Phase Transition Solutions in Geometrically Constrained Magnetic Domain Wall Models Shouxin Chen School of Mathematics Henan University Kaifeng, Henan 475004, PR of China Yisong Yang Department of Mathematics Polytechnic Institute of New York University Brooklyn, NY 11201, USA Abstract Recent work on magnetic phase transition in nanoscale systems indicates that new physical phenomena, in particular, the Bloch wall width narrowing, arise as a consequence of geometrical confinement of magnetization and leads to the introduction of geometrically constrained domain wall models. In this paper, we present a systematic mathematical analysis on the existence of the solutions of the basic governing equations in such domain wall models. We show that, when the cross section of the geometric constriction is a simple step function, the solutions may be obtained by minimizing the domain wall energy over the constriction and solving the Bogomol’nyi equation outside the constriction. When the cross section and potential density are both even, we establish the existence of an odd domain wall solution realizing the phase transition process between two adjacent domain phases. When the cross section satisfies a certain integrability condition, we prove that a domain wall solution always exists which links two arbitrarily designated domain phases.

1

Introduction

Instantons, monopoles, dyons, vortices, and kinks are topological solitons which are solutions of systems of partial differential equations representing stable and smoothly changing structures which are localized in space and often owe their stability to associated topological characteristics. These solitons find applications in a wide range 1

of areas including particle physics, cosmology, condensed-matter, biology, superfluids and superconductivity, and magnetism. Kinks, also known as domain walls, are one-dimensional solitons describing the simplest but widely useful phase transition process between two distinct ground states in a mathematically precise way. Besides, although low dimensional, kinks or domain walls may serve as interpolation objects, and sometimes play crucial roles, for the understanding of various physical processes involving all spectrum of higher dimensional solitons such as instantons, monoples, dyons, vortices, strings, and membranes. For example, Zel’dovich and Khlopov  and Preskill  observed that there is a serious conflict between grand unified field theory (GUT) and standard cosmology: GUT symmetry breaking in the early universe would produce an over-abundance of monopoles which would overclose the universe by many orders of magnitude. This is known as the cosmological monopole problem . A recent proposal for the resolution of the monopole problem is due to Dvali, Liu, and Vachaspati  who consider a theory in which both monopoles and domain walls are present and domain walls are able to sweep up monopoles and antimonopoles and diffuse the monopole charge. Such a mechanism leads to an annihilation between monopole and antimonopole charge, thus solving the monopole problem. See also [21, 22] and references therein. In the classical context of magnetism, a magnetic domain, or simply a domain, is referred to as a region within a material which has uniform magnetization so that the moments of all the atoms are aligned with one another. The region separating two distinctly aligned magnetic domains is called a domain wall where the magnetization rotates coherently from the direction in one domain to that in the next domain so that it realizes a gradual reorientation of individual moments across a finite distance. In other words, a domain wall unambiguously describes a phase transition. It is well known that the classical Bloch wall situation in which the magnetization changes from an orientation angle θ = −π/2 to the opposite orientation angle θ = π/2, say, may be modeled by the sine-Gordon equation  whose static limit reads θ00 +

K sin(2θ) = 0, 2

−∞ < x < ∞,

(1.1)

so that the two phases or ground states are attained as the two boundary conditions given by θ(−∞) = −π/2 and θ(∞) = π/2 and the associated total energy Z ∞ E(θ) = {(θ0(x))2 + K cos2 θ(x)} dx, (1.2) −∞

is to be minimized by the solution of the equation (1.1). See  for more realistic generalizations of this problem. The sine-Gordon equation even in its original wave equation form can completely be integrated [24, 27]. In the present paper, we aim at developing a mathematical existence theory for the solutions of the geometrically constrained domain wall models proposed by Bruno  which does not allow an explicit integration and presents familiar difficulties for direct minimization due to lack of compactness . 2

Note that geometrically constrained solitons naturally arise in many other areas of physics, especially when gravitation is considered and are known to lead to new structures and consequences. For example, reformulated as a curved-space version of the Abelin Higgs model over the Poincar´e hyperbolic half space, Witten  was able to find a large class of Yang–Mills instantons; Gibbons et al showed that the presence of monopoles designates a lower bound for the ADM mass in terms of the monopole topological charge ; Kibble [15, 16] and Vilenkin and Shellard  interpreted vortices as heavily massive solitons called cosmic strings which are responsible for matter-accretion for galaxy formation in the early universe. In the context of cosmic strings, it has been established [34, 35, 36] that a gravitationally coupled curved-space metric gives rise to obstructions to the existence of multiple-charge vortices not seen in the flat-space model . In other words, nontrivial geometry introduces extra constraints into the problem not present otherwise. Motivated by modeling magnetic nanostructures, in particular, the question how micromagnetic structures respond to geometric constraints, Bruno  introduces a geometrically constrained magnetic domain wall model defined by the total energy Z ∞ E(θ) = {[A(θ0)2 + F (θ)]S(x)} dx, (1.3) −∞

where F (θ) ≥ 0 is a general potential function whose global minimum value, 0, is attained at two specially designated adjacent phases or ground states θ = ±θ0 (θ0 > 0), A > 0 is a constant, and S(x) > 0 is a given profile function of x, called the constriction cross section. Let θ1 ∈ (−θ0, θ0) be such that F attains its global maximum value over [−θ0, θ0] at η (say). Hence, if θ(x) is a configuration function connecting the two ground states ±θ0 at ±∞, it creates a lump for the bare energy density H(x) = [A(θ0)2 + F (θ)](x) (1.4) in an interval, where θ(x) passes through η. In the classical situation, S is constantvalued so that the energy (1.3) is translation-invariant and there is no mechanism to dictate the location of the lump. However, when S depends on space coordinate x so that the translation-invariance is broken, the lump tends to lock itself in the region where S stays low. As a consequence, the domain wall, which minimizes the total energy (1.3), is seen to be geometrically constrained  near the global minimum of S, referred to as the constriction , and can be obtained as a solution of the Euler–Lagrange equation of (1.3): θ00 +

S 0 (x) 0 1 dF (θ) θ − = 0. S(x) 2A dθ

(1.5)

In general, (1.5) cannot be integrated and one has to resort to approximations and truncations  to achieve a certain level of understanding of the solutions of it. Due to the relevance of geometrically constrained domain wall models to nanoscale magnetism  including applications to issues concerning domain walls pinned at 3

structural constrictions [14, 17, 20, 30], electron transport and magnetoresistance [3, 7, 11, 25, 30], and spin modulation in magnetic semiconductors [14, 19, 31], we develop here a systematic existence theory for solutions of the basic equation (1.5) governing such geometrically constrained magnetic domain walls. An outline of the rest of the paper is as follows. In Section 2, we review the three geometric constriction models of Bruno  and establish that when the cross section is defined by a simple step function (Bruno’s Model I), a least energy domain wall solution can be constructed by minimizing the domain wall energy over the constriction and solving the Bogomol’nyi equation outside the constriction. In Section 3, we consider the situation when the cross section is a smoothly changing function. We establish a general existence theorem for a domain wall solution by a dynamical shooting method, assuming that both the cross section and the potential density are even functions. These conditions cover Models II and III in  as special examples. In Section 4, we prove a uniqueness theorem for the domain wall solution obtained in Section 3. The conditions for this uniqueness theorem are fulfilled by Model III but not Model II, in . In Section 5, we present an existence theorem for domain wall solutions realizing the phase transition process between two arbitrary domain phases which do not have to be neighboring states as in the sine-Gordon model [18, 24]. The integrability condition obtained for the cross section is satisfied by Models II and III in . In Section 6, we state our conclusions. Although domain walls are all one-dimensional solitons governed by ordinary differential equations, there are still some important existence problems left unresolved, especially in the areas of high-energy physics and cosmology [5, 9, 12, 21, 22, 26, 28, 29]. Our study here may provide new methods for and insight into some of those more complicated problems.

2

Piecewise Defined Domain Wall

√ With the rescaling x 7→ Ax and omitting a proportionality constant, we may rewrite the energy (1.1) in its normalized form Z ∞ E(θ) = {[θ0(x)]2 + F (θ(x))]S(x)} dx, (2.1) −∞

where the potential density function F satisfies the standard condition F ≥ 0,

F (±θ0) = 0,

θ0 > 0.

(2.2)

To motivate our study, we first recall the three concrete models considered in . Model I. The constriction cross section function S is defined by  S0 , |x| ≤ a, S(x) = (2.3) S1 , |x| > a, 4

where and in the sequel 0 < S0 < S1 and a > 0. Furthermore, Model II and Model III are defined by  x2  S(x) = S0 1 + 2 , (2.4) a and x S(x) = S0 cosh , (2.5) a respectively. For all the three models, the most natural admissible class C over which the energy (2.1) is to be minimized so that a phase transition from the two domains given as −θ0 and θ0 is realized may be given by absolutely continuous functions satisfying the anticipated boundary conditions at ±∞ spelled out in (2.2) and the finite-energy assumption. That is, C = {θ(x) | θ(±∞) = ±θ0 and E(θ) < ∞}.

(2.6)

We start from the simplest case, Model I. Since the cross section function is discontinuous at the points x = ±a, the equation (1.5) may only be considered piecewise in (−a, a) and outside it. Therefore, we are led to constructing a domain wall profile θ(x) which is continuous everywhere in (−∞, ∞) and satisfies the equation (1.5) in the intervals (−∞, −a), (−a, a), and (a, ∞), separately, and realizes the asymptotic condition lim θ(x) = −θ0,

x→−∞

lim θ(x) = θ0.

x→∞

(2.7)

For this purpose, we use α and β to denote the intermediate values of the domain orientation parameter θ at the two boundary points x = −a and x = a of the constriction such that θ(−a) = α,

θ(a) = β,

−θ0 < α < β < θ0.

(2.8)

We may assume that α and β are not zeros of the potential function F (θ). It is not hard to prove that finite-energy condition prohibits the intervals (−θ0, α] and [β, θ0) to contain additional zeros of F (θ) (see the proof of Proposition 5.2). In other words, we have the necessary condition F (θ) > 0,

θ ∈ (−θ0, α] ∪ [β, θ0).

(2.9)

Since it is difficult to invoke a direct minimization process (cf. the discussions in Section 5 and ) in the noncompact intervals (−∞, −a] and [a, ∞), we will use the well-known indirect method of Bogomol’nyi . Thus, we can rewrite the energy of Model I as  Z α Z θ0 p  Z −a Z ∞  p E(θ) = 2S1 + F (s) ds + S1 + (θ0 − F (θ))2 dx β −∞ a 0 Z −θ a +S0 {(θ0)2 + F (θ)} dx. (2.10) −a

5

Using standard methods, it is seen that the minimization problem Z a n o 0 2 min Ea (θ) Ea(θ) = {(θ ) + F (θ)} dx < ∞ and θ satisfies (2.8)

(2.11)

−a

has a solution. On the other hand, in order to minimize the other θ-dependent functionals on the left-hand side of (2.10), we demand that θ satisfy the Bogomol’nyi equation p θ0 − F (θ) = 0 (2.12) in the intervals (−∞, −a) and (a, ∞), whose solutions are formally expressed by the formulas Z θ Z θ ds ds p p = x + a, x < −a; = x − a, x > a, (2.13) F (s) F (s) α β

respectively. Consequently, in order to fulfill the asymptotic condition (2.7), we arrive at the necessary and sufficient condition Z α Z θ0 ds ds p p = ∞, = ∞, (2.14) F (s) F (s) −θ0 β

to be imposed near ±θ0 . (Such a condition is general enough to accommodate all classical examples such as the double-well type F (θ) = λ(θ2 − θ02)2 and the sineGordon type F (θ) = K cos2 θ.) For a broader range of applications, we may also consider the exceptional cases when (2.14) or part of it is violated. For example, if Z θ0 ds p < ∞, (2.15) F (s) β

we can use (2.13) to see that there is some finite number b > a such that θ(b) = θ0. In this situation, we set θ(x) = θ0,

x > b;

θ(x) is defined implicitly by (2.13) when a ≤ x ≤ b.

(2.16)

In particular, such constructed θ satisfies the asymptotic condition (2.7) and enjoys the same minimization property as described before. An interesting question asks whether the domain wall phase transition may be achieved totally within the constriction −a < x < a. In other words, whether it is possible to minimize the energy with a configuration function connecting the ground states −θ0 and θ0 at the finite boundary points x = −a and x = a already. In fact, here we show that, again using the Bogomol’nyi method, such a situation will happen under a specific critical condition. For simplicity, we assume that F (θ) > 0,

θ ∈ (−θ0, θ). 6

(2.17)

Moreover, the above discussion indicates that, in order to reach ±θ0 finitely, we need to require the convergence condition Z θ0 ds p < ∞. (2.18) F (s) −θ0 With (2.18), we may apply the method of Bogomol’nyi as before to decompose the energy as Z a Z ∞  Z −a p E(θ) = S1 +S0 +S1 (θ0 − F (θ))2 dx +2S1

−∞ Z θ0

−θ0

a

−a

p

F (s) ds + 2(S0 − S1)

Z

β

α

p

F (s) ds.

(2.19)

Since S0 < S1, the left-hand side of (2.19) is minimized by taking α → −θ0 , β → θ0, and setting θ to satisfy (2.12) in the intervals (−∞, −a), (−a, a), and (a, ∞), separately. This can consistently be done with θ(x) = −θ0,

x < −a;

and taking θ(x) to be implicitly defined by Z θ ds p = x, F (s) θ1

θ(x) = θ0,

x > a;

−a < x < a,

(2.20)

(2.21)

where θ0 and a satisfy the critical condition Z θ0 ds p = 2a, F (s) −θ0

(2.22)

in addition to (2.18) and θ1 is uniquely determined by Z θ1 Z θ0 ds ds p p = = a. F (s) F (s) −θ0 θ1

(2.23)

The construction above can be relaxed to cover the boundary values α and β in the closed set −θ0 ≤ α ≤ β ≤ θ0. The associated energy is obviously a continuous function of α and β, say I(α, β). Let αa and βa be such that I(αa, βa) = min{I(α, β) | − θ0 ≤ α ≤ β ≤ θ0 },

αa ≤ βa.

(2.24)

Such a pair, αa and βa, determines the least energy domain wall solution among all other solutions constructed from an arbitrary pair, α and β, just described. Subsequently, we focus on such a ‘canonical’ solution, say θa. In view of (2.19), we see that our calculation clearly shows that, as S1 → S0 , the least energy (canonical) solution approaches that given by the Bogomol’nyi equation 7

(2.12) over the full real line which recovers the classical domain wall solution without constriction as it should. In terms of the least energy solution, we can now describe what happens when S1 → ∞. For this purpose, let Θ be any test function so that Θ(x) = −θ0 ,

x ≤ −a;

Θ(x) = θ0 ,

x ≥ a.

(2.25)

Using (2.10), we have E(Θ) = S0

Z

a −a

{(Θ0)2 + F (Θ)} dx

≥ I(−θ0, θ0) ≥ E(θa)  Z αa Z ≥ 2S1 + −θ0

θ0 βa

p

F (s) ds,

(2.26)

which allows us to conclude that αa → −θ0

and βa → θ0

as S1 → ∞.

(2.27)

In other words, when S1 is sufficiently large, the least energy domain wall solution tends to be confined within the constriction. We may summarize our results as follows. Theorem 2.1. Consider the domain wall phase transition between the two ground phases −θ0 and θ0 defined as the isolated zeros of the nonnegative potential function F (θ) in Model I with the interval [−a, a] as the constriction. For any pair of numbers α and β satisfying −θ0 < α < β < θ0 so that the positivity condition (2.9) holds, the phase transition can be realized by such a domain wall solution θ(x) that it is continuous over (−∞, ∞), piecewise satisfies the domain wall equation in the intervals (−∞, −a), (−a, a), and (a, ∞), respectively, and is obtained by solving the minimization problem (2.11) over the constriction and the Bogomol’nyi equation (2.12) outside the constriction. Under the strengthened positivity condition (2.17), the domain wall solution may be realized by a least energy configuration among all other solutions constructed so that, as S1 → S0 , the solution approaches the domain wall solution in the classical situation without a constriction, and, as S1 → ∞, the solution is approximately confined within the constriction. Furthermore, if the convergence condition (2.18) holds, then there is a critical length 2a for the constriction given by (2.22) so that the gradual phase change is totally realized in the constriction where the cross section function S takes its lower value. In other words, in this situation, energetically the constriction completely locks the gradual magnetic phase change in it. We next consider the situation when the cross section S varies continuously as a function of x. As special examples of our study, we will be able to construct solutions of Models II and III of Bruno . Two complementary methods will be presented in the subsequent sections: a dynamical shooting method and a minimization method. 8

3

Domain Wall with Symmetric Cross Section

Motivated by Models II and III of Bruno  reviewed in the previous section, we assume in this section that S(x) is even and satisfies S(x) ≥ S0 > 0,

(3.1)

where S0 is a constant. To save space, we write Fθ = dF/dθ. Accordingly, for the potential density function F (θ), we assume that F (θ) is even, F (θ) ≥ 0, F (θ0) = 0, and Fθ (θ) < 0 for 0 < θ < θ0,

(3.2)

for some number θ0 > 0. Note that the conditions on F are general enough to cover all well known classical models. For example, F (θ) = K cos2 θ is the sine-Gordon model and F (θ) = λ(θ2 − 1)2 is the double-well model. The two distinct adjacent domain phases between which we are to find a gradual phase transition solution or domain wall are consequently realized as θ = −θ0,

θ = θ0 .

(3.3)

For convenience, we assume S and F are sufficiently differentiable. The Euler–Lagrange equation of (2.1) is 1 S(x)θ00 + S 0(x)θ0 − S(x)Fθ (θ) = 0. 2

(3.4)

In order to realize the anticipated phase transition, we need to get a solution of (3.4) which interpolates the two domain states given in (3.3). That is, we need to solve (3.4) subject to the boundary condition θ(±∞) = ±θ0.

(3.5)

We can state our existence theorem concerning (3.4) and (3.5) as follows. Theorem 3.1. Assume that the conditions (3.1) and (3.2) hold for the cross section S and potential density F , respectively. Then the equation (3.4) subject to the boundary condition (3.5) has a solution. Furthermore, such a solution may be obtained as an odd function of x due to the fact that F and S are both even. Our method of proof will be based on a dynamical shooting approach. The details of this method may provide useful guides for numerical simulation. First, we observe that, since the behavior of F (θ) beyond θ0 is of no interest, we may assume that Fθ (θ) ≥ 0 for θ > θ0 for convenience because otherwise we can always extend F (θ) in this way. Besides, since both F (θ) and S(x) are even, it suffices to prove the existence of an odd solution of (3.4) and (3.5), indeed. In other words, we need only to construct a solution of (3.4) over 0 < x < ∞ subject to the boundary condition θ(0) = 0,

θ(∞) = θ0 , 9

(3.6)

because a solution of (3.4) (over −∞ < x < ∞) and (3.5) may be obtained from a solution of (3.4) (over 0 < x < ∞) and (3.6) by using the extension θ(x) = −θ(−x) for x < 0.

(3.7)

In order to solve the boundary value problem (3.4) and (3.6), we consider instead the initial value problem 1 S(x)θ00 + S 0 (x)θ0 − S(x)Fθ (θ) = 0, 2 θ(0) = 0,

0 < x < ∞,

θ0 (0) = a.

(3.8)

The shooting method amounts to show that, when a > 0 is suitably chosen, the initial value problem (3.8) has a global solution which satisfies the desired boundary condition (3.6). Hence, we obtain an existence result for the two-point boundary value problem (3.4) and (3.5). Denote by θ(x; a) the unique solution of the initial value problem (3.8) defined over its interval of existence. Define the sets of shooting data as A− = {a > 0 | there exists x > 0 so that θ0 (x; a) < 0}, A0 = {a > 0 | θ0 (x; a) > 0 and θ(x; a) ≤ θ0 for all x > 0}, A+ = {a > 0 | θ0 (x; a) > 0 for all x > 0 and θ(x; a) > θ0 for some x}. Lemma 3.2. We have the disjoint union R+ = (0, ∞) = A− ∪ A0 ∪ A+ . Proof. If a 6∈ A− , then θ0 (x; a) ≥ 0 for all x > 0 (over the interval of existence of the solution). If there is a point x0 > 0 (in the interval of existence) such that θ0 (x0; a) = 0, then Fθ (θ(x0; a)) 6= 0, otherwise we reach an equilibrium point of the differential equation in (3.8) at finite x0 > 0 which is against the uniqueness theorem for the initial value problem of ordinary differential equations. Using Fθ (θ(x0 ; a)) 6= 0 but θ0 (x0; a) = 0 in the differential equation in (3.8), we see that either θ00 > 0 or θ00 < 0 at x0 . Therefore, when x is close to x0, we must have θ0 (x; a) < 0 for either x < x0 or x > x0 , which contradicts the assumption that a 6∈ A− . Consequently, θ0 (x; a) > 0 for all x (in the interval of existence of the solution) and the lemma follows. Lemma 3.3. The set A− is nonempty and open. Proof. From (3.8) we have 1 S(x)θ (x; a) = S(0)a + 2 0

Z

x

S(y)Fθ (θ(y; a)) dy.

(3.9)

0

If a 6∈ A− , then θ0 (x; a) > 0 for all x > 0 (in the interval of existence). In particular, (3.9) implies that S(x)θ0(x; a) ≤ S(0)a (3.10) 10

before θ(x; a) goes beyond θ0 because we have assumed that Fθ (θ) < 0 for 0 < θ < θ0. Hence

(3.11)

S(0)a S0

(3.12)

S0 = inf {S(x)} > 0.

(3.13)

0 < θ0 (x; a) < where we recall that x≥0

From (3.12), we get 0 < θ(x; a) <

S(0)a x, S0

x > 0.

(3.14)

In the interval of existence of the solution, say (0, K), we have either 0 < K < ∞ and θ(x; a) → ∞ as x → K or K = ∞ and θ(x; a) → ∞ or a finite number, say θ∞ > 0, as x → ∞. Using (3.9) and (3.11), we see that we must have θ∞ ≥ θ0. Thus, we conclude that, in any case, there is an interval [x1, x2] so that θ(x1 ; a) =

θ0 , 4

θ(x2; a) =

θ0 , 2

θ0 θ0 < θ(x; a) < 4 2

for x1 < x < x2 .

(3.15)

In view of (3.12) and (3.15), we have x2 − x1 ≥

θ 0 S0 . 4aS(0)

(3.16)

Inserting (3.16) into (3.9), we get Z 1 x2 0 < S(x2 )θ (x2 ) = S(0)a + S(y)Fθ (θ(y; a)) dy 2 0 Z 1 x2 S(y)Fθ (θ(y; a)) dy < S(0)a + 2 x1 Z x2 1 < S(0)a − 1 inf 1 {|Fθ (θ)|} S(y) dy 2 4 θ0 <θ< 2 θ0 x1 0

≤ S(0)a −

inf 1 {|Fθ (θ)|}

1 θ <θ< 2 θ0 4 0

θ0S02 , 8aS(0)

(3.17)

which will fail to be valid when a > 0 is made small. In other words, we have proved that A− contains an interval of the form (0, ε) for ε > 0 sufficiently small. In particular, A− is not empty. The openness of A− follows from the continuous dependence theorem for the solution of an ordinary differential equation on its initial data. Lemma 3.4. The set A+ is nonempty and open. 11

Proof. For any given x0 > 0, let a > 0 satisfy 1 a> sup {|Fθ (θ)|} 2S(0) 0≤θ≤θ0

Z

x0

S(y) dy.

(3.18)

0

Again, from (3.9), we have Z x Z Z y 1 1 x 1 θ(x; a) = S(0)a dy + S(z)Fθ (θ(z; a)) dzdy. 2 0 S(y) 0 0 S(y)

(3.19)

We see from this relation that we can make a > 0 large enough so that it satisfies (3.18) and θ(x; a) goes beyond θ0 before x = x0 . From (3.9) and (3.18), we see that θ(x; a) > 0 for all x in the interval of existence of the solution because Fθ (θ) ≥ 0 for θ > θ0. This proves that A+ contains an interval of the form (δ, ∞) for δ > 0 being sufficiently large. In particular, A+ is nonempty. To see that A+ is open, we take a point a1 ∈ A+ . For a small positive number σ, we can find a unique x1 > 0 so that θ(x1 ; a) = θ0 + σ. By the continuous dependence theorem for the initial value problem of an ordinary differential equation, we see that there is an open neighborhood N ⊂ R+ of a1 so that for any a ∈ N , we have θ0 (x; a) > 0 for 0 ≤ x ≤ x1 and θ(x1 ; a) > θ0 + σ/2 (say). Beyond x1, we still have θ0 (x; a) > 0 because Fθ (θ) ≥ 0 for θ > θ0. This proves N ⊂ A+ and the openness of A+ follows. Lemma 3.5. The set A0 is nonempty. Proof. This follows from the connectedness of R+ and the openness and non-emptiness of the sets A− and A+ . Proposition 3.6. For any a ∈ A0, the solution of the initial value problem (3.8) solves the boundary value problem given by (3.4) and (3.6), over 0 < x < ∞. Proof. By the definition of A0 , we see that the solution of (3.8) for a ∈ A0, say θ(x), satisfies lim θ(x) = θ∞ ≤ θ0 . (3.20) x→∞

Of course, θ∞ > 0. If θ∞ < θ0 , we see from (3.9) that θ0 (x) will not be able to stay positive when x is large enough because the cross section S(x) is bounded from below (cf. (3.13)), which contradicts the definition of A0. Therefore, we must have θ∞ = θ0 in (3.20).

12

4

Uniqueness of Domain Wall

We now consider the uniqueness problem. Let θ(x; a) be the solution of the initial value problem (3.8) defined in its interval of existence. Then θ(x; a) is differentiable with respect to the parameter a. Set θa =

∂θ , ∂a

w = θ0 .

(4.1)

In view of (3.8), the functions θa and w satisfy the initial value problems S0 0 1 θa − Fθθ (θ)θa = 0, S 2 θa (0) = 0, θa0 (0) = 1, θa00 +

x > 0, (4.2)

i S 0 0 h S 0 0 1 w + − Fθθ (θ) w = 0, S S 2 w(0) = a, w0(0) = 0.

w00 +

(4.3)

We observe that w can only have isolated zeros. Otherwise, if x0 > 0 is a zero of w which is not isolated, then w0 (x0) = 0. Hence we have θ0 (x0) = θ00 (x0) = 0 in the differential equation in (3.8). Consequently, we are led to F 0(θ(x0)) = 0. In other words, θ(x0 ) is an equilibrium of the differential equation and θ(x) ≡ θ(x0 ), which is impossible (we actually showed that, whenever θ0(x0 ) = 0, then θ00(x0) 6= 0). We will need to compare θa with w. In the interval of the form (0, x0 ) in which w = θ0 > 0, we consider the ratio θa (4.4) η= . w Then, using the definitions of θa and w, we see that η satisfies (w2 η 0)0 = −

 S0 

η(0) = 0,

η 0(0) =

S

(w2 η 0 ) + 1 . a

 S 0 0 S

θa w,

0 < x < x0 , (4.5)

Lemma 4.1. Suppose that SS 00 − (S 0 )2 ≥ 0,

∀x > 0.

(4.6)

Then, in the interval (0, x0) where θ0 (x) > 0, we have θa(x) > 0 as well. Furthermore, for arbitrarily small ε > 0, there is some constant δ(ε) > 0 such that θa(x) ≥ δ(ε)θ0(x)

for all x ∈ (ε, x0).

13

(4.7)

Proof. From the initial condition in (4.2), we see that θa(x) > 0 when x > 0 is small. Suppose otherwise that there is an x1 ∈ (0, x0 ) such that θa(x1 ) = 0 but θa (x) > 0 for x ∈ (0, x1 ). Let us rewrite the differential equation in (4.5) as (S[w2η 0 ])0 =

(SS 00 − [S 0]2 ) θaw, S

0 < x < x0 .

(4.8)

Since the right-hand side of (4.8) stays non-negatively when x ∈ [0, x1], we obtain S(x)w2 (x)η 0(x) ≥ S(0)w2 (0)η 0 (0) = S(0)a > 0

(4.9)

everywhere in [0, x1], which implies that θa (x) θa (ε) = η(x) ≥ η(ε) = w(x) w(ε)

(4.10)

in [ε, x1] when ε > 0 is sufficiently small. Recall that θa (0) = 0, θa0 (0) = 1, w(0) = a. We see that θa (ε) > 0 for ε > 0 small. (4.11) δ(ε) = w(ε) Combining (4.10) and (4.11), we have θa(x) ≥ δ(ε)w(x), x ∈ [ε, x1]. In particular, we see that θa (x1) > 0 and the assumption x1 < x0 is false. In other words, we have proved that θa (x) > 0 in (0, x0) and (4.7) holds in (0, x0 ) as stated. Lemma 4.2. Under the condition (4.6), there is a number a2 > 0 such that A+ = (a2, ∞). Proof. Since A+ is open, it suffices to show that if (b1 , b2) ⊂ A+ where b2 < ∞, then b2 ∈ A + . For any a ∈ (b1 , b2), let x0(a) > 0 be the first point where the solution θ(x; a) of (3.8) crosses the horizontal axis θ = θ0 in the (x, θ)-plane. Of course, θ0(x0 (a); a) > 0.

θ(x0(a); a) = θ0,

(4.12)

Applying the implicit function theorem to the equation θ(x0(a); a) = θ0, we see that x0(a) is a differentiable function of a in the interval (b1, b2 ) and by Lemma 4.1 d θa(x0(a); a) x0(a) = − 0 < 0. da θ (x0(a); a) So x0(a) is monotone decreasing. Using (3.19), we get Z x 1 θ(x; a) ≤ aS(0) dy 0 S(y)

(4.13)

(4.14)

for x ∈ (0, x0 (a)] because θ(x; a) ≤ θ0 there. Taking x = x0 (a) in (4.14), we find Z x0 (a) 1 θ0 ≤ aS(0) dy, (4.15) S(y) 0 14

which specifies a positive lower bound for x0(a). Consequently, x0(b2 ) ≡ lim− x0 (a)

(4.16)

a→b2

is a well defined positive number. By continuity, we have θ(x0 (b2); b2 ) = θ0 . Since x0(b2 ) > 0, we have θ0 (x0(b2 ); b2) 6= 0 otherwise θ(x; b2) ≡ θ0. This means θ(x; b2) > θ0 for x sufficiently close to x0(b2 ). As a result, we have b2 ∈ A+ as expected. Lemma 4.3. Under the condition (4.6), there is a positive number a1 ≤ a2 where a2 is as stated in Lemma 4.2, so that A− = (0, a1 ). Proof. We show that, if (b1, b2) ⊂ A− where b1 > 0, then b1 ∈ A− . For a ∈ (b1 , b2), let x1(a) be the left-most local maximum point of θ(x; a). Then θ(x1(a); a) < θ0 . Otherwise, we must have θ(x1(a); a) > θ0 and θ(x; a) is nondecreasing over (0, x1(a)). Let x01 be the unique point in (0, x1 (a)) so that θ(x01; a) = θ0. Then θ0 (x01; a) > 0. Since Fθ (θ) > 0 for θ > θ0 , we see that θ0(x; a) > 0 for all x > x01, a contradiction. Inserting the results 0 < θ(x1 (a); a) < θ0 ,

θ0(x1 (a); a) = 0

(4.17)

into the differential equation in (3.8), we obtain θ00(x1 (a); a) < 0. By the implicit function theorem, we see that x1(a) is a differentiable function of a in (b1, b2). Set m(a) = θ(x1(a); a), a ∈ (b1, b2). (4.18)

Then 0 < m(a) < θ0. Since θ0 (x; a) > 0 for 0 < x < x1(a), we have θa(x; a) > 0 for 0 < x < x1 (a) as well (see Lemma 4.1). By continuity, we have θa (x1(a); a) ≥ 0. Therefore, d d m(a) = θ0 (x1(a); a) x1(a) + θa(x1 (a); a) = θa (x1(a); a) ≥ 0. da da

(4.19)

Hence m(a) is nondecreasing for a ∈ (b1, b2 ). In particular, the limit m(b1) ≡ lim+ m(a)

(4.20)

a→b1

exists and 0 ≤ m(b1) < θ0. Inserting θ0 (x1(a); a) = 0 in (3.9), we get 1 S(0)a + 2

Z

x1 (a)

S(y)Fθ (θ(y; a)) dy = 0.

(4.21)

0

Hence, for b1 > 0, the above implies that there is a constant c0 > 0 such that x1(a) ≥ c0 > 0 for a > b1 . 15

We claim that {x1(a)} stays bounded as a → b1 . Suppose otherwise that the opposite is true. Going to a subsequence if necessary, we may assume that x1(a) → ∞ as a → b1. On the other hand, (3.9) says that the family of functions {θ(x; a)}a>b1 is equicontinuous and uniformly bounded over any interval of the form [0, K] for any K > 0 when a is sufficiently close to b1 (so that x1(a) > K) because of (4.17). Hence, going to a subsequence if necessary, we may assume that the limit ˜ θ(x) = lim θ(x; a) a→b+ 1

(4.22)

˜ exists for any x > 0. Inserting (4.22) into (3.9), we see that θ(x) solves (3.8) with 0 0 ˜ a = b1 > 0 and θ˜ (x) ≥ 0, θ(x) ≤ θ0 for all x > 0 because θ (x; a) > 0, θ(x; a) < θ0 in (0, x1 (a)). Hence, by our earlier discussion in the existence part of this problem, we see that b1 ∈ A0. However, using the property m(a0) ≤ m(a00) < θ0 , we derive

a0 < a00,

a0, a00 ∈ (b1, b2),

˜ lim θ(x) ≤ lim+ m(a) < θ0,

x→∞

(4.23) (4.24)

a→b1

which contradicts Proposition 3.6. So the claim is proved. Going to a subsequence if necessary, we may assume that x1(b1 ) ≡ lim+ x1(a)

(4.25)

a→b1

exists. Of course, x1(b1) > 0. For the solution θ(x; a), we see that θ0 (x; a) > 0 for x ∈ (0, x1(b1 ) − ε) when a is close to b1 from the right and ε > 0 is arbitrarily small. We claim that m(b1) > 0. Otherwise, if m(b1) = 0, then 0 < θ(x1(b1) − ε; a) < θ(x1(a); a) = m(a) implies θ(x1(b1 ) − ε; b1) = lim+ θ(x1(b1 ) − ε; a) = 0.

(4.26)

a→b1

So we arrive at the trivial solution θ = 0 at finite location x = x1(b1) − ε > 0, which is impossible. Now, letting a → b+ 1 , by continuity again, we get the solution θ(x; b1 ) satisfying θ0(x; b1) ≥ 0, x ∈ (0, x1 (b1)), θ0 (x1(b1); b1 ) = 0; 0 < m(b1) = θ(x1 (b1); b1) < θ0.

(4.27)

Using (4.27) in the differential equation in (3.8), we see that θ00(x1(b1 ); b1) < 0. Consequently, θ0 (x; b1) < 0 for x slightly above x1(b1 ), which finally proves b1 ∈ A− . The fact that a1 ≤ a2 follows from the fact that A− ∩ A+ = ∅.

16

Lemma 4.4. With the notation and condition of Lemmas 4.2 and 4.3, we have A0 = [a1, a2]. Moreover, if F satisfies the condition Fθθ (θ) ≥ 0 for θ near θ0, then a1 = a2, provided that the integral Z ∞ 1 dx (4.28) S(x) 0 is divergent. Proof. The fact that A0 = [a1, a2] has already been established. Hence, we focus on the second statement of the lemma. For each a ∈ A0, we know that θ0 (x) > 0 for all x > 0. Thus, θa (x) > 0 for all x > 0 (Lemma 4.1). Let xδ (a) > 0 be such that θ0 − δ < θ(x; a) < θ0,

x > xδ (a).

(4.29)

Here δ > 0 is a small number so that Fθθ (θ) ≥ 0 for θ0 − δ ≤ θ ≤ θ0 . Case 1: There is some x0 > xδ (a) such that θa0 (x0 ) ≥ 0. We rewrite the differential equation in (4.2) as (Sθa0 )0 = 12 SFθθ (θ)θa . Using θa (x) > 0, we have, after integration, Z 1 x 0 0 S(x)θa (x) − S(x0)θa (x0) = S(y)Fθθ (θ(y))θa(y) dy ≥ 0, x > x0, (4.30) 2 x0 which implies θa0 (x) ≥ 0. As a consequence, the limit θa(∞; a) ≡ lim θa(x; a) x→∞

(4.31)

is a well-defined positive quantity. Case 2: θa0 (x) < 0 for all x > xδ (a). Then the limit in (4.31) is again well defined. We can show that it is still a positive quantity, which is less obvious. To see this, we form the ratio ξ=

w . θa

(4.32)

Using the differential equation in (4.2) and (4.3), we have (S[θa2ξ 0 ])0 = −

(SS 00 − [S 0]2 ) θa w. S

Integrating (4.33) and using the initial conditions in (4.2) amd (4.3), we get Z x (SS 00 − [S 0]2) 2 0 S(x)θa (x)ξ (x) = −S(0)a − θaw dy S 0 ≤ −S(0)a, x > 0. 17

(4.33)

(4.34)

For x2 > x1 > xδ (a), we integrate the right-hand side of (4.34) over the interval (x1, x2 ) to get Z w(x2) w(x1 ) S(0)a x2 1 ξ(x2 ) = ≤ − dy, (4.35) θa (x2) θa (x1) θa2 (x1) x1 S(y) where we have used the property that θa(x1 ) ≥ θa(x) > 0 for x ∈ [x1, x2]. In view of the divergence of (4.28), we derive from (4.35) the conclusion that w(x2 )/θa (x2) becomes negative when x2 is sufficiently large, which is false. This shows that Case 2 cannot happen. Now consider the interval [a1, a2 ] = A0 . Since θ(∞; a) = θ0 for any a ∈ A0, we obtain by using Fatou’s Lemma the relation lim (θ(x; a2) − θ(x; a1)) Z a2 = lim θa(x; a) da x→∞ a 1 Z a2 ≥ lim θa(x; a) da a1 x→∞ Z a2 = θa(∞; a) da,

0 =

x→∞

(4.36)

a1

where we have inserted (4.31) as the integrand for the integral on the right-hand side of (4.36). Since we have seen that θa(∞; a) is positive, we must have a1 = a2 in (4.36). So the lemma follows. In conclusion, we have established the following uniqueness result. Theorem 4.5. Suppose the cross section S(x) and R ∞ the potential density function F (θ) 00 0 2 satisfy the conditions (i) SS − (S ) ≥ 0, (ii) 0 dx/S(x) = ∞, (iii) Fθθ (θ) ≥ 0 for θ sufficiently close to θ0 from below. Then there exists a unique positive number a such that the solution of (3.8) gives rise to the unique solution of the equation (3.4) subject to the boundary condition (3.6). In particular, the equation (3.4) subject to the boundary condition (3.5) has a unique odd solution. To close this section, we make two remarks. (i) The information stated in Lemmas 4.2 and 4.3 will be useful for numerical computation of solutions. (ii) The condition (4.6) is satisfied by Model III but not satisfied by Model II. Indeed, for Model III, the cross section gives us SS 00 − (S 0)2 = S02 /d2 .

5

Existence via Direct Minimization

In this section, we consider the geometrically constrained domain wall models without any symmetry assumption. In other words, we study the situation when the cross section S and the potential density F are not assumed to be even functions. In particular, if θ1 and θ2 are two arbitrary global minima of F so that F (θ) ≥ 0,

F (θ1) = 0, 18

F (θ2) = 0,

θ1 < θ2 ,

(5.1)

we aim at establishing the existence of a domain wall solution realizing the phase transition process from the domain θ = θ1 to θ = θ2. That is, we want to obtain a solution of the equation (3.4) satisfying the boundary condition θ(−∞) = θ1,

θ(∞) = θ2 .

(5.2)

For this purpose, we use a direct minimization method, instead of the shooting method presented in Section 3. We modify the admissible class (2.6) into C1 = {θ(x) | θ(−∞) = θ1, θ(∞) = θ2, E(θ) < ∞}.

(5.3)

Our existence theorem may be stated as follows. Theorem 5.1. Suppose that the cross section S satisfies the integrability condition Z ∞ 1 dx < ∞. (5.4) −∞ S(x) Then the boundary value problem consisting of (3.4) and (5.2) has a solution which is the solution of the direct minimization problem η ≡ inf{E(θ) | θ ∈ C1 }.

(5.5)

Furthermore, the solution θ obtained satisfies θ1 < θ(x) < θ2 for all x. Besides, if we assume that there is another point θ0 between θ1 and θ2 such that F (θ) increases over (θ1, θ0 ) and decreases over (θ0 , θ2), then the solution is strictly increasing. Proof. The existence proof is straightforward. Let {Θn } be a minimizing sequence of the problem (5.5). We may assume E(Θn ) ≤ η + 1 (say) for all n. We have Z ∞ |Θn (x) − θ2| ≤ |Θ0n (y)| dy x Z ∞ 1 1/2 Z ∞ 1/2 0 2 ≤ dy S(y)(Θn (y)) dy S(y) x x Z ∞ 1 1/2 p ≤ η+1 dy . (5.6) S(y) x Hence we see that {Θn } fulfills the boundary condition (5.2) uniformly rapidly at x = ∞. Similar result holds for {Θn } at x = −∞. Besides, it is easily seen that {Θn } is bounded in W 1,2(−K, K) for any K > 0. Using a diagonal subsequence argument, we can show that {Θn } has a subsequence which converges in a well defined sense to a solution of (5.5). To verify the stated pointwise bounds, we first show the weaker statement, θ1 ≤ θ(x) ≤ θ2 for −∞ < x < ∞. Otherwise, assume there is an x0 ∈ (−∞, ∞) so that θ(x0) > θ2 or θ(x0 ) < θ1 . For definiteness, we assume θ(x0) > θ2. By continuity, we 19

can find x1 , x2 ∈ (−∞, ∞], x1 < x2, so that θ(x1) = θ(x2 ) = θ2 and θ(x) > θ2 for x1 < x < x2 . Define  θ2 when x1 < x < x2, ˜ θ(x) = (5.7) θ(x) when x 6∈ (x1 , x2). ˜ < E(θ) = η, which contradicts the definition of η (see (5.5)). Then θ˜ ∈ C1 and E(θ) So θ1 ≤ θ(x) ≤ θ2 for all x ∈ (−∞, ∞). In order to establish the strict result, θ1 < θ(x) < θ2 for x ∈ (−∞, ∞), we assume otherwise that there is an x0 ∈ (−∞, ∞) such that θ(x0 ) = θ2 (say). Hence θ(x) attains its global maximum at x0. As a consequence, θ0 (x0) = 0. However, θ = θ2 is an equilibrium point of the equation. So by the uniqueness theorem, we have θ(x) ≡ θ2 for all x ∈ (−∞, ∞), which contradicts θ(−∞) = θ1 < θ2 . Now assume that there is another point θ0 between θ1 and θ2 such that F (θ) increases over (θ1 , θ0) and decreases over (θ0, θ2 ). Thus θ1 , θ0, θ2 are equilibria of the equation (3.4) for which θ0 is unstable but θ1, θ2 are stable. We are to find a monotone phase transition between the two stable equilibria. Let θ(x) be a solution just obtained by solving (5.5). We first claim that there is exactly one point, say x0, in (−∞, ∞) so that θ(x0) = θ0 . Suppose otherwise that there are at least two points, x1 , x2 ∈ (−∞, ∞), so that θ(x1) = θ(x2 ) = θ0 . Since θ(−∞) = θ1 < θ0 < θ2 = θ(∞), we see that the graph of the function θ(x) will have a twist across the the horizontal line θ = θ0 . In particular, there is at least a third point, x3 ∈ (−∞, ∞) such that θ(x3) = θ0. Since θ = θ0 is an equilibrium of the equation, we see that θ0(x) 6= 0 for x = x1, x2 , x3. Modifying these three points if necessary, we may further assume that these points are the left most such points. Hence we have θ0(x1 ) > 0, θ0 (x2 ) < 0, θ0 (x3 ) > 0. We also have θ(x) > θ0 for x ∈ (x1 , x2) and θ(x) < θ0 for x ∈ (x2 , x3). Let y1 ∈ (x1 , x2) and y2 ∈ (x2 , x3) be such that θ(y1 ) = sup {θ(x)}, θ(y2) = inf {θ(x)}. (5.8) x∈(x2,x3 )

x∈(x1 ,x2 )

If F (θ(y1)) ≤ F (θ(y2)), then the properties of F imply that F (θ(y1)) ≤ F (θ(x))) for all x ∈ (y1 , y3) where y3 satisfies y3 = inf{x > y2 | θ(x) = θ(y1)}. Of course, y3 > x3. Now define the function  θ(y1), x ∈ (y1, y3), ˜ θ(x) = θ(x), x 6∈ (y1, y3).

(5.9)

(5.10)

˜ < E(θ), which is a contradiction. If F (θ(y1)) > F (θ(y2)), we choose Then E(θ) y4 = sup{x < y1 | θ(x) = θ(y2 )}. 20

(5.11)

˜ Then y4 < x1. As in (5.10), we define θ˜ such that θ(x) = θ(y2) for x ∈ (y4, y2) and ˜ ˜ θ(x) = θ(x) elsewhere, we again have E(θ) < E(θ), another contradiction. We next assert that θ(x) is nondecreasing over the entire interval (−∞, ∞). Suppose otherwise that there are points x1, x2 ∈ (−∞, ∞), x1 < x2, satisfying θ0 (x) < 0 for x ∈ (x1, x2 ). Let x0 be the unique point in (−∞, ∞) such that θ(x0) = θ0. Of course, x0 6= x1 otherwise it violates the uniqueness of x0. If x0 < x1, set n o y1 = sup x ∈ (−∞, x2) | θ(x) = sup {θ(x)} . (5.12) x∈(−a,x2)

We have y1 > x0. Since θ(y1 ) < θ2, we can find a point y2 ∈ (y1, ∞) so that θ(y2 ) = θ(y1 ) and θ(x) < θ(y1 ) for x ∈ (y1, y2). Because y1 > x0 , we have θ(x) > θ0 for x > y1. Hence, in particular, F (θ(y1)) < F (θ(x)) for x ∈ (y1, y2) since F (θ) decreases in (θ0, θ2 ). Define  θ(y1), x ∈ (y1, y2), ˜ θ(x) = (5.13) θ(x), x 6∈ (y1, y2). ˜ < E(θ), which is a contradiction to the definition of We see immediately that E(θ) θ. If x0 > x1, then x2 < x0. Of course, θ(x) < θ0 for all x ∈ (−∞, x0). Like before, we can find two points y1, y2 ∈ (−∞, x0), y1 < y2, so that θ(y1) = θ(y2 ) and θ(x) > θ(y1 ) for x ∈ (y1, y2 ). Since F (θ) increases in (θ1 , θ0), we have F (θ(x)) > F (θ(y1)). Modifying θ into θ˜ through (5.13), we get the same contradiction. Finally, we show that θ(x) is strictly increasing. In fact, if it is not strictly increasing, there exists x1 < x2 such that θ(x1 ) = θ(x2 ). Hence θ(x) is constant in the interval (x1, x2). Inserting this result into the equation (3.4), we see that Fθ (θ(x1 )) = 0, which contradicts the fact that θ(x1) 6= θ0, hence Fθ (θ(x1)) 6= 0. The proof of the theorem is complete. Note that (5.4) is valid for both Models II and III. Although the existence proof of Theorem 5.1 is simple, it has an interesting implication that the integrability condition (5.4) for the geometric constriction cross section S allows a phase transition between two arbitrary domain phases to take place. For example, suppose that the potential density F has a sequence of distinct domain phases, say {θn }. The theorem says that there is always a domain wall solution which links θi to θj for any pair i, j, i 6= j, and minimizes the total energy (2.1). Such a phase transition solution may not exist when (5.4) is violated. A classical example is the sine-Gordon model defined by the energy (1.2) for which the domain phases are π θn = (2n + 1) , n = 0, ±1, ±2, · · · . (5.14) 2 It is well known that there can only exist a domain wall solution which links two neighboring domain phases [18, 24]. Below, we give a proof of this fact in its general situation.

21

Proposition 5.2. Let θ1 , θ2, θ3 be three distinct zeros of the nonnegative potential density function F (θ) so that θ2 lies between θ1 and θ3. Then the domain wall energy Z ∞ {(θ0)2 + F (θ)} dx (5.15) E(θ) = ∞

cannot have a finite-energy critical point, θ, satisfying the boundary condition θ(−∞) = θ1 ,

θ(∞) = θ3.

(5.16)

Proof. Suppose otherwise that θ(x) is such a critical point. Then it satisfies the Euler–Lagrange equation of (5.15): 1 θ00 − Fθ (θ) = 0, 2

−∞ < x < ∞.

(5.17)

Since E(θ) < ∞, there is a sequence {xn }, xn → −∞ as n → ∞ such that θ0 (xn ) → 0 as n → ∞. Let x0 ∈ (−∞, ∞) be such that θ(x0) = θ2 . Multiplying (5.17) by θ0 and integrating over [xn , x0], we have Z x0 Z x0   d 0 2 0 ([θ (x)] ) dx = F (θ(x)) dx. (5.18) dx xn xn As a consequence, we obtain [θ0(x0 )]2 =

lim [θ0(xn )]2 + F (θ2) − lim F (θ(xn ))

n→∞

n→∞

= F (θ2) − F (θ1) = 0.

(5.19)

In other words, θ(x) satisfies the initial condition θ(x0 ) = θ2, θ0 (x0 ) = 0 at x = x0. Since θ2 is an equilibrium of the equation (5.17), we deduce that θ(x) ≡ θ2 for all x by the uniqueness theorem for the solution of the initial value problem of an ordinary differential equation. This contradicts the boundary condition (5.16) assumed for θ.

6

Conclusions

In this paper, we have obtained a series of existence results for the solutions of the geometrically constrained magnetic domain wall models  and observed a wide range of mathematical properties associated with the presence of constrictions described in terms of a cross section function, S. Our results establish that (i) When the cross section is a step function as described in Model I in , the domain wall solution may be obtained by minimizing the domain wall energy over the constriction and solving the Bogomol’nyi equation outside the constriction. This procedure can be done in such a way that the total energy is minimized among all other so-constructed field configurations connecting the two ground phases. Furthermore, the least energy solution is approximately confined within the constriction when the 22

cross section maintains a sufficiently high level outside the constriction. Besides, under a specific critical condition relating the potential function, and constriction width, and the ground state levels, the domain wall phase transition may totally be realized within the constriction. (ii) If the cross section S and potential density F are such that the conditions (3.1) and (3.2) hold, which cover Models II and III in  as special cases, the domain wall equation (3.4) subject to the boundary condition (3.5) always has an odd solution. Such a solution may be obtained constructively by a dynamical shooting method. (iii) Under some additional conditions (such as those stated in Theorem 4.5 which cover Model III in ), the equation (3.4) subject to the boundary condition (3.5) has a unique odd solution. (iv) For any two zeros, θ1 and θ2, of the nonnegative potential density F , the domain wall equation (3.4) has a solution satisfying the boundary condition (5.2), provided that the cross section S satisfies the integrability condition (5.4). Furthermore, such a solution may be obtained as a solution of the direct minimization problem (5.5). On the other hand, if (5.4) fails, the equation (3.4) may not have a finite-energy solution satisfying (5.2) when there is another zero of F between θ1 and θ2. An example is given with S(x) = constant. In other words, in such a situation, a phase transition may only be realized between two neighboring domains. Acknowledgments. The research of SC was supported in part by Natural Science Funds of Henan Education Office (grants 2007110004 and 2008A110002) and that of YY by National Science Foundation under grant DMS–0406446.

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25

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