonstationary solution decreases as a result of the evolution, the corresponding dissipation rate being, in the general case, dF = R≡ dt
∞
−∞
∞ =− −∞
δF ∂ 2 δF dx δu ∂x 2 δu
∂ δF ∂x δu
2 ,
(4)
where integration by parts was used. The CH equation has a stationary solution that satisfies the boundary conditions (b.c.) lim u(x) = u± ,
x→±∞
(5)
u± being two stable zeroes of the nonlinear function f (u). This solution is commonly known as a kink, alias a front solution separating two distinct phases. The aim of this Letter is to study driven motion of the front. To this end, it is possible to introduce a steadily moving inhomogeneity in the medium, which pulls the front hard enough to set it into motion. In processes like directional solidification (see, e.g., Ref. [5]) such a dragging inhomogeneity may appear quite naturally, and it may be included into one of the terms in the function f (u) in Eq. (1). In this work, we focus on the accordingly modified equation, ∂ 2 ∂ 2u ∂u 3 =− 2 + b(x − vt)u − u , (6) ∂t ∂x ∂x 2 with a function b(x − vt) representing for the moving inhomogeneity that may drag the kink. This choice is the simplest possible one, as the dragging inhomogeneity, moving at a constant velocity v, is introduced in the linear term. A basic issue to be addressed is to find a maximum (critical) value of v such that the inhomogeneity is still able to pull the kink with the velocity v for a given form of b(x − vt). The problem of pulling the front was earlier addressed in other models of phase transitions. In particular, the problem of the motion of an interface and breakdown of perturbation theory associated with it were discussed in reviews [6] for models of phase transitions other than the CH equation. In different physical contexts, similar problems are known too, for instance, the capture and drag of an internal-wave soli-
ton by a moving dipole (representing a submarine vehicle) in the ocean [7]. The Letter is organized as follows. In Section 2.1, we develop a perturbation theory approach to search for the above-mentioned critical velocity. In Section 2.2, we give a physical interpretation of the perturbative results, treating the kink as a quasiparticle. In Section 3, where we take, as a practically relevant example, very steep inhomogeneities, we present results of numerical experiments that verify our analytical approximations.
2. Estimate of the critical velocity 2.1. The perturbation theory To develop a perturbation theory aimed at finding the critical velocity, we adopt the following two assumptions: first, the inhomogeneity in Eq. (6) is of the form b(ξ ) = 1 + b1 (ξ ), where is a small parameter measuring the strength of the inhomogeneity, and ξ ≡ x − vt is the traveling coordinate, and second, the kink velocity v is of order , i.e., v ≡ v, ¯ so that ξ = x − vt, ¯ where v¯ is O(1). We then seek for a solution in the form u(x, t) = u0 (ξ ) + u1 (ξ ) + · · · .
(7)
In the reference frame moving at the velocity v, Eq. (6) becomes d 2 d 2u du 3 = 2 v (8) + b(ξ )u − u . dξ dξ dξ 2 We substitute the expansion (7) into Eq. (8), to obtain, at the zeroth order, d 2 d 2 u0 3 + u − u (9) 0 0 = 0. dξ 2 dξ 2 Eq. (9) has a well-known [8] kink solution √ u0 (ξ ) = tanh(ξ/ 2 ). At the next order in , we obtain d2 d 2 u1 du0 2 = 2 b 1 u0 + + u1 − 3u0 u1 . v¯ dξ dξ dξ 2
(10)
(11)
This equation may be integrated twice in ξ to obtain 2 d u1 2 + 1 − 3u0 u1 , −vU ¯ 0 + b 1 u0 = − (12) dξ 2
B.A. Malomed et al. / Physics Letters A 295 (2002) 267–272
where √ √ U0 = u0 dξ ≡ 2 ln cosh(ξ/ 2 ) .
(13)
In line with our choice of integration constants in Eq. (9), integration constants have been set to zero in Eq. (12). Eq. (12) is a linear inhomogeneous ordinary differential equation. Therefore, we must derive a consistency condition for the existence of a solution satisfying the boundary conditions. This will be done by means of the Fredholm alternative. To this end, let ua be a solution to the adjoint homogeneous equation. To make the notation more compact, we define the linear Sturm–Liouville operator L, 2 ≡ d u + 1 − 3u20 u, Lu 2 dξ
acting in the space of twice differentiable functions. Using this notation, Eq. (12) becomes 1 = F, Lu
(14)
with F ≡ −vU ¯ 0 + b 1 u0 . In its functional space, the operator L is selfadjoint. The consistency condition for the existence of a solution to Eq. (14), satisfying the chosen b.c., is then given by the Fredholm alternative as F, ua = 0, and the where ua is a zero mode of the operator L, inner product is defined as
The expression (15) can be simplified by means of the integration by parts. Using b.c. and the definition U0 ≡ u0 , we obtain
∞ −1 ∞ 1 2 2 v¯ = (16) b1 u0 dξ u0 dξ , 2 −∞
2.2. The kink as a particle A more physically transparent approach to the determination of the critical velocity can be developed, treating the pulled kink as a quasiparticle. To this end, we neglect, for the time being, the perturbation accounting for the dragging force, integrate Eq. (8) twice, and obtain d 2u 3 + u − u = v u(ξ ) dξ, dξ 2 which is further substituted into the variational derivative of the Lyapunov functional (3), (2) yielding 2 d u δF 3 =− + u − u = −v u(ξ ) dξ. δu dξ 2
U (ξ )V (ξ ) dξ.
−∞
The translational invariance implies that the ua = is a solution of the homogeneous equation Lua = 0, i.e., the zero mode to be inserted into the consistency condition. The consistency condition then becomes
∞ −1 ∞ u0
v¯ = −∞
b1 u0 u0 dξ
U0 u0 dξ
−∞
which is a final result of the perturbation theory. It is worth noting in closing this section that in the above integral the unperturbed solution u0 is poised at the front’s center ζ = 0 (see below), whereas the inhomogeneity is centered around ξ = 0.
Using this result and Eq. (4), we arrive at the following expression for the free-energy dissipation rate corresponding to the moving kink,
∞ U, V =
269
,
(15)
−∞
Eq. (15) imposes a relation on the pulling velocity v¯ and the shape b1 (ξ ) of the moving drag (inhomogeneity). Note that the function b1 (ξ ) may be a localized one, as well as periodic or stochastic, as the integrals in Eq. (15) converge, which follows from the expressions (10) and (13).
∞ R = −v
2
u20 dξ,
(17)
−∞
where integration by parts has been applied, and the solution u(ξ ) is replaced by the lowest-order approximation u0 (ξ ) in the end. We now assume that this dissipation rate may be modeled as originating from a friction force, ffr = −av, with some friction coefficient α. Under this assumption, the energy-dissipation rate is R = −αv 2 , equating which to the expression (17) ∞ yields α = −∞ u20 dξ . For the actual form (10) of u0 (ξ ), an elementary calculation gives √ α = 2 2 (1 − ln 2) ≈ 0.8679. (18)
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We now restore the drag force in the model, which corresponds to the following extra term in the Lyapunov functional accounting for the interaction between the kink and moving inhomogeneity, 1 Fint = − 2
∞ −∞
b1 (x − vt)u20 (x − ζ ) + 2b0 u0 (x − ζ )u1 (x − ζ ) dx,
(19)
ζ being the instantaneous coordinate of the kink’s center. The term (19) is obtained by considering the leading-order (O()) corrections to the functional. Notice, however, that the latter term in the integrand in Eq. (19) only yield a constant contribution, independent of ζ . In the framework of the quasiparticle approximation, the term (19) added to the Lyapunov functional gives rise to a pulling force fpull = dFint /dζ . Then, the critical velocity is that at which this force and the friction force are in equilibrium. It is straightforward to check that this approach yields exactly the same value (16) which was found above by means of a more formal perturbation theory.
3. Analytical and numerical results for fronts pulled by steep inhomogeneities As a simplest example, one may take a very steep dragging inhomogeneity of the form b1 (x − vt) = δ(x − vt). This term models a localized impurity, similar to the above-mentioned approximation for the internal-wave solitons in the ocean, with a submarine vehicle (that could trap and drag the soliton) modeled by a delta-like dipole [7]. In this case, it is easy to produce analytical results for the critical velocity. Taking the zeroth-order approximation (10) for the kink, one immediately evaluates the interaction term (19) in the Lyapunov function, 2 ζ − vt √ Fint (ζ ) = − tanh (20) 2 (recall ζ is the coordinate of the kink’s center), and the pulling force is then found in the form fpull ≡
dFint dζ
ζ − vt 2 ζ − vt sech , = tanh √ (21) √ 2 2 √ (max) whose maximum value is fpull = 2/3 5. The critical velocity may is obtained from the (max) . With regard to the expression condition avcr = fpull (18) for the friction coefficient α, this yields vcr = √ ≈ 0.344 3 10 (1 − ln 2)
(22)
(the same result follows from the more general expression (16)). In order to check these analytical predictions, the inhomogeneous CH equation was solved numerically using a finite-difference scheme, with the steep inhomogeneity approximated as b1 = sech(x − vt), which turns out to be steep enough (see results displayed below). Varying the value of v, we observed that, for v above the critical value vcr predicted by the analytical approximation (22), the kink remains immobile, developing strong fluctuations in its rear, which finally destroy the kink. This can be seen in Fig. 1(a), where the evolution of the kink is shown for v = 2vcr and = 0.1. Additionally, in Fig. 1(b) the motion of the kink’s center is shown in the (x, t) plane. We calculate the position of the center of the kink by (approximately) evaluating
∞
∞ xc = −∞
−1 |u| dx
x|u| dx
.
(23)
−∞
The accuracy of the method was also confirmed by numerical interpolation schemes, similar to the ones used in [9], evaluating the zero crossing of the field u. As is seen from the figures, the kink cannot overcome the friction force and inertia, and, barring a short transient (when it tries to trail behind the inhomogeneity), it remains immobile, until it gets destroyed. On the contrary, when v is taken below vcr , the kink is able to follow the inhomogeneity. In this case, the kink moves, remaining essentially undistorted, as is seen in Fig. 2(a) (for v = vcr /2 and = 0.1). Furthermore, as is shown in Fig. 2(b), where the motion of the kink’s center is shown, after a short transient the kink assumes a value of the velocity fluctuating about v. In extensive numerical experiments, we have seen that the behavior described above is generic for a wide
B.A. Malomed et al. / Physics Letters A 295 (2002) 267–272
271
(a)
(a)
(b)
(b)
Fig. 1. (a) The evolution of the front for v = 2vcr and = 0.1. The spatial profile of the front is shown at t = 0, t = 5, and t = 10. (b) The position of the front as a function of time in the same case.
Fig. 2. The same as in Fig. 1 for v = 0.5vcr and = 0.1. The spatial profile is shown here at t = 0, t = 20, and t = 40.
range of values of v, provided that is small enough. This is expected, since our analytical results were obtained through a perturbative approach based on the smallness of . In order to understand what happens as is increased, we have found the maximum velocity of the inhomogeneity, at which the kink could still
be dragged by it (i.e., the critical velocity), as a function of from direct simulations, see the result in Fig. 3. It is evident that the analytical prediction for vcr , given by Eq. (22), is in excellent agreement with the numerical results for values of up to 0.25, whereas above this value the perturbation theory
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B.A. Malomed et al. / Physics Letters A 295 (2002) 267–272
ulations, with the conclusion that the simulations confirm the leading-order analytical prediction well. If the perturbation is strong, the kink is found to be more robust than it might be expected from the perturbative results.
Acknowledgements
Fig. 3. The analytically predicted (solid line) and numerically found (star symbols) critical dragging velocity vcr vs. the inhomogeneity strength .
This work was partially supported by PENED99 Grant No. 99ED527 from the General Secretariat of Research and Technology (Greece). Constructive discussions with Profs. K. Hizanidis (The National Technical University of Athens) and N. Alikakos (The University of Athens) are acknowledged. B.A.M. appreciates hospitality of the Department of Physics at the University of Athens.
References predicts a critical velocity which is smaller than that observed in the numerical experiments. The latter result implies that, as a matter of fact, the kink is a more robust object than it might be expected on the basis of the perturbation theory.
4. Conclusions In this Letter, we studied the possibility of pulling a front by a moving inhomogeneity in the context of the Cahn–Hilliard equation. An analytical prediction for the critical velocity of the inhomogeneity required to pull the front (kink) was obtained, using both the perturbation theory and quasiparticle approximation for the front. An example of a steep dragging inhomogeneity was studied in detail by means of direct sim-
[1] J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 (1958) 258. [2] D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, Chapman Hall, 1992. [3] A.A. Golovin, S.H. Davis, A.A. Nepomnyashchy, Physica D 122 (1998) 202; C. Chevallard, C. Clerc, P. Coullet, G.J. Gilli, Eur. Phys. J. E 1 (2000) 179; H. Calisto, M. Clerc, R. Rojas, E. Tirapegui, Phys. Rev. Lett. 85 (2000) 3805. [4] O. Penrose, in: J.M. Ball (Ed.), Material Instabilities in Continuum Mechanics, Clarendon, 1988. [5] K.A. Jackson, J.D. Hunt, Acta Metall. 13 (1965) 1212; R. Triveldi, K. Somboonsuk, Acta Metall. 33 (1985) 1061. [6] W. Van Saarloos, Phys. Rep. 301 (1998) 9; U. Ebert, W. Van Saarloos, Phys. Rep. 337 (2000) 139. [7] B.A. Malomed, Physica D 32 (1988) 393. [8] T. Sugiyama, Progr. Theor. Phys. 61 (1979) 1550. [9] R. Boesch, C.R. Willis, M. El-Batanouny, Phys. Rev. B 40 (1989) 2284.
Physics Letters A 295 (2002) 267–272 www.elsevier.com/locate/pla
Pulled fronts in the Cahn–Hilliard equation B.A. Malomed a , D.J. Frantzeskakis b , H.E. Nistazakis b , A.N. Yannacopoulos b , P.G. Kevrekidis c,∗ a Department of Inderdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel b Department of Physics, University of Athens, Panepistimiopolis, GR 157 84, Athens, Greece c Department of Mathematics and Statistics, University of Massachusetts, Lederle Graduate Research Tower,
Amherst, MA 01003-4515, USA Received 19 September 2001; accepted 10 December 2001 Communicated by A.R. Bishop
Abstract A possibility of pulling a front by a moving inhomogeneity is considered in the context of the Cahn–Hilliard equation, which is a generic model of nonequilibrium phase-separation processes. The critical (maximum) velocity of the inhomogeneity, at which it is still able to steadily drag the front, is found in an analytical approximation, using both perturbation theory and a quasiparticle description of the front. A case of steep inhomogeneities is studied in detail by means of direct simulations, showing that the analytical prediction for the critical velocity is in very good agreement with numerical results for small and moderate values of the inhomogeneity’s strength. If the driving velocity exceeds the critical value, the kink is eventually destroyed. If the perturbation is strong, the simulations show that the actual critical velocity is larger than a formally extended analytical value, i.e., the kink turns out to be more robust than it is expected from the perturbative results. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction A well-established model for the study of phase transitions in binary alloys is the Cahn–Hilliard (CH) equation [1,2], ∂ 2 ∂ 2u ∂u =− 2 (1) + f (u) , ∂t ∂x ∂x 2 where u is the relative concentration of the two species in the alloy, and f (u) is a polynomial nonlinearity usually taken in the form f (u) = bu − u3 , where b is a constant coefficient. In fact, the CH equation is * Corresponding author.
E-mail address: [email protected] (P.G. Kevrekidis).
a fairly universal one, as it describes nonequilibrium processes in a number of different physical systems, see papers [3] and references therein. The CH equation is well known (see, e.g., Ref. [4]) to possess a Lyapunov functional F , such that the equation can be written in the gradient form, ∂ 2 δF ∂u = 2 , (2) ∂t ∂x δu where δ/δu is the Frechet (variational) derivative of the functional with respect to the function u. For Eq. (1), the Lyapunov functional is ∞ 2 1 ∂u − F (u) dx, F= 2 ∂x −∞
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 1 7 3 - 1
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268
B.A. Malomed et al. / Physics Letters A 295 (2002) 267–272
where F (u) ≡ f (u) du. The value of the Lyapunov functional corresponding to any nonstationary solution decreases as a result of the evolution, the corresponding dissipation rate being, in the general case, dF = R≡ dt
∞
−∞
∞ =− −∞
δF ∂ 2 δF dx δu ∂x 2 δu
∂ δF ∂x δu
2 ,
(4)
where integration by parts was used. The CH equation has a stationary solution that satisfies the boundary conditions (b.c.) lim u(x) = u± ,
x→±∞
(5)
u± being two stable zeroes of the nonlinear function f (u). This solution is commonly known as a kink, alias a front solution separating two distinct phases. The aim of this Letter is to study driven motion of the front. To this end, it is possible to introduce a steadily moving inhomogeneity in the medium, which pulls the front hard enough to set it into motion. In processes like directional solidification (see, e.g., Ref. [5]) such a dragging inhomogeneity may appear quite naturally, and it may be included into one of the terms in the function f (u) in Eq. (1). In this work, we focus on the accordingly modified equation, ∂ 2 ∂ 2u ∂u 3 =− 2 + b(x − vt)u − u , (6) ∂t ∂x ∂x 2 with a function b(x − vt) representing for the moving inhomogeneity that may drag the kink. This choice is the simplest possible one, as the dragging inhomogeneity, moving at a constant velocity v, is introduced in the linear term. A basic issue to be addressed is to find a maximum (critical) value of v such that the inhomogeneity is still able to pull the kink with the velocity v for a given form of b(x − vt). The problem of pulling the front was earlier addressed in other models of phase transitions. In particular, the problem of the motion of an interface and breakdown of perturbation theory associated with it were discussed in reviews [6] for models of phase transitions other than the CH equation. In different physical contexts, similar problems are known too, for instance, the capture and drag of an internal-wave soli-
ton by a moving dipole (representing a submarine vehicle) in the ocean [7]. The Letter is organized as follows. In Section 2.1, we develop a perturbation theory approach to search for the above-mentioned critical velocity. In Section 2.2, we give a physical interpretation of the perturbative results, treating the kink as a quasiparticle. In Section 3, where we take, as a practically relevant example, very steep inhomogeneities, we present results of numerical experiments that verify our analytical approximations.
2. Estimate of the critical velocity 2.1. The perturbation theory To develop a perturbation theory aimed at finding the critical velocity, we adopt the following two assumptions: first, the inhomogeneity in Eq. (6) is of the form b(ξ ) = 1 + b1 (ξ ), where is a small parameter measuring the strength of the inhomogeneity, and ξ ≡ x − vt is the traveling coordinate, and second, the kink velocity v is of order , i.e., v ≡ v, ¯ so that ξ = x − vt, ¯ where v¯ is O(1). We then seek for a solution in the form u(x, t) = u0 (ξ ) + u1 (ξ ) + · · · .
(7)
In the reference frame moving at the velocity v, Eq. (6) becomes d 2 d 2u du 3 = 2 v (8) + b(ξ )u − u . dξ dξ dξ 2 We substitute the expansion (7) into Eq. (8), to obtain, at the zeroth order, d 2 d 2 u0 3 + u − u (9) 0 0 = 0. dξ 2 dξ 2 Eq. (9) has a well-known [8] kink solution √ u0 (ξ ) = tanh(ξ/ 2 ). At the next order in , we obtain d2 d 2 u1 du0 2 = 2 b 1 u0 + + u1 − 3u0 u1 . v¯ dξ dξ dξ 2
(10)
(11)
This equation may be integrated twice in ξ to obtain 2 d u1 2 + 1 − 3u0 u1 , −vU ¯ 0 + b 1 u0 = − (12) dξ 2
B.A. Malomed et al. / Physics Letters A 295 (2002) 267–272
where √ √ U0 = u0 dξ ≡ 2 ln cosh(ξ/ 2 ) .
(13)
In line with our choice of integration constants in Eq. (9), integration constants have been set to zero in Eq. (12). Eq. (12) is a linear inhomogeneous ordinary differential equation. Therefore, we must derive a consistency condition for the existence of a solution satisfying the boundary conditions. This will be done by means of the Fredholm alternative. To this end, let ua be a solution to the adjoint homogeneous equation. To make the notation more compact, we define the linear Sturm–Liouville operator L, 2 ≡ d u + 1 − 3u20 u, Lu 2 dξ
acting in the space of twice differentiable functions. Using this notation, Eq. (12) becomes 1 = F, Lu
(14)
with F ≡ −vU ¯ 0 + b 1 u0 . In its functional space, the operator L is selfadjoint. The consistency condition for the existence of a solution to Eq. (14), satisfying the chosen b.c., is then given by the Fredholm alternative as F, ua = 0, and the where ua is a zero mode of the operator L, inner product is defined as
The expression (15) can be simplified by means of the integration by parts. Using b.c. and the definition U0 ≡ u0 , we obtain
∞ −1 ∞ 1 2 2 v¯ = (16) b1 u0 dξ u0 dξ , 2 −∞
2.2. The kink as a particle A more physically transparent approach to the determination of the critical velocity can be developed, treating the pulled kink as a quasiparticle. To this end, we neglect, for the time being, the perturbation accounting for the dragging force, integrate Eq. (8) twice, and obtain d 2u 3 + u − u = v u(ξ ) dξ, dξ 2 which is further substituted into the variational derivative of the Lyapunov functional (3), (2) yielding 2 d u δF 3 =− + u − u = −v u(ξ ) dξ. δu dξ 2
U (ξ )V (ξ ) dξ.
−∞
The translational invariance implies that the ua = is a solution of the homogeneous equation Lua = 0, i.e., the zero mode to be inserted into the consistency condition. The consistency condition then becomes
∞ −1 ∞ u0
v¯ = −∞
b1 u0 u0 dξ
U0 u0 dξ
−∞
which is a final result of the perturbation theory. It is worth noting in closing this section that in the above integral the unperturbed solution u0 is poised at the front’s center ζ = 0 (see below), whereas the inhomogeneity is centered around ξ = 0.
Using this result and Eq. (4), we arrive at the following expression for the free-energy dissipation rate corresponding to the moving kink,
∞ U, V =
269
,
(15)
−∞
Eq. (15) imposes a relation on the pulling velocity v¯ and the shape b1 (ξ ) of the moving drag (inhomogeneity). Note that the function b1 (ξ ) may be a localized one, as well as periodic or stochastic, as the integrals in Eq. (15) converge, which follows from the expressions (10) and (13).
∞ R = −v
2
u20 dξ,
(17)
−∞
where integration by parts has been applied, and the solution u(ξ ) is replaced by the lowest-order approximation u0 (ξ ) in the end. We now assume that this dissipation rate may be modeled as originating from a friction force, ffr = −av, with some friction coefficient α. Under this assumption, the energy-dissipation rate is R = −αv 2 , equating which to the expression (17) ∞ yields α = −∞ u20 dξ . For the actual form (10) of u0 (ξ ), an elementary calculation gives √ α = 2 2 (1 − ln 2) ≈ 0.8679. (18)
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We now restore the drag force in the model, which corresponds to the following extra term in the Lyapunov functional accounting for the interaction between the kink and moving inhomogeneity, 1 Fint = − 2
∞ −∞
b1 (x − vt)u20 (x − ζ ) + 2b0 u0 (x − ζ )u1 (x − ζ ) dx,
(19)
ζ being the instantaneous coordinate of the kink’s center. The term (19) is obtained by considering the leading-order (O()) corrections to the functional. Notice, however, that the latter term in the integrand in Eq. (19) only yield a constant contribution, independent of ζ . In the framework of the quasiparticle approximation, the term (19) added to the Lyapunov functional gives rise to a pulling force fpull = dFint /dζ . Then, the critical velocity is that at which this force and the friction force are in equilibrium. It is straightforward to check that this approach yields exactly the same value (16) which was found above by means of a more formal perturbation theory.
3. Analytical and numerical results for fronts pulled by steep inhomogeneities As a simplest example, one may take a very steep dragging inhomogeneity of the form b1 (x − vt) = δ(x − vt). This term models a localized impurity, similar to the above-mentioned approximation for the internal-wave solitons in the ocean, with a submarine vehicle (that could trap and drag the soliton) modeled by a delta-like dipole [7]. In this case, it is easy to produce analytical results for the critical velocity. Taking the zeroth-order approximation (10) for the kink, one immediately evaluates the interaction term (19) in the Lyapunov function, 2 ζ − vt √ Fint (ζ ) = − tanh (20) 2 (recall ζ is the coordinate of the kink’s center), and the pulling force is then found in the form fpull ≡
dFint dζ
ζ − vt 2 ζ − vt sech , = tanh √ (21) √ 2 2 √ (max) whose maximum value is fpull = 2/3 5. The critical velocity may is obtained from the (max) . With regard to the expression condition avcr = fpull (18) for the friction coefficient α, this yields vcr = √ ≈ 0.344 3 10 (1 − ln 2)
(22)
(the same result follows from the more general expression (16)). In order to check these analytical predictions, the inhomogeneous CH equation was solved numerically using a finite-difference scheme, with the steep inhomogeneity approximated as b1 = sech(x − vt), which turns out to be steep enough (see results displayed below). Varying the value of v, we observed that, for v above the critical value vcr predicted by the analytical approximation (22), the kink remains immobile, developing strong fluctuations in its rear, which finally destroy the kink. This can be seen in Fig. 1(a), where the evolution of the kink is shown for v = 2vcr and = 0.1. Additionally, in Fig. 1(b) the motion of the kink’s center is shown in the (x, t) plane. We calculate the position of the center of the kink by (approximately) evaluating
∞
∞ xc = −∞
−1 |u| dx
x|u| dx
.
(23)
−∞
The accuracy of the method was also confirmed by numerical interpolation schemes, similar to the ones used in [9], evaluating the zero crossing of the field u. As is seen from the figures, the kink cannot overcome the friction force and inertia, and, barring a short transient (when it tries to trail behind the inhomogeneity), it remains immobile, until it gets destroyed. On the contrary, when v is taken below vcr , the kink is able to follow the inhomogeneity. In this case, the kink moves, remaining essentially undistorted, as is seen in Fig. 2(a) (for v = vcr /2 and = 0.1). Furthermore, as is shown in Fig. 2(b), where the motion of the kink’s center is shown, after a short transient the kink assumes a value of the velocity fluctuating about v. In extensive numerical experiments, we have seen that the behavior described above is generic for a wide
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271
(a)
(a)
(b)
(b)
Fig. 1. (a) The evolution of the front for v = 2vcr and = 0.1. The spatial profile of the front is shown at t = 0, t = 5, and t = 10. (b) The position of the front as a function of time in the same case.
Fig. 2. The same as in Fig. 1 for v = 0.5vcr and = 0.1. The spatial profile is shown here at t = 0, t = 20, and t = 40.
range of values of v, provided that is small enough. This is expected, since our analytical results were obtained through a perturbative approach based on the smallness of . In order to understand what happens as is increased, we have found the maximum velocity of the inhomogeneity, at which the kink could still
be dragged by it (i.e., the critical velocity), as a function of from direct simulations, see the result in Fig. 3. It is evident that the analytical prediction for vcr , given by Eq. (22), is in excellent agreement with the numerical results for values of up to 0.25, whereas above this value the perturbation theory
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ulations, with the conclusion that the simulations confirm the leading-order analytical prediction well. If the perturbation is strong, the kink is found to be more robust than it might be expected from the perturbative results.
Acknowledgements
Fig. 3. The analytically predicted (solid line) and numerically found (star symbols) critical dragging velocity vcr vs. the inhomogeneity strength .
This work was partially supported by PENED99 Grant No. 99ED527 from the General Secretariat of Research and Technology (Greece). Constructive discussions with Profs. K. Hizanidis (The National Technical University of Athens) and N. Alikakos (The University of Athens) are acknowledged. B.A.M. appreciates hospitality of the Department of Physics at the University of Athens.
References predicts a critical velocity which is smaller than that observed in the numerical experiments. The latter result implies that, as a matter of fact, the kink is a more robust object than it might be expected on the basis of the perturbation theory.
4. Conclusions In this Letter, we studied the possibility of pulling a front by a moving inhomogeneity in the context of the Cahn–Hilliard equation. An analytical prediction for the critical velocity of the inhomogeneity required to pull the front (kink) was obtained, using both the perturbation theory and quasiparticle approximation for the front. An example of a steep dragging inhomogeneity was studied in detail by means of direct sim-
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