ANALYTICAL SOLUTIONS OF SKYRME MODEL

. Exact analytic solutions of the kink soliton equation obtained in a recent interesting study of the classical Skyrme model deﬁned on a simple spherically symmetric background are presented. By a variational method, the existence of spherically symmetric monopole solutions are proved. In particular, all ﬁnite-energy kink solitons must be Bogomool’nyi–Prasad–Sommerﬁeld are showed. Moreover, together with numerical analysis, we can clearly see the validity of our theoretical results.

from some recent earlier work [1,4,5,6], a static ansatz describing a spherically symmetric Skyrmion in the background [6]. The second-order governing equation takes a complicated form but it may be solved by a much simpler first-order equation whose solutions interpolate adjacent minima of the one-dimensional Skyrme energy density. The importance of the work of Canfora, et al [7] is that it makes an explicit construction of kink-like solutions in the classical Skyrme model possible, which hints our research.
In the present paper, exact spherically symmetric solutions of the Skyrme model with both a non-trivial winding number and a finite soliton mass (topological charge) are studied. We will prove that the first-order equation of Canfora, et al and the second-order equation of the one-dimensional kink energy of Canfora, et al are actually equivalent under the finite-energy condition. Therefore we obtain the sharp properties of all finite-energy solutions of the Skyrme model within the Canfora, et al kink ansatz [7].
Let µ, ν denote the Minkowski spacetime indices with metric g = (g µν ) = diag{−1, 1, 1, 1}. The action of the SU (2) Skyrme system in four dimensional spacetime is where K, λ > 0 are the coupling parameters, R µ = U −1 ∇ µ U = R i µ t i for an SU (2)valued map U , t i (i = 1, 2, 3) are generators of the Lie algebra of SU (2), and F µν = [R µ , R ν ]. The first term of the Skyrme action (1) is mandatory to describe poins while the second is the only covariant term leading to second order field equations in time which supports the existence of Skyrmions in four dimensions. The SU (2)valued scalar U (x µ ) is represented through the expression where 1 denotes the 2 × 2 identity matrix and Y 0 , Y i are scalar functions over spacetime. Thus, the hedgehog ansatz describing a spherically symmetric Skyrmion can be written in terms of the unit vectors andn 1 = sin θ cos φ,n 2 = sin θ sin φ andn 3 = cos θ. So, the Skyrme field equations of the action (1) (or the associated Euler-Lagrange equations) are We will supply variational method analogous to that of [8,9,23,24], for the Skyrme field equations, which is different from the analytic method in [7]. Our study here may provide insight into some of those more complicated problem.
The rest of this paper is organized as follows. In section 2, we prove the existence of the Skyrme model solutions by a variational method. Furthermore, we obtain the sharp properties of an energy-minimizing solution. In section 3, we show that the solutions of the second-order equations also satisfies the reduced self-dual equations, establishing the equivalence of the Skyrme field equations and the BPS monopoles equations under the radial symmetry assumption. Moreover, we demonstrate a few concrete numerical examples to illustrate the effectiveness of our results.

2.
Existence of variational solution. In this section, we first consider the existence of the Skyrme model solution. The reduced one-dimensional energy functional (1) is as follows where R 0 > 0 is much larger than the radius of the proton. It is clear that the Euler-Lagrange equations of (5) is where α = dα dx . We note that for the ansatz (3), the Skyrme field equations (4) reduce to the single ordinary differential equation for the Skyrmion profile α (6). The reduced energy density is In view of finite-energy condition, we impose the boundary conditions The winding number for the hedgehog ansatz in (3) is where (x 1 , x 2 ) correspond to the limits in the range of spatial direction, that can be taken as (−∞, ∞). This winding number takes integer values n, for boundary We need to find solutions to equation (6) subject to the boundary conditions (8), which amounts to solving a two-point boundary value problem which seems difficult. We will solve the problem by the variational method. The admissible space A is defined by A = {α| α is absolutely continuous on every compact subinterval of (−∞, ∞) so that it satisfies the boundary condition (8) and E(α) < ∞}. (10) With above preparation, we can state our main results as follows.
has a solution.
Proof. Consider For α n ∈ A, we can take a minimizing sequence {α n }. Without loss of generality, we may assume that: Besides, the form of the energy E given in (5) indicates that we may assume Otherwise we may modify the sequence to fulfill the above inequality meanwhile without enlarging the energy. We may get that the sequence {α n } is bounded in W 1,2 (−N, N ) for any integer N ≥ 2 . Using weak compactness, we may assume that α n (in fact, a subsequence in it) is weakly convergent in W 1,2 (−N, N ). Applying a diagonal subsequence argument, we may assume there is a α ∈ W 1,2 weakly in W 1,2 (−N, N ) and strongly in C[−N, N ] for any N = 2, 3, · · · . Consequently we see that α is absolutely continuous in any compact subinterval of (−∞, ∞) and obeys (13) as well. Furthermore, in order to show that α ∈ A , we need to have α(−∞) = 0 and α(∞) = π . In fact, 0 ≤ α(x) ≤ π, we may suppose that there is an We deduce from (15) and (16) Similarly, there is a 0 < x 2 < ∞ such that 99π 100 ≤ α(x) < π when x ∈ (x 2 , ∞). Therefore we may get 0 < π−α ≤ π 100 and (π−α) 2 ≤ C 2 sin 2 (π−α), where C 2 is the appropriate constant. Then we know that π − α ∈ W 1,2 (x 2 , ∞) which immediately leads to α(∞) = π. Let where H(α) denoting the energy density defined in (7). Using the weak lower semicontinuity property of the functional we obtain the inequality Thus we see that α fulfills the complete boundary conditions (8). Therefore α ∈ A, and (19) allows us to obtains E(α) = E m . That is, α is found to be a solution of (11). As a consequence, α is a finite-energy solution of (6) and (8). The proof of theorem (2.1) is now completed.
Next, we will establish some sharp properties of energy-minimizing solution.
Theorem 2.2. The energy-minimizing solution of (6) and (8) α satisfies: has the property of convex function as |x| → ∞; (iv) α(x) holds the following asymptotic estimates where b 1 = 1 and > 0 is small enough. Proof. (i) Let α be the energy-minimizing solution of (6)- (8), If there is a point x 0 ∈ (−∞, ∞) such that α(x 0 ) = 0, then α (x 0 ) = 0 since x 0 is a minimum point for the function α. Applying the uniqueness theorem of the initial value problem of an ordinary differential equation, we have α(x) = 0 for all x ∈ (−∞, ∞), which contradicts the fact α(∞) = π. This proves On the other hand, we can show that Indeed, if there is a point x 0 > 0 such that α(x 0 ) = π, then x 0 is a maximum point of α and we have α (x 0 ) < 0. Inserting these into (6), we arrive at a contradiction. Thus (23) is valid.
Summarizing the above results, we can see that the energy-minimizing solution α(x) is convex as x → −∞ and is concave as x → ∞.
A direct calculation, we can obtain γ(x) satisfying Similarly, we choose a comparison function where b 2 = 2 R 2 0 +2λ > 0, > 0 is small enough and C > 0 is a suitable constant. Then for any given > 0 sufficiently small, there exists large enough x > 0 such that x > x , we can get for ∀x > x . Setting Φ = γ − β 2 , then we can choose C > 0 large enough such that Φ(x ) = γ(x ) − β 2 (x ) ≤ 0 for the fixed x > 0. Thus Similar discussions with above, we have Thus for any Then we complete the proof of theorem (2.2).
3. The solution of BPS monopoles equations. We next turn to the energy E(α), the classical equations coincide with the equations of the BPS monopole. In fact, one is to find solutions to (6) subject to the boundary condition (8), which amounts to solving a two-point boundary value problem which seems difficult. Fortunately, Canfora, Correa, and Zanelli [7] identify a nontrivial kink charge Q given by so that they obtain through a BPS trick [3,20] the following expression for the total energy functional (5) for |Q| = ∓Q, which leads them to arrive at the conclusion that the energy lower bound in (40) is attained when α satisfies the first-order equation Equation in (41) are actually the self-dual BPS monopoles equations (6). Proof. It is straightforward to check that (41) implies (6). It will take the some effort, however, to show that the converse is also true. In other words, we shall prove that any finite-energy solution of (6) satisfies (41) as well. That is, the first-order equation (41) and the second-order equation (6) are actually equivalent. Equation (6) can be rewritten as Let α be a finite-energy solution of (42) and set Then, in view of (42), we have Therefore, if there is some x 0 ∈ (−∞, ∞) such that P + (x 0 ) = 0 or P − (x 0 ) = 0, then applying the uniqueness theorem for the initial value problem of an ordinary differential equation we obtain P + ≡ 0 or P − ≡ 0, which implies α must satisfy one of the equations stated in (41).
In the following, we give a few numerical examples visually to observe the above theoretical results.  The value of the solution of (41) at the point x i will be denoted by α i . We will use the classical four order Runge-Kutta method to implement our computation.
We first compute the solution in (−L, L) with L = 8. The value of α M , i.e., the approximate value of α(L) is 3.0909 which is away from the desired value of the second boundary condition in (8). We then take L = 10, the computation result of the solution at the right endpoint is 3.1386. Although this value is very close to the expected value of the boundary, the approximate value of solution α(x) did not exhibit a stable trend near the right endpoint. Now we let L = 15, Fig.1 presents the curve of α(x). Computation results show that the approximate value of α(x) are all 3.1416 when x near the right endpoint. Within the allowable error range, which implies the agreement with the theoretical results. Similar conclusion can be obtained when L ≥ 15 by further computer experiments.
We have solved numerically equation (41). We can clearly see many of the properties of energy-minimizing solution α(x) from the graph. Furthermore, the numerical examples which also demonstrate the effectiveness of our theory results.