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The μ-derivative and its applications to finding exact solutions of the Cahn–Hilliard, Korteveg–de Vries, and Burgers equations
Journal of Colloid and Interface Science 290 (2005) 310–317 www.elsevier.com/locate/jcis
Priority communication
The µ-derivative and its applications to finding exact solutions of the Cahn–Hilliard, Korteveg–de Vries, and Burgers equations Vlad Mitlin Mitlin and Associates,10329 Azuaga St., San Diego, CA 92129, USA Received 1 March 2005; accepted 30 July 2005 Available online 19 August 2005
Abstract A new transformation termed the µ-derivative is introduced. Applying it to the Cahn–Hilliard equation yields dynamical exact solutions. It is shown that the µ-transformed Cahn–Hilliard equation can be presented in a separable form. This transformation also yields dynamical exact solutions and separable forms for other nonlinear models such as the modified Korteveg–de Vries and the Burgers equations. The general structure of a nonlinear partial differential equation that becomes separable upon applying the µ-derivative is described. 2005 Published by Elsevier Inc. Keywords: Fluid dynamics; Nonlinear science
1. Introduction Phase separation in systems with one order parameter is described by the Cahn–Hilliard equation [1,2]. This equation is typically presented in the following form: δF (v) ∂v = Λ∇ 2 , ∂t δv F= W (v) + K(∇v)2 d x.
(1.1) (1.2)
Here, v is the order parameter describing the state of the system; W is a free energy density of a homogeneous system; K is the gradient energy coefficient; and Λ is the Onsager coefficient. The function W (v) having two inflection points results in an instability of homogeneous states between those inflection points. The breakup dynamics of such an unstable state is termed the spinodal decomposition. Cook [3] extended the Cahn–Hilliard model by accounting for thermal fluctuations. Metiu et al. [4] showed that the Cahn–Hilliard model represents one of two basic scenarios of the phase transition dynamics. Specifically, the phase transition dynamics in a system with conserved order paraE-mail address: [email protected]. 0021-9797/$ – see front matter 2005 Published by Elsevier Inc. doi:10.1016/j.jcis.2005.07.069
meter is described by model (1.1), while the phase transition dynamics in the system with nonconserved order parameter is described by Eq. (1.1) with the ∇ 2 term on the r.h.s. of this equation replaced by −1. In statistical physics the Cahn– Hilliard equation is often referred to as model B, while its nonconserved-order-parameter counterpart is referred to as model A [5,6]. The dynamics of spinodal decomposition is characterized by a complex modulated structure of the order parameter. It was extensively studied numerically (see, for example, [7–11]). Analytical studies of the Cahn–Hilliard equation were also performed: for example, the instability of spatially periodical steady-state solutions was proved in [12,13]. However, up till now, no dynamical exact solutions of the Cahn–Hilliard equation had yet been found. Obviously, this equation is one of the most complex in nonlinear physics and standard methods simply do not work for it. The author has worked on problems related to the Cahn– Hilliard equation for many years [14–20], and developing a method of finding exact solutions of this equation appears to him quite important. In this paper we introduce a nonlinear transformation that yields dynamical exact solutions for the Cahn–Hilliard equation. As this work progressed, we realized that its scope is much broader than was initially
V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
intended. It turns out that we developed a new analytical method of solving nonlinear partial differential equations of a certain class. Examples of such equations are the Burgers equation [21] and the mKdV equation [22]. We also found that for equations from this class, this new transformation yields separable forms, i.e., allows one to present them as pairs of ordinary differential equations. The paper is structured as follows: in Section 2 we define this transformation and derive its main properties. In Section 3 this transformation is used to obtain exact solutions of the Burgers equation. In Section 4 it is used to obtain exact solutions of the modified Korteveg–de Vries (mKdV) equation. In Section 5 it is used to obtain exact solutions of the Cahn–Hilliard equation. A general structure of a partial differential equation yielding a separable form upon applying the new transformation is described in Section 6.
311
H (v, . . .) = −h(u, . . .)/u, v, H (v, . . .) ⇒ u, h(u, . . .) , µ
(2.8) u = 1/v,
h(u, . . .) = −H (v, . . .)/v.
(2.9)
In other words, a conservation law with a field variable u and the rate h µ-transforms into another conservation law with a field variable v and the rate H related to u and h, as shown in Eq. (2.8). Conversely, the conservation law with a field variable v and the rate H µ-transforms into the initial conservation law with a field variable u and the rate h. There is a restriction on the solutions of Eq. (2.6). Namely, since r is an argument of u and since u is the first derivative of r, u should be of the same sign in the region of solution.
3. Exact solutions of the Burgers equation 2. The µ-derivative Consider the following general equation with the structure of a conservation law: ut = hx ,
h = h(u, ux , uxx , . . .).
(2.1)
The transformation we are going to introduce is somewhat unconventional. Specifically, let us change independent variables so that u would depend on t and r where rx = u. Let us define the µ-derivative (where µ stands for “metamorphosis”) as follows: D(f (u)) (2.2) ≡ f (u) t − f (u) r rt , rx ≡ u. Dt It is easy to show the µ-derivative has the main properties of a derivative; i.e., Dg D(f ± g) Df = ± , Dt Dt Dt Dg D(f g) Df = g+ f, Dt Dt Dt D(f (g)) Dg = fg . Dt Dt Applying the µ-derivative to Eq. (2.1) yields Du = hr u − hur = u2 (h/u)r , Dt where h = h u, ur u, (ur u)r u, . . . .
(2.3) (2.4)
ut = hx ,
h = −u2 + ux .
(3.1)
Applying the µ-derivative yields Du = hr u − hur Dt or, equivalently,
(3.2)
Du (3.3) = u2 (−u + ur )r . Dt As the r.h.s. of Eq. (3.3) is homogeneous with respect to u, it is natural to seek its solutions in the form of a product of time- and r-dependent functions. Introducing this product into Eq. (3.3) reveals an important property of the µderivative, i.e., the µ-transformed Burgers equation allows for separation of variables, even though r itself depends on time,
(2.5)
u = q(r)b(t):
(2.6)
where Q is an eigenvalue. The equation for the time-dependent part of the solution, b, is easily integrated,
(2.7)
One can see that the structure of nonlinearity of the original equation (2.1) has been modified in Eqs. (2.6) and (2.7). In the following sections it will be shown that the new structure allows one to derive some exact solutions of Eq. (2.6) that is not possible for Eq. (2.1). Equations (2.1)–(2.7) allow one to formulate the following duality property of this transform: u, h(u, . . .) ⇒ v, H (v, . . .) , v = 1/u, µ
The Burgers equation is one-dimensional reduction of the Navier–Stokes equation. It can be, in appropriate variables, presented as follows:
bt = −Qb3 ,
1 1 − 2 = −2Qt, ∗2 b b
b=
q(−q + qr )r = −Q,
b∗ 1 + 2Qb∗2 t
,
(3.4)
(3.5)
where b∗ = b(0). The equation for the r-dependent part of the solution, q, can be integrated by the following change of variables: q = v exp r,
w = 1/ exp r.
Combining Eqs. (3.4) and (3.6) yields vww = −Q/v, vw = ± −2Q ln v/v ∗ ,
(3.6)
(3.7)
where v ∗ = v(1). Further analysis depends on the sign of Q.
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V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
If Q is positive, Eq. (3.7) yields 1=
v∗ 2Q(w − 1)2
1 v/v ∗
Combining Eqs. (3.11) and (3.12) yields the following expressions for x and u in terms of the variable w:
dv . √ − ln v
(3.8)
1=
4v ∗ erf − ln(v/v ∗ ) . 2 Qπ(w − 1)
∗
v = v / exp erf 2 erf(w) = √ π
−1
(3.9)
Qπ(1 − w)2 4v ∗
2 ,
w exp(−w 2 ) dw.
(3.10)
0
Next, using Eq. (3.5) yields u = b(t)v(w)/w,
r = − ln w,
u∗ = b∗ v ∗ ,
(3.11)
where u∗ is u at x = 0 and t = 0. Combining Eqs. (3.5) and (3.11) yields rx = b(t)v(w)/w,
dw , v(w)
u = −wx /w = b(t)v(w)/w. (3.13)
Introducing Eq. (3.10) into Eqs. (3.13) yields the solution in the form of the following pair of equations:
Now v can be expressed as a function of w:
1 w
Equation (3.8) can be rewritten as follows:
1 x= b(t)
wx = −b(t)v(w).
(3.12)
u∗ , (3.14) 2 2 w 1 + 2Qb∗2 t exp erf−1 Qπ(1−w) ∗ 4u
1 2 1 + 2Qb∗2 t Qπ(1 − w )2 −1 dw exp erf . x= u∗ 4u∗ w (3.15) Figs. 1–3 present an example of solutions (3.14) and (3.15). One can see that depending on a small change in the value of parameter u∗ the wave profiles look very different. This is because the system is in the vicinity of the bifurcation value of parameter Q, Q∗ , determined from the following relation:
u=
Q∗ π (3.16) = 1. 4u∗ As a result, wave profiles shown in Fig. 1 are defined for all positive x; they are upper-bounded and bent downward. Wave profiles shown in Fig. 3 are space-localized, unbounded, and bent upward. Fig. 2 shows wave profiles in the bifurcation point; they are nearly straight lines.
Fig. 1. An example of solutions (3.14) and (3.15) at Q = 0.25, u∗ = π/16 − 0.001, b∗ = 1, and t = 0, 0.6, 1.2, and 1.8.
V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
Fig. 2. An example of solutions (3.14) and (3.15) at Q = 0.25, u∗ = π/16, b∗ = 1, and t = 0, 0.6, 1.2, and 1.8.
Fig. 3. An example of solutions (3.14) and (3.15) at Q = 0.25, u∗ = π/16 + 0.001, b∗ = 1, and t = 0, 0.6, 1.2, and 1.8.
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V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
In the case of negative Q, Eq. (3.10) is replaced by the following equation
2 Qπ(1 − w)2 ∗ −1 − , v = v exp erfi (3.17) 4v ∗ and we have the solution in the form of the following pair of equations:
2 2 Qπ(1 − w) u∗ − , exp erfi−1 u= 4u∗ w 1 + 2Qb∗2 t (3.18) ∗2 1 + 2Qb t x= u∗
2 1 )2 Qπ(1 − w × dw / exp erfi−1 − . (3.19) 4u∗ w
In Eqs. (3.17)–(3.19) we defined 2 erfi(w) = i · erf(i · w) = √ π
w exp w 2 dw.
(3.20)
The procedure of solving the equation for r is quite nonobvious. It is based on presenting the equation Q + q q 2 + (q 2 )rr r = 0 (4.6) in the form 1 1 −Qs = 2 + 2, sr rr sr where s = − q 2 + (q 2 )rr ,
These solutions, unlike the ones shown in Figs. 1–3, are time-localized.
q = Q/sr .
(4.8)
Changing variables in Eq. (4.7) yields 2fss = f 3 + Qsf,
(4.9)
where f=
1 . sr
(4.10)
Further consideration depends on the sign of Q. At positive Q, changing variables in Eq. (4.9) one more time as s = ak,
0
(4.7)
f = cg,
a = (2/Q)1/3 ,
c = Q1/3 /25/6
(4.11)
yields 4. Exact solutions of the mKdV equation
gkk = 2g 3 + kg.
The µ-derivative transformation is applicable to other well-known nonlinear equations. As another example, consider one of the main equations of the soliton theory, the mKdV equation [23]:
Equation (4.12) is a particular case of one of the six Painleve equations, specifically, the second one [24]. As is known [23], the appearance of a Painleve transcendent in the process of solving an equation of the soliton theory means that we are on the right track. Thus, we have
ut = hx ,
g = P (k) = P (k, 0),
h = −u3 − 2uxx .
(4.1)
Applying the µ-derivative to Eq. (4.1) yields Du = hr u − hur Dt or, equivalently,
Du = −u2 u2 + (u2 )rr r . (4.3) Dt As we did in the case of the Burgers equation, let us now seek solutions of Eq. (4.3) in the form of a product of timeand r-dependent functions. This yields u = q(r)b(t): bt = −Qb4 , Q + q q 2 + (q 2 )rr r = 0, (4.4) where Q is an eigenvalue. The equation for b(t) is easily integrated, 1 1 − = −3Qt, b∗3 b3 where b∗ = b(0).
b=
(1 + 3Qb∗3 t)1/3
,
(4.13)
where (4.2)
b∗
(4.12)
(4.5)
Pkk (k, j ) = 2P 3 (k, j ) + kP (k, j ) + j.
(4.14)
Next, introducing Eqs. (4.8), (4.10), and (4.11) into Eq. (4.13) yields f = cP (s/a) = q/Q.
(4.15)
Combining Eqs. (4.8) and (4.15) yields √ a p + prr = − 2 2 P −1 p , p = (q/cQ)2 . c Q Integrating Eq. (4.16) yields √ ∗ p 4a ∗2 2 pr = P −1 (p )p dp , p − p + c2 Q2 √
(4.16)
(4.17)
p
where p ∗ = p(0). Equation (4.17) is equivalent to the following equation:
V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
315
Fig. 4. An example of solution (4.20) at u∗ = 1, Q = 0.1, b∗ = 1, and t = 0, 3, 15, and 63.
4a 2ux = b2 (t) 2 2 c2 Q2 c Q
u∗ /b(t)cQ
approximate solution (4.20) overestimates the wave front advancement, compared to the exact solution (4.19). At negative Q the solution has the form
P −1 (p )p dp
u/b(t)cQ
u∗4 u4 + 4 − 4 4 4 b (0)c Q b (t)c4 Q4
1/2 .
(4.18)
cQ x= b(t)
Solving Eq. (4.18) yields cQ x= b(t)
u∗ /b(t)cQ
u∗ /b(t)cQ 2
ac Q u/b(t)cQ
+
2
u 4 u∗4 − 4b4 (0) 4b4 (t)
P
+ −1
(p )p dp
−1/2
(4.19)
Equation (4.19) presents a solution of the mKdV equation as a dependence of x on u and t. At small Q, solution (4.19) takes the form U ∗ X= U
dU , √ U ∗4 −U 4
X = xb(t)/2cQ,
−u∗ /b(t)cQ 2
Pi−1 (p )p dp
2
ac Q
−u /b(t)cQ
−u/b(t)cQ
u 4 u∗4 − 4b4 (0) 4b4 (t)
−1/2
du ,
(4.21)
where
u /b(t)cQ
du .
−u∗ /b(t)cQ
Pi(k) = i · P (i · k):
Pikk (k) = 2Pi3 (k) − kPi(k)
(4.22)
c = (−Q)1/3 /25/6 .
(4.23)
and a = (−2/Q)1/3 ,
This solution has a finite lifetime, unlike solutions (4.19) and (4.20).
U = u/b(t).
(4.20) Figs. 4 and 5 show examples of solutions (4.20) and (4.19) at u∗ = 1, Q = 0.1, and t = 0, 10, 20, and 30. These solutions exist for all t and are localized in space, the localization area expanding with time, the maximum value decreasing with time. We found that the larger the Q, the more the
5. Exact solutions of the Cahn–Hilliard equation In this section the µ-derivative is applied to the Cahn– Hilliard equation in the case of the Ginsburg–Landau free energy functional. Specifically, we will consider Eq. (1.1) at W (v) = −av 2 /2 + v 4 /4,
K = 1,
Λ = 1.
(5.1)
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V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
Fig. 5. An example of solution (4.19) at u∗ = 1, Q = 0.1, b∗ = 1, and t = 0, 3, 15, and 63.
The Cahn–Hilliard equation also allows for separation of variables upon applying the µ-derivative. As this equation has the highest order of spatial derivatives among all the equations considered in this paper, this is the hardest problem to be discussed. Consider the Cahn–Hilliard equation with W (v) given by Eq. (5.1) at the critical point, e.g., at a = 0. Upon applying the µ-derivative, this equation takes the form
Changing variables in Eq. (5.6) yields
Dv 3 = v − v(v 2 )rr rr v 2 . (5.2) Dt Let us seek solutions of Eq. (5.2) in the form of a product of time- and r-dependent functions. This yields v = b(t)q(r): bt = −Qb5 , −Q = q q 3 − q(q 2 )rr rr , (5.3) where Q is an eigenvalue. The solution for b(t) is
Solving Eq. (5.8) for s yields
b=
b∗ (1 + 4Qb∗4 t)1/4
,
−sp/Q3 = (p 3 − 2pss )s ,
Combining Eqs. (5.3) and (5.5) yields the following equation for s: 1 1 1 . −s/Q3 = 3 − (5.6) sr sr2 rr r sr
(5.7)
So far, we found only one solution of Eq. (5.7); e.g., at Q < 0, p = s/ −3Q3 . (5.8) s 2 = 2r −3Q3 ,
q 2 = 2r −Q/3.
(5.9)
Combining Eqs. (5.9) and (5.2) yields v 2 = b2 2r −Q/3.
(5.10)
Finally, introducing Eq. (5.3) into Eq. (5.10) yields
(5.4)
where b∗ = b(0). To construct a solution for q(r) introduce an auxiliary variable s: Q = qsr , s = − q 3 − q(q 2 )rr r . (5.5)
p = 1/sr = q/Q.
v=x −
Qb∗4 . 3(1 + 4Qb∗4 t)
(5.11)
It is interesting that the Burgers equation also has this kind of solution; e.g., u=
x/X . 1 + 2t/X
(5.12)
Solution (5.11) has a finite lifetime. Also, it satisfies Eq. (1.1) at any a.
V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
6. Discussion
physics as well. We hope to present more examples of the µ-derivative usage in future publications.
In this paper we present a new method of solving nonlinear partial differential equations in the form of a conservation law. The method consists in applying the µ-derivative to a given equation. A common feature of the equations considered in this paper is that their µ-transforms can be presented in a separable form. For an equation to have a separable form upon applying the µ-derivative, it has to have a certain structure. Specifically, the function h in Eq. (2.1) should be presentable as the sum of N terms: the ith term is a “product” of ni d/dx and mi u terms. The condition imposed on the equation whose µ-transform is separable is n1 + m1 = · · · = nN + mN .
(6.1)
The hardest task following the separation of variables in a µ-transformed equation is solving an equation for its rdependent part. Below, we will outline a method that appears to be general for solvable µ-transformed equations. Specifically, if Eq. (2.6) is separable, its r-dependent part can be presented in the following form, h(q, . . .) −Q = q (6.2) , q r where Q is an eigenvalue and q(r) is the r-dependent part of the solution. We found that rather than solving Eq. (6.2) for q it is often better to try to find a solution in terms of the auxiliary variable s defined as follows: s=−
h(q, . . .) , q
317
q=
Q . sr
Acknowledgment I am grateful to Carole for all her help in preparation of this paper.
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
(6.3)
Combining Eqs. (6.2) and (6.3) yields the following equation for s: Q Q , . . . = 0. s +h (6.4) sr sr Unlike Eq. (6.2), Eq. (6.4) often turns to be solvable. We believe that the µ-derivative will be useful for finding exact solutions of other nonlinear equations of mathematical
[17] [18] [19] [20] [21] [22] [23] [24]
J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 31 (1959) 688. J.W. Cahn, Acta Metall. 9 (1961) 795. H.E. Cook, Acta Metall. 18 (1970) 297. H. Metiu, K. Kitahara, J. Ross, J. Chem. Phys. 65 (1976) 393. P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. J.D. Gunton, M. SanMiguel, P. Sahni, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, vol. 8, Academic Press, New York, 1983, pp. 276–466. Y. Oono, S. Puri, Phys. Rev. Lett. 58 (1987) 836. O.T. Valls, G.F. Mazenko, Phys. Rev. B 38 (1988) 520. T.M. Rogers, R.C. Desai, Phys. Rev. B 39 (1989) 11956. G. Brown, A. Chakrabarti, J. Chem. Phys. 98 (1993) 2451. J. Garcia-Ojalvo, A.M. Lacasta, J.M. Sancho, R. Toral, Europhys. Lett. 42 (1998) 125. A. Novick-Cohen, J. Stat. Phys. 38 (3/4) (1985) 707. A.B. Givental, Teor. Mat. Fiz. 76 (3) (1988) 462; Theor. Math. Phys. (USSR) 76 (3) (1988) 989. V.S. Mitlin, L.I. Manevich, I.Ya. Erukhimovich, Zh. Eksp. Teor. Fiz. 88 (2) (1985) 495; Sov. Phys. JETP 61 (2) (1985) 290. V.S. Mitlin, Zh. Eksp. Teor. Fiz. 95 (1989) 1826; Sov. Phys. JETP 68 (1989) 1056. V.S. Mitlin, Zh. Eksp. Teor. Fiz. 98 (2) (1990) 4554; Sov. Phys. JETP 71 (2) (1990) 308. V.S. Mitlin, L.I. Manevich, J. Polym. Sci. B 28 (1990) 1. V.S. Mitlin, Nonlinear Dynamics of Reservoir Mixtures, CRC Press, Boca Raton, 1993. V.S. Mitlin, N.V. Petviashvili, Phys. Lett. A 192 (1994) 323. V.S. Mitlin, Phys. Lett. A 327 (2004) 455. J.M. Burgers, Adv. Appl. Mech. 1 (1948) 171. R.M. Miura, J. Math. Phys. 9 (1968) 1202. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. E. Hille, Ordinary Differential Equations in the Complex Domain, Dover, New York, 1997.
Priority communication
The µ-derivative and its applications to finding exact solutions of the Cahn–Hilliard, Korteveg–de Vries, and Burgers equations Vlad Mitlin Mitlin and Associates,10329 Azuaga St., San Diego, CA 92129, USA Received 1 March 2005; accepted 30 July 2005 Available online 19 August 2005
Abstract A new transformation termed the µ-derivative is introduced. Applying it to the Cahn–Hilliard equation yields dynamical exact solutions. It is shown that the µ-transformed Cahn–Hilliard equation can be presented in a separable form. This transformation also yields dynamical exact solutions and separable forms for other nonlinear models such as the modified Korteveg–de Vries and the Burgers equations. The general structure of a nonlinear partial differential equation that becomes separable upon applying the µ-derivative is described. 2005 Published by Elsevier Inc. Keywords: Fluid dynamics; Nonlinear science
1. Introduction Phase separation in systems with one order parameter is described by the Cahn–Hilliard equation [1,2]. This equation is typically presented in the following form: δF (v) ∂v = Λ∇ 2 , ∂t δv F= W (v) + K(∇v)2 d x.
(1.1) (1.2)
Here, v is the order parameter describing the state of the system; W is a free energy density of a homogeneous system; K is the gradient energy coefficient; and Λ is the Onsager coefficient. The function W (v) having two inflection points results in an instability of homogeneous states between those inflection points. The breakup dynamics of such an unstable state is termed the spinodal decomposition. Cook [3] extended the Cahn–Hilliard model by accounting for thermal fluctuations. Metiu et al. [4] showed that the Cahn–Hilliard model represents one of two basic scenarios of the phase transition dynamics. Specifically, the phase transition dynamics in a system with conserved order paraE-mail address: [email protected]. 0021-9797/$ – see front matter 2005 Published by Elsevier Inc. doi:10.1016/j.jcis.2005.07.069
meter is described by model (1.1), while the phase transition dynamics in the system with nonconserved order parameter is described by Eq. (1.1) with the ∇ 2 term on the r.h.s. of this equation replaced by −1. In statistical physics the Cahn– Hilliard equation is often referred to as model B, while its nonconserved-order-parameter counterpart is referred to as model A [5,6]. The dynamics of spinodal decomposition is characterized by a complex modulated structure of the order parameter. It was extensively studied numerically (see, for example, [7–11]). Analytical studies of the Cahn–Hilliard equation were also performed: for example, the instability of spatially periodical steady-state solutions was proved in [12,13]. However, up till now, no dynamical exact solutions of the Cahn–Hilliard equation had yet been found. Obviously, this equation is one of the most complex in nonlinear physics and standard methods simply do not work for it. The author has worked on problems related to the Cahn– Hilliard equation for many years [14–20], and developing a method of finding exact solutions of this equation appears to him quite important. In this paper we introduce a nonlinear transformation that yields dynamical exact solutions for the Cahn–Hilliard equation. As this work progressed, we realized that its scope is much broader than was initially
V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
intended. It turns out that we developed a new analytical method of solving nonlinear partial differential equations of a certain class. Examples of such equations are the Burgers equation [21] and the mKdV equation [22]. We also found that for equations from this class, this new transformation yields separable forms, i.e., allows one to present them as pairs of ordinary differential equations. The paper is structured as follows: in Section 2 we define this transformation and derive its main properties. In Section 3 this transformation is used to obtain exact solutions of the Burgers equation. In Section 4 it is used to obtain exact solutions of the modified Korteveg–de Vries (mKdV) equation. In Section 5 it is used to obtain exact solutions of the Cahn–Hilliard equation. A general structure of a partial differential equation yielding a separable form upon applying the new transformation is described in Section 6.
311
H (v, . . .) = −h(u, . . .)/u, v, H (v, . . .) ⇒ u, h(u, . . .) , µ
(2.8) u = 1/v,
h(u, . . .) = −H (v, . . .)/v.
(2.9)
In other words, a conservation law with a field variable u and the rate h µ-transforms into another conservation law with a field variable v and the rate H related to u and h, as shown in Eq. (2.8). Conversely, the conservation law with a field variable v and the rate H µ-transforms into the initial conservation law with a field variable u and the rate h. There is a restriction on the solutions of Eq. (2.6). Namely, since r is an argument of u and since u is the first derivative of r, u should be of the same sign in the region of solution.
3. Exact solutions of the Burgers equation 2. The µ-derivative Consider the following general equation with the structure of a conservation law: ut = hx ,
h = h(u, ux , uxx , . . .).
(2.1)
The transformation we are going to introduce is somewhat unconventional. Specifically, let us change independent variables so that u would depend on t and r where rx = u. Let us define the µ-derivative (where µ stands for “metamorphosis”) as follows: D(f (u)) (2.2) ≡ f (u) t − f (u) r rt , rx ≡ u. Dt It is easy to show the µ-derivative has the main properties of a derivative; i.e., Dg D(f ± g) Df = ± , Dt Dt Dt Dg D(f g) Df = g+ f, Dt Dt Dt D(f (g)) Dg = fg . Dt Dt Applying the µ-derivative to Eq. (2.1) yields Du = hr u − hur = u2 (h/u)r , Dt where h = h u, ur u, (ur u)r u, . . . .
(2.3) (2.4)
ut = hx ,
h = −u2 + ux .
(3.1)
Applying the µ-derivative yields Du = hr u − hur Dt or, equivalently,
(3.2)
Du (3.3) = u2 (−u + ur )r . Dt As the r.h.s. of Eq. (3.3) is homogeneous with respect to u, it is natural to seek its solutions in the form of a product of time- and r-dependent functions. Introducing this product into Eq. (3.3) reveals an important property of the µderivative, i.e., the µ-transformed Burgers equation allows for separation of variables, even though r itself depends on time,
(2.5)
u = q(r)b(t):
(2.6)
where Q is an eigenvalue. The equation for the time-dependent part of the solution, b, is easily integrated,
(2.7)
One can see that the structure of nonlinearity of the original equation (2.1) has been modified in Eqs. (2.6) and (2.7). In the following sections it will be shown that the new structure allows one to derive some exact solutions of Eq. (2.6) that is not possible for Eq. (2.1). Equations (2.1)–(2.7) allow one to formulate the following duality property of this transform: u, h(u, . . .) ⇒ v, H (v, . . .) , v = 1/u, µ
The Burgers equation is one-dimensional reduction of the Navier–Stokes equation. It can be, in appropriate variables, presented as follows:
bt = −Qb3 ,
1 1 − 2 = −2Qt, ∗2 b b
b=
q(−q + qr )r = −Q,
b∗ 1 + 2Qb∗2 t
,
(3.4)
(3.5)
where b∗ = b(0). The equation for the r-dependent part of the solution, q, can be integrated by the following change of variables: q = v exp r,
w = 1/ exp r.
Combining Eqs. (3.4) and (3.6) yields vww = −Q/v, vw = ± −2Q ln v/v ∗ ,
(3.6)
(3.7)
where v ∗ = v(1). Further analysis depends on the sign of Q.
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If Q is positive, Eq. (3.7) yields 1=
v∗ 2Q(w − 1)2
1 v/v ∗
Combining Eqs. (3.11) and (3.12) yields the following expressions for x and u in terms of the variable w:
dv . √ − ln v
(3.8)
1=
4v ∗ erf − ln(v/v ∗ ) . 2 Qπ(w − 1)
∗
v = v / exp erf 2 erf(w) = √ π
−1
(3.9)
Qπ(1 − w)2 4v ∗
2 ,
w exp(−w 2 ) dw.
(3.10)
0
Next, using Eq. (3.5) yields u = b(t)v(w)/w,
r = − ln w,
u∗ = b∗ v ∗ ,
(3.11)
where u∗ is u at x = 0 and t = 0. Combining Eqs. (3.5) and (3.11) yields rx = b(t)v(w)/w,
dw , v(w)
u = −wx /w = b(t)v(w)/w. (3.13)
Introducing Eq. (3.10) into Eqs. (3.13) yields the solution in the form of the following pair of equations:
Now v can be expressed as a function of w:
1 w
Equation (3.8) can be rewritten as follows:
1 x= b(t)
wx = −b(t)v(w).
(3.12)
u∗ , (3.14) 2 2 w 1 + 2Qb∗2 t exp erf−1 Qπ(1−w) ∗ 4u
1 2 1 + 2Qb∗2 t Qπ(1 − w )2 −1 dw exp erf . x= u∗ 4u∗ w (3.15) Figs. 1–3 present an example of solutions (3.14) and (3.15). One can see that depending on a small change in the value of parameter u∗ the wave profiles look very different. This is because the system is in the vicinity of the bifurcation value of parameter Q, Q∗ , determined from the following relation:
u=
Q∗ π (3.16) = 1. 4u∗ As a result, wave profiles shown in Fig. 1 are defined for all positive x; they are upper-bounded and bent downward. Wave profiles shown in Fig. 3 are space-localized, unbounded, and bent upward. Fig. 2 shows wave profiles in the bifurcation point; they are nearly straight lines.
Fig. 1. An example of solutions (3.14) and (3.15) at Q = 0.25, u∗ = π/16 − 0.001, b∗ = 1, and t = 0, 0.6, 1.2, and 1.8.
V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
Fig. 2. An example of solutions (3.14) and (3.15) at Q = 0.25, u∗ = π/16, b∗ = 1, and t = 0, 0.6, 1.2, and 1.8.
Fig. 3. An example of solutions (3.14) and (3.15) at Q = 0.25, u∗ = π/16 + 0.001, b∗ = 1, and t = 0, 0.6, 1.2, and 1.8.
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In the case of negative Q, Eq. (3.10) is replaced by the following equation
2 Qπ(1 − w)2 ∗ −1 − , v = v exp erfi (3.17) 4v ∗ and we have the solution in the form of the following pair of equations:
2 2 Qπ(1 − w) u∗ − , exp erfi−1 u= 4u∗ w 1 + 2Qb∗2 t (3.18) ∗2 1 + 2Qb t x= u∗
2 1 )2 Qπ(1 − w × dw / exp erfi−1 − . (3.19) 4u∗ w
In Eqs. (3.17)–(3.19) we defined 2 erfi(w) = i · erf(i · w) = √ π
w exp w 2 dw.
(3.20)
The procedure of solving the equation for r is quite nonobvious. It is based on presenting the equation Q + q q 2 + (q 2 )rr r = 0 (4.6) in the form 1 1 −Qs = 2 + 2, sr rr sr where s = − q 2 + (q 2 )rr ,
These solutions, unlike the ones shown in Figs. 1–3, are time-localized.
q = Q/sr .
(4.8)
Changing variables in Eq. (4.7) yields 2fss = f 3 + Qsf,
(4.9)
where f=
1 . sr
(4.10)
Further consideration depends on the sign of Q. At positive Q, changing variables in Eq. (4.9) one more time as s = ak,
0
(4.7)
f = cg,
a = (2/Q)1/3 ,
c = Q1/3 /25/6
(4.11)
yields 4. Exact solutions of the mKdV equation
gkk = 2g 3 + kg.
The µ-derivative transformation is applicable to other well-known nonlinear equations. As another example, consider one of the main equations of the soliton theory, the mKdV equation [23]:
Equation (4.12) is a particular case of one of the six Painleve equations, specifically, the second one [24]. As is known [23], the appearance of a Painleve transcendent in the process of solving an equation of the soliton theory means that we are on the right track. Thus, we have
ut = hx ,
g = P (k) = P (k, 0),
h = −u3 − 2uxx .
(4.1)
Applying the µ-derivative to Eq. (4.1) yields Du = hr u − hur Dt or, equivalently,
Du = −u2 u2 + (u2 )rr r . (4.3) Dt As we did in the case of the Burgers equation, let us now seek solutions of Eq. (4.3) in the form of a product of timeand r-dependent functions. This yields u = q(r)b(t): bt = −Qb4 , Q + q q 2 + (q 2 )rr r = 0, (4.4) where Q is an eigenvalue. The equation for b(t) is easily integrated, 1 1 − = −3Qt, b∗3 b3 where b∗ = b(0).
b=
(1 + 3Qb∗3 t)1/3
,
(4.13)
where (4.2)
b∗
(4.12)
(4.5)
Pkk (k, j ) = 2P 3 (k, j ) + kP (k, j ) + j.
(4.14)
Next, introducing Eqs. (4.8), (4.10), and (4.11) into Eq. (4.13) yields f = cP (s/a) = q/Q.
(4.15)
Combining Eqs. (4.8) and (4.15) yields √ a p + prr = − 2 2 P −1 p , p = (q/cQ)2 . c Q Integrating Eq. (4.16) yields √ ∗ p 4a ∗2 2 pr = P −1 (p )p dp , p − p + c2 Q2 √
(4.16)
(4.17)
p
where p ∗ = p(0). Equation (4.17) is equivalent to the following equation:
V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
315
Fig. 4. An example of solution (4.20) at u∗ = 1, Q = 0.1, b∗ = 1, and t = 0, 3, 15, and 63.
4a 2ux = b2 (t) 2 2 c2 Q2 c Q
u∗ /b(t)cQ
approximate solution (4.20) overestimates the wave front advancement, compared to the exact solution (4.19). At negative Q the solution has the form
P −1 (p )p dp
u/b(t)cQ
u∗4 u4 + 4 − 4 4 4 b (0)c Q b (t)c4 Q4
1/2 .
(4.18)
cQ x= b(t)
Solving Eq. (4.18) yields cQ x= b(t)
u∗ /b(t)cQ
u∗ /b(t)cQ 2
ac Q u/b(t)cQ
+
2
u 4 u∗4 − 4b4 (0) 4b4 (t)
P
+ −1
(p )p dp
−1/2
(4.19)
Equation (4.19) presents a solution of the mKdV equation as a dependence of x on u and t. At small Q, solution (4.19) takes the form U ∗ X= U
dU , √ U ∗4 −U 4
X = xb(t)/2cQ,
−u∗ /b(t)cQ 2
Pi−1 (p )p dp
2
ac Q
−u /b(t)cQ
−u/b(t)cQ
u 4 u∗4 − 4b4 (0) 4b4 (t)
−1/2
du ,
(4.21)
where
u /b(t)cQ
du .
−u∗ /b(t)cQ
Pi(k) = i · P (i · k):
Pikk (k) = 2Pi3 (k) − kPi(k)
(4.22)
c = (−Q)1/3 /25/6 .
(4.23)
and a = (−2/Q)1/3 ,
This solution has a finite lifetime, unlike solutions (4.19) and (4.20).
U = u/b(t).
(4.20) Figs. 4 and 5 show examples of solutions (4.20) and (4.19) at u∗ = 1, Q = 0.1, and t = 0, 10, 20, and 30. These solutions exist for all t and are localized in space, the localization area expanding with time, the maximum value decreasing with time. We found that the larger the Q, the more the
5. Exact solutions of the Cahn–Hilliard equation In this section the µ-derivative is applied to the Cahn– Hilliard equation in the case of the Ginsburg–Landau free energy functional. Specifically, we will consider Eq. (1.1) at W (v) = −av 2 /2 + v 4 /4,
K = 1,
Λ = 1.
(5.1)
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V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
Fig. 5. An example of solution (4.19) at u∗ = 1, Q = 0.1, b∗ = 1, and t = 0, 3, 15, and 63.
The Cahn–Hilliard equation also allows for separation of variables upon applying the µ-derivative. As this equation has the highest order of spatial derivatives among all the equations considered in this paper, this is the hardest problem to be discussed. Consider the Cahn–Hilliard equation with W (v) given by Eq. (5.1) at the critical point, e.g., at a = 0. Upon applying the µ-derivative, this equation takes the form
Changing variables in Eq. (5.6) yields
Dv 3 = v − v(v 2 )rr rr v 2 . (5.2) Dt Let us seek solutions of Eq. (5.2) in the form of a product of time- and r-dependent functions. This yields v = b(t)q(r): bt = −Qb5 , −Q = q q 3 − q(q 2 )rr rr , (5.3) where Q is an eigenvalue. The solution for b(t) is
Solving Eq. (5.8) for s yields
b=
b∗ (1 + 4Qb∗4 t)1/4
,
−sp/Q3 = (p 3 − 2pss )s ,
Combining Eqs. (5.3) and (5.5) yields the following equation for s: 1 1 1 . −s/Q3 = 3 − (5.6) sr sr2 rr r sr
(5.7)
So far, we found only one solution of Eq. (5.7); e.g., at Q < 0, p = s/ −3Q3 . (5.8) s 2 = 2r −3Q3 ,
q 2 = 2r −Q/3.
(5.9)
Combining Eqs. (5.9) and (5.2) yields v 2 = b2 2r −Q/3.
(5.10)
Finally, introducing Eq. (5.3) into Eq. (5.10) yields
(5.4)
where b∗ = b(0). To construct a solution for q(r) introduce an auxiliary variable s: Q = qsr , s = − q 3 − q(q 2 )rr r . (5.5)
p = 1/sr = q/Q.
v=x −
Qb∗4 . 3(1 + 4Qb∗4 t)
(5.11)
It is interesting that the Burgers equation also has this kind of solution; e.g., u=
x/X . 1 + 2t/X
(5.12)
Solution (5.11) has a finite lifetime. Also, it satisfies Eq. (1.1) at any a.
V. Mitlin / Journal of Colloid and Interface Science 290 (2005) 310–317
6. Discussion
physics as well. We hope to present more examples of the µ-derivative usage in future publications.
In this paper we present a new method of solving nonlinear partial differential equations in the form of a conservation law. The method consists in applying the µ-derivative to a given equation. A common feature of the equations considered in this paper is that their µ-transforms can be presented in a separable form. For an equation to have a separable form upon applying the µ-derivative, it has to have a certain structure. Specifically, the function h in Eq. (2.1) should be presentable as the sum of N terms: the ith term is a “product” of ni d/dx and mi u terms. The condition imposed on the equation whose µ-transform is separable is n1 + m1 = · · · = nN + mN .
(6.1)
The hardest task following the separation of variables in a µ-transformed equation is solving an equation for its rdependent part. Below, we will outline a method that appears to be general for solvable µ-transformed equations. Specifically, if Eq. (2.6) is separable, its r-dependent part can be presented in the following form, h(q, . . .) −Q = q (6.2) , q r where Q is an eigenvalue and q(r) is the r-dependent part of the solution. We found that rather than solving Eq. (6.2) for q it is often better to try to find a solution in terms of the auxiliary variable s defined as follows: s=−
h(q, . . .) , q
317
q=
Q . sr
Acknowledgment I am grateful to Carole for all her help in preparation of this paper.
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
(6.3)
Combining Eqs. (6.2) and (6.3) yields the following equation for s: Q Q , . . . = 0. s +h (6.4) sr sr Unlike Eq. (6.2), Eq. (6.4) often turns to be solvable. We believe that the µ-derivative will be useful for finding exact solutions of other nonlinear equations of mathematical
[17] [18] [19] [20] [21] [22] [23] [24]
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