September 13, 1997

Exixtence of String-like Solitons.

Vieri Benci^{∗}, Donato Fortunato^{∗∗} ∗Dipartimento di Matematica Applicata ”U. Dini” Università di Pisa, via Bonanno 25/b 56126 PISA (Italy) ∗∗Dipartimento di Matematica Università di Bari, via Orabona 4 70125 BARI (Italy)

I. Introduction

A soliton is a solution of a field equation whose energy travels as a localized packet and which preserves its form under perturbations. The literature on solitons is enormous and we refer to (^{4}), (^{7}) and (^{9}) for an extensive review. However the research has mainly been concentrated on one dimensional spatial cases with few exceptions. Here we are concerned with solitons in three space dimensions using the methods and the results of (^{1}), (^{2}) and (^{3}). In higher dimensions solitons do not need to have a bump structure; they might have a more complicated structure as a string or a knot in the sense that the mass-energy might be concentrated around a linear structure embedded in R^{3}.

These string-like structures seem to be relevant for a large class of phenomena. For example, today most phycicists agree that the fundamental interactions are described by strings. In this model the different elementary particles correspond to the vibrational excitations of a fundamental string. Also strings and knots occur in many other physical, chemical and byological systems as statistical physics, superfluids, nematic liquid cristals, molecular biology etc . Despite the clear relevance of these structures, their existence and properties have not been much investigated.

The only exception we know is the work Faddeev and Niemi (cf.^{6} and its references). They study a field model using numerical simulations and get a strong evidence of the existence of string-like structures both knotted and unknotted. Their model is related to the following energy functional

E_{F N}(u) = E_{2}(u) + e^{2}E_{4}(u) = ∫_{R^{3}} |∇u|^{2} dx + e^{2} ∫_{R^{3}} F^{2}dx

where u(x) is a field of vectors of unit lenght, e is a coupling constant and and F^{2} is a function of fourth degree in the field derivatives \frac{∂u}{∂x_{i}} . This is the most general 3-dimensional Lagrangian which admits a relativistic invariant dynamics and involves only terms up to the fourth degree in the field derivatives.

In this paper we consider perturbations of E_{2}(u) different from e^{2}E_{4}(u) and we obtain a field equation for which it is possible to prove the existence of string-like solitons just exploiting a theorem of.^{3} More precisely we consider the energy functional

E(u) = E_{2}(u) + ε^{4}E_{6}(u)

where

E_{6}(u) = \frac{1}{6} ∫_{R^{3}} |∇u|^{6} dx.

Even if E_{6}(u) has an higher degree than E_{4}(u), it has good properties which help to obtain existence results. To our knowledge these are the only existence results of string-like solitons for a Lorentz invariant field equation in three space dimensions.

II. The model

In our model a field is a function

ψ : R^{3,1}→S^{2,1}

where R^{3,1} is the Minkowsky space-time (the space and time coordinates will be respectively denoted by x and t) and the target space S^{2,1} is defined as follows:

S^{2,1} = {ψ = (ψ_{1}, .., ψ_{4}) ∈ R^{3,1} | ∑_{i=1}^{3} ψ_{i}^{2} − ψ_{4}^{2} = 1} .

The points of S^{2,1} can be interpreted as the space-like 4-dimensional vectors having the Lorentzian lenght equal to 1. Notice that S^{2,1} is a Lorentzian manifold with the Lorentzian structure induced by the Minkowskian metric of R^{3,1}.

The scalar

σ = ∑_{i,k=1}^{3} (\frac{∂ψ_{k}}{∂x_{i}} )^{2} − ∑_{i=1}^{3} (\frac{∂ψ_{4}}{∂x_{i}} )^{2} − ∑_{k=1}^{3} (\frac{∂ψ_{k}}{∂t} )^{2} + (\frac{∂ψ_{4}}{∂t} )^{2}

is Lorentz invariant. Since we require the Lorentz invariance of our model, we shall consider Lagrangian densities of the form

L(∇ψ, ψ_{t}) = −\frac{1}{2}α(σ)  (1)

where α : R → R.

The action functional related to (1) is

S(ψ) = ∫_{R^{3,1}} L(∇ψ, ψ_{t}) dxdt = − ∫_{R^{3,1}} \frac{1}{2}α(σ) dxdt.

So the Euler-Lagrange equations are

D_{t} [α′ (σ) \frac{∂ψ_{k}}{∂t} ] − ∑_{i=1}^{3} D_{x_{i}} [α′ (σ) \frac{∂ψ_{k}}{∂x_{i}} ] = 0 , k = 1, ..., 4  (2)

where D_{t} and D_{x_{i}} denote the covariant derivatives with respect to the Lorentz metric in S^{2,1}.

We look for static solutions of the (2) having the form

ψ = (u_{1}(x), u_{2}(x), u_{3}(x), 0), x ∈ R^{3}

where u : R^{3} → S^{2} ⊂ R^{3} solves the equation

∑_{i=1}^{3} D_{x_{i}} [α′ (|∇u|^{2}) \frac{∂u_{k}}{∂x_{i}} ] = 0 , k = 1, 2, 3  (3)

with the asymptotic condition

lim_{|x|→∞} u(x) = ξ_{0}  (4)

where ξ_{0} ∈ S^{2} is a distinguished point fixed once for ever.

Here, D_{x_{i}} denotes the standard covariant derivative with respect to the S^{2} usual Riemannian metric and

|∇u|^{2} = ∑_{i,k=1}^{3} (\frac{∂u_{k}}{∂x_{i}} )^{2} .

Thus in any point x, u(x) is a unit vector. At ∞, u(x) approaches the constant value ξ_{0} which can be interpreted as the vacuum state.

u(x) ≡ ξ_{0} is the trivial solution; we will look for nontrivial solutions characterized by topological constraints which force the field u(x) away from its vacuum value in a suitable bounded spatial region. In this region the angles change, the spatial gradient is not zero and it gives the energy to the field.

If u = u(x_{1}, x_{2}, x_{3}) is a solution of the (3) we obtain dynamic soliton-like solution of the (2) just making a Lorentz transformation. Namely, if we take v = (v, 0, 0), we get the following solution of the (2):

ψ_{1}(x_{1}, x_{2}, x_{3}, t) = γ u_{1} (γ (x_{1} − vt) , x_{2}, x_{3}) ,

ψ_{2}(x_{1}, x_{2}, x_{3}, t) = u_{2} (γ (x_{1} − vt) , x_{2}, x_{3}) ,

ψ_{3}(x_{1}, x_{2}, x_{3}, t) = u_{3} (γ (x_{1} − vt) , x_{2}, x_{3}) ,

ψ_{4}(x_{1}, x_{2}, x_{3}, t) = −\frac{γv}{c^{2}} u_{1} (γ (x_{1} − vt) , x_{2}, x_{3})

where

γ = \frac{1}{\sqrt{1 − (\frac{v}{c} )^{2}}} .

Observe that the (3) are the Euler-Lagrange equations relative to the energy for functions independent of t

E(u) = ∫ [\frac{1}{2}α (|∇u|^{2})] dx.  (5)

The simplest equation of type (2) is obtained by choosing

α(σ) = σ  (6)

Then the equation (3) for the static solutions becomes the equation of harmonic maps with values in S^{2}

Δu + |∇u|^{2} u = 0

Considering the asymptotic condition (4), it is well known that the only solution is the trivial one u(x) ≡ ξ_{0}.

To get the existence of nontrivial solutions, it is necessary to assume α to be nonlinear:

α(σ) = σ + a_{2}σ^{2} + a_{3}σ^{3} + ...

Now let us look for the simplest choice of α. A priori the simplest choice is

α(σ) = σ + a_{2}σ^{2},

however this is a bad choice if we consider the evolution equation (2); in fact α′(σ) might be negative for some values of σ and this is a ”disaster” for the (2). Then, it is natural to consider the next simplest choice, namely

α(σ) = σ + \frac{1}{3}ε^{4}σ^{3}  (7)

being ε > 0 a parameter having the dimention of a lenght.

Then the energy functional (5) becomes

E(u) = E_{2}(u) + ε^{4}E_{6}(u) = \frac{1}{2} ∫_{R^{3}} |∇u|^{2} + \frac{ε^{4}}{6} ∫_{R^{3}} |∇u|^{6} dx  (8)

III. Existence results

First, we analyze the topological structure of our problem. We identify R^{3} ∪ {∞} with S^{3} with a distinguished point and we consider continuous functions u : R^{3} → S^{2} with the asymptotic condition lim_{|x|→∞} u(x) = ξ_{0}; each

of them identifies a homotopy class [u] ∈ π_{3}(S^{2}).

We recall that π_{3}(S^{2}) ≃ Z and every homotopy class is characterized by a topological invariant namely the Hopf invariant. The Hopf invariant is defined as follows: let ω = (dn ∧ dn, n) be the volume form on S^{2} and let ω^{∗} be its pull-back on S^{3} by u. Since ω is closed, also ω^{∗} is closed. Then, since S^{3} is 2-connected, ω^{∗} is exact. Let χ be its primitive. Then, the Hopf invariant can be defined as follows:

Q_{H}(u) = ∫ ω ∧ χ;

Q_{H} is an integer number which labels the homotopy class [u] .

Now let us define the following function space

Λ = {u ∈ C(R^{3}, S^{2}) | lim_{|x|→∞} u(x) = ξ_{0}, ∇u ∈ L_{loc}^{1}, ∥u∥_{Λ} < +∞}

where the gradient is intended in the sense of distributions and

∥u∥_{Λ} = (∫_{R^{3}} |∇u|^{2} dx)^{\frac{1}{2}} + (∫_{R^{3}} |∇u|^{6} dx)^{\frac{1}{6}} .

Standard arguments from calculus of variations show that Λ is the natural space where to study the energy functional E. In particular the energy is finite on any function of Λ and E(u) is a functional of class C^{1}.

Using the Hopf invariant, Λ can be decomposed into connected components as follows:

Λ = ⋃_{q∈Z} Λ_{q}  (9)

Each connected component Λ_{q} = {u ∈ Λ : Q_{H}(u) = q} is called sector. We have the following theorem

Theorem 1 There is q ≠ 0 such that E achieves the minimum in Λ_{±q} and the minimizer u_{±q} is of class C^{2}

Proof. In (^{3}) the authors study the functional (8) for maps u on R^{3} and taking values in an arbitrary riemannian manifold M (of course in this case the gradient ∇ in (8) is taken with respect to the riemannian structure of M). Clearly also in this case it can be introduced a configuration space Λ which is decomposed into connected components Λ_{q}, q belonging to the homotopy group π_{3}(M). If π_{3}(M) ≠ 0, it can be shown (see theorems 1.1, 3.5 of (^{3}) ) that (8) has C^{2} critical points belonging to Λ_{±q}, with q ≠ 0. Then, since π_{3}(S^{2}) ≃ Z, the conclusion follows.

IV. Some properties of the solutions

In the previous section we have seen that our problem has nontrivial solutions. We assume that u_{q} is one of them and we analyse some of its properties and in particular its string-like structure.

Since u_{q} is C^{2}, we can apply Sard lemma; then, for a.e. ξ ∈ S^{2} − ξ_{0}, the set u_{q}^{−1}(ξ) ia a one-dimensinal manifold in R^{3} which will be denoted by Γ_{q}. Moreover, since by the (4) u_{q}^{−1}(ξ) is bounded, Γ_{q} has the topological structure of S^{1} (or the union of a finite number of copies of S^{1}).

Now let ξ be a regular value for u_{q}, and assume that ξ is very far from ξ_{0} (e.g. ξ = −ξ_{0}, provided that −ξ_{0} is a regular value). On each x in Γ_{q} the vector u(x) points in a direction (almost) opposite to that of the vacuum state ξ_{0}; in order to reach ξ_{0} it must ”rotate”; this rotation makes ∇u ≠ 0 and gives the energy of the soliton. Clearly this energy accumulates near Γ_{q}. Since Γ_{q} is a one-dimensional manifold, it can be considered a string and the field energy is its mass.

The energy cost to change the shape of the string ensures that the string can vibrate around its equilibrium state but it cannot be broken. Thus the string persists unless it encounters a string of opposite Hopf invariant.

By theorem 1, for some q ∈ Z − 0 we have a local minimizer u_{q} and the corresponding Γ_{q} consists of one connected component. We do not know which sectors Λ_{q} contain minimizers and how many they are. Probably only numerical computations can answer to these questions. If q ≠ ±1, then Γ_{q} is a knot.

Now let us make some considerations on the size of Γ_{q}.

The equation (3) with α given by the (7) reduces to

∑_{i=1}^{3} D_{x_{i}} [(1 + ε^{4} |∇u|^{4}) \frac{∂u_{k}}{∂x_{i}} ] = 0, k = 1, 2, 3

which, if we do not make use of the covariant derivative, can be written as follows

Δ u + |∇u|^{2} u + ε^{4} (Δ_{6}u + |∇u|^{6} u) = 0  (10)

where

Δ_{6}u = ∑_{i=1}^{3} \frac{∂}{∂x_{i}} (|\frac{∂u}{∂x_{i}} |^{4} \frac{∂u}{∂x_{i}} )

Let U_{q}(x) be a solution of the equation (10) with ε = 1, namely

Δ U_{q} + |∇U_{q}|^{2} U_{q} + Δ_{6}U_{q} + |∇U_{q}|^{6} U_{q} = 0

then, it is easy to check that

u_{q}(x) = U_{q} (\frac{x}{ε})

is a solution of (10). Thus ε, which has the dimension of a lenght, is proportional to the diameter of Γ_{q}. Letting ε → 0, the string Γ_{q} becomes smaller and also the global mass goes to 0 with ε :

E(u_{q}) = ∫_{R^{3}} (\frac{1}{2}|∇u_{q}|^{2} + ε^{4} \frac{1}{6}|∇u_{q}|^{6}) dx

= ∫_{R^{3}} (\frac{ε^{−2}}{2} |∇U_{q}|^{2} + ε^{4} \frac{ε^{−6}}{6} |∇U_{q}|^{6}) ε^{3}dx

= ε ∫_{R^{3}} (\frac{1}{2}|∇U_{q}|^{2} + \frac{1}{6}|∇U_{q}|^{6}) dx = ε E(U_{q})

However, the mass density,

e(u_{q}) = \frac{1}{2}|∇u_{q}|^{2} + ε^{4} \frac{1}{6}|∇u_{q}|^{6}

= \frac{ε^{−2}}{2} |∇U_{q}|^{2} + ε^{4} \frac{ε^{−6}}{6} |∇U_{q}|^{6} = \frac{e(U_{q})}{ε^{2}}

for ε → 0, tends to ∞.

So, if ε is small, the string Γ_{q} is small, but its mass density is very large.

References

1V. Benci, D. Fortunato, L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension 3, to appear in Reviews in Mathematical Physics.

2V. Benci, D. Fortunato, L. Pisani, Remarks on topological solitons, Topological Methods in nonlinear Analysis, 7 (1996), 349-367.

3V. Benci, F. Giannoni, P. Piccione, A variational problem for manifold valued functionals, preprint.

4R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, New York, 1982.

5M.J. Esteban, A Direct Variational Approach to Skyrme’s Model for Meson Fields, Comm. Math. Phys. 105 (1986), 571-591.

6L. Faddeev, A.J. Niemi, Stable knot-like structures in classical field theory, Nature, 387 (1997), 58-61.

7L. Faddeev, L.A. Takhtajan, Hamiltonian methods in the theory of solitons, Springer, Berlin, 1987.

8M.J. Esteban, P.L. Lions, Skyrmions and Symmetry, Asymptotic Anal. 1 (1988), 187-192.

9C. Rebbi, G. Soliani, Solitons and particles, World Scientific, Singapore, 1984.

10T.H.R. Skyrme, A non-linear field theory, Proc. Roy. Soc. A260 (1961), 127-138.