Einstein-Yang-Mills-aether theory with nonlinear axion field:
Decay of color aether and the axionic dark matter production

We work with the multiplet of real vector fields $U^{j(a)}$, which appear in the adjoint representation of the SU(N) group Rubakov . We assume that the vector fields satisfy the condition $U^{m(a)}U^{n(b)}g_{mn}G_{(a)(b)}{=}1$. The group index $a$ runs in the range $a=1,2,...N^{2}{-}1$. $G_{(a)(b)}$ is the metric in the group space; it is defined as follows

$$G_{(a)(b)}=\left({\bf t}_{(a)},{\bf t}_{(b)}\right)\equiv 2{\rm Tr}\ {\bf t}_{(a)}{\bf t}_{(b)}$$

(3)

via the SU(N) group generators ${\bf t}_{(a)}$, which satisfy the commutation relations

$$\left[{\bf t}_{(a)},{\bf t}_{(b)}\right]=if^{(c)}_{\ (a)(b)}{\bf t}_{(c)}\,.$$

(4)

This formula involves the SU(N) group constants $f^{(a)}_{\ (b)(c)}$, which satisfy the Jacobi identity

$$f^{(a)}_{\ (b)(c)}f^{(c)}_{\ (e)(h)}{+}f^{(a)}_{\ (e)(c)}f^{(c)}_{\ (h)(b)}{+}f^{(a)}_{\ (h)(c)}f^{(c)}_{\ (b)(e)}{=}0\,.$$

(5)

The structure constants with three subscripts

$$f_{(c)(a)(b)}\equiv G_{(c)(d)}f^{(d)}_{\ (a)(b)}={-}2i{\rm Tr}\left[{\bf t}_{(a)},{\bf t}_{(b)}\right]{\bf t}_{(c)}$$

(6)

are antisymmetric with respect to transposition of any two indices. The Lagrange multiplier $\lambda$ appeared in front of the term $\left(U^{m}_{(a)}U^{(a)}_{m}{-}1\right)$ stands to guarantee that this term is equal to zero. The SU(N) symmetric extended covariant derivative Akhiezer in the application to the vector field yields

$$D_{k}U^{j(a)}=\nabla_{k}U^{j(a)}+gf^{(a)}_{\ (b)(c)}A^{(b)}_{k}U^{j(c)}\,,$$

(7)

where $\nabla_{k}$ is the spacetime covariant derivative. It includes the coupling constant $g$ and the multiplet of co-vector potentials $A^{(a)}_{m}$, describing the gauge field attributed to the SU(N) group. The metric $G_{(a)(b)}$ and the structure constants $f^{(d)}_{\ (a)(c)}$ satisfy the conditions

$$D_{m}G_{(a)(b)}=0\,,\qquad D_{m}f^{(a)}_{\ (b)(c)}=0\,,$$

(8)

i.e., they are the gauge covariant constant tensors in the group space:

The multiplet of tensors $F^{(a)}_{mn}$ is expressed via $A^{(a)}_{m}$ as follows:

$$F^{(a)}_{mn}=\nabla_{m}A^{(a)}_{n}-\nabla_{n}A^{(a)}_{m}+gf^{(a)}_{\ (b)(c)}A^{(b)}_{m}A^{(c)}_{n}\,.$$

(9)

This multiplet of tensors describes the gauge field strength and is indicated by the term Yang-Mills tensors. The dual tensors ${}^{*}\!F^{ik(a)}$

$${}^{*}\!F^{ik(a)}=\frac{1}{2}\epsilon^{ikls}F^{(a)}_{ls}\,,$$

(10)

are based on the universal Levi-Civita pseudo tensor $\epsilon^{ikls}=\frac{E^{ikls}}{\sqrt{-g}}$ ($E^{0123}=1$). When we use the definition (9), we obtain the identity

$$D_{k}{}^{*}\!F^{ik(a)}\equiv\nabla_{k}\ ^{*}\!F^{ik(a)}+gf^{(a)}_{\ (b)(c)}A^{(b)}_{k}{{}^{*}\!F^{ik(c)}}=0\,.$$

(11)

The pseudoscalar (axion) field is described by the term $\phi$ (here we use the dimensionless quantity $\phi$ instead of $\theta$). $\Psi_{0}$ is the coupling constant describing the interaction of the axions with the gauge field. According to the theory of axions, the Lagrangian of the axionically active system has to be invariant with respect to discrete symmetry $\phi\to\phi+2\pi n$ ($n$ is an integer), that is why we consider the potential of the axion field $V(\phi)$ to be of the periodic form

$$V(\phi)=2m^{2}_{A}\left(1-\cos{\phi}\right)\,.$$

(12)

The term $\frac{1}{4}\sin{\phi}{}^{*}\!F^{(a)}_{mn}F^{mn}_{(a)}$ in the unified invariant (1) also satisfies this discrete symmetry. As well as, the function $\sin{\phi}$ is odd, thus it changes the sign, when $t\to-t$, as the properties of the pseudoscalar field require. When $\phi\to 0$ the mentioned term converts into $\frac{1}{4}\phi{}^{*}\!F^{(a)}_{mn}F^{mn}_{(a)}$ recovering the original term introduced by Peccei and Quinn PQ .

The structure of the constitutive tensor

$${\cal K}^{ijmn}_{(a)(b)}=\cos{\phi}\left[G_{(a)(b)}\left(C_{1}g^{ij}g^{mn}{+}C_{2}g^{im}g^{jn}{+}C_{3}g^{in}g^{jm}\right)+C_{4}U^{i}_{(a)}U^{j}_{(b)}g^{mn}\right]+$$

(13)

$${+}\sin{\phi}\left\{C_{5}G_{(a)(b)}\epsilon^{ijmn}{+}C_{6}\left[U_{s(a)}\left(U^{i}_{(b)}\epsilon^{jsmn}{+}U^{j}_{(b)}\epsilon^{isnm}\right){+}U_{s(b)}\left(U^{i}_{(a)}\epsilon^{jsmn}{+}U^{j}_{(a)}\epsilon^{isnm}\right)\right]\right\}\,,$$

which we use in the Lagrangian, can be explained as follows. First, when $\phi=0$, and there is only one unit vector field $U^{i}$, this term recovers the Jacobson’s constitutive tensor J1 . Second, this tensor holds the form when we use the transformation $i\to j$, $m\to n$, $a\to b$ simultaneously, i.e., it supports the internal symmetry of the kinetic term ${\cal K}^{ijmn}_{(a)(b)}D_{i}U^{(a)}_{m}D_{j}U^{(b)}_{n}$. Third, this constitutive tensor includes $\phi$ as the argument of the trigonometric functions only, thus supporting the requirement of the discrete symmetry, prescribed by the axion field. Clearly, without the pseudoscalar field, i.e., when $\phi=0$, the Jacobson’s constitutive tensor could not contain the Levi-Civita pseudo-tensor $\epsilon^{ijmn}$. In other words, axionic extension of the Einstein-Yang-Mills-aether theory allows us to add principally new terms and new phenomenological constants $C_{5}$ and $C_{6}$. As it was mentioned in color1 there are also a lot of possibilities to introduce new coupling constants, however, we omit them, since after the decay of the color aether, when all the vector fields become parallel in the group space, the mentioned terms disappear and the corresponding coupling constants become hidden. To conclude, in addition to the coupling constants $C_{1}$, $C_{2}$, $C_{3}$ and $C_{4}$ introduced in the classical work J1 , we consider two new coupling constants $C_{5}$ and $C_{6}$, which are appropriate just for the axionic extension of the theory.

Variation of the action functional (2) with respect to $A^{(a)}_{i}$ gives the extended Yang-Mills equations

$$D_{k}\left\{{\cal L}^{\prime}({\cal I})\left[F^{ik}_{(a)}+\sin{\phi}\ {}^{*}\!F^{ik}_{(a)}\right]\right\}=\Gamma^{i}_{(a)}\,,$$

(14)

where the prime denotes the derivative with respect to the argument of the function. The color current $\Gamma^{i}_{(a)}$

$$\Gamma^{i}_{(a)}=\frac{g}{\kappa}f^{(d)}_{\ (c)(a)}U^{(c)}_{m}{\cal K}^{ijmn}_{(d)(b)}D_{j}U^{(b)}_{n}$$

(15)

appears because of variation of the second term in the right-hand side of (7) with respect to $A_{i}^{(a)}$. If we deal with the quasi-linear model, i.e., ${\cal L}={\cal I}$, keeping in mind (11) we can rewrite (14) in the form

$$\nabla_{k}F^{ik}_{(a)}+gf^{(c)}_{\ (b)(a)}A^{(b)}_{k}F^{ik}_{(c)}=$$

$$=-{}^{*}\!F^{ik}_{(a)}\cos{\phi}\nabla_{k}\phi+\frac{g}{\kappa}f^{(d)}_{\ (c)(a)}U^{(c)}_{m}{\cal K}^{ijmn}_{(d)(b)}\left(\nabla_{j}U^{(b)}_{n}+gf^{(b)}_{\ (h)(e)}A^{(h)}_{j}U^{(e)}_{n}\right)\,.$$

(16)

The first term in the right-hand side of this equation for the gauge field can be interpreted as the source associated with the axion field, and the second term relates to the contribution of the color aether. In other words, we deal with two independent sources, which prescribe the details of the Yang-Mills field evolution.

Variation of the action functional (2) with respect to the Lagrange multiplier $\lambda$ gives the normalization condition $U^{k}_{(a)}U_{k}^{(a)}=1$. Variation with respect to the vector fields $U^{(a)}_{m}$ yields the balance equation

$$D_{i}{\cal J}^{im}_{(a)}=\lambda\ U^{m}_{(a)}+{\cal I}^{m}_{(a)}\,.$$

(17)

Here the tensors ${\cal J}^{im}_{(a)}$ can be presented in the form

$${\cal J}^{im}_{(a)}\equiv{\cal K}^{ijmn}_{(a)(b)}D_{j}U^{(b)}_{n}=$$

(18)

$$=\cos{\phi}\left[C_{1}D^{i}U^{m}_{(a)}{+}C_{2}g^{im}D_{n}U^{n}_{(a)}{+}C_{3}D^{m}U^{i}_{(a)}+C_{4}U^{i}_{(a)}U^{j}_{(b)}D_{j}U^{m(b)}\right]{-}2C_{5}\sin{\phi}\ \Omega^{im}_{(a)}-$$

$${-}C_{6}\sin{\phi}\left[2\Omega^{ms(b)}\left(U_{s(a)}U^{i}_{(b)}+U_{s(b)}U^{i}_{(a)}\right)\ +\epsilon^{ismn}D_{j}U_{n}^{(b)}\left(U_{s(a)}U^{j}_{(b)}+U_{s(b)}U^{j}_{(a)}\right)\right]\,.$$

In this formula we used the new definition for the multiplet of pseudotensors

$$\Omega^{lm(b)}\equiv\frac{1}{2}\epsilon^{jlmn}D_{j}U_{n}^{(b)}\,,$$

(19)

which are skew-symmetric with respect to the indices $l$ and $m$, and play the roles of color vortex operators in our context. The term ${\cal I}^{m}_{(a)}$ in (17) appeared due to variation of the extended constitutive tensor with respect to the vector fields; its structure is the following:

$${\cal I}^{l}_{(d)}\equiv\frac{1}{2}D_{i}U_{m}^{(a)}D_{j}U_{n}^{(b)}\frac{\delta}{\delta U^{(d)}_{l}}\ {\cal K}^{ijmn}_{(a)(b)}=C_{4}\cos{\phi}D^{l}U^{n}_{(d)}\ U^{j}_{(b)}D_{j}U_{n}^{(b)}+$$

$$+2C_{6}\sin{\phi}\left[\Omega^{lm(b)}U^{p}_{(b)}D_{p}U_{m(d)}{+}\Omega^{lm}_{(d)}U^{p}_{(b)}D_{p}U_{m}^{(b)}+U_{s(b)}\left(\Omega^{sm}_{(d)}D^{l}U_{m}^{(b)}{+}\Omega^{sm(b)}D^{l}U_{m(d)}\right)\right]\,.$$

(20)

The Lagrange multiplier can be found as

$$\lambda=U_{m}^{(a)}\left[D_{i}{\cal J}^{im}_{(a)}-{\cal I}^{m}_{(a)}\right]\,,$$

(21)

when the corresponding terms are already calculated.

Variation of the action functional with respect to the pseudoscalar field $\phi$ gives the equation

$$\nabla^{k}\nabla_{k}\phi+m^{2}_{A}\sin{\phi}=-\frac{1}{4\Psi^{2}_{0}}\cos{\phi}{\cal L}^{\prime}({\cal I}){}^{*}\!F^{(a)}_{mn}F^{mn}_{(a)}+{\cal G}\,,$$

(22)

where the first term in the right-hand side describes the source related to the gauge field, and the pseudoscalar ${\cal G}$ given by

$${\cal G}=\frac{1}{2\kappa\Psi^{2}_{0}}\left\{\sin{\phi}\left[C_{1}D_{n}U^{(a)}_{m}D^{n}U_{(a)}^{m}{+}C_{2}D^{m}U^{(a)}_{m}D_{n}U_{(a)}^{n}{+}C_{3}D_{n}U^{(a)}_{m}D^{m}U_{(a)}^{n}{+}\right.\right.$$

(23)

$$\left.\left.+C_{4}U^{p}_{(a)}D_{p}U^{(a)}_{m}U^{q}_{(b)}D_{q}U^{(b)m}\right]-\right.$$

$$\left.{-}2\cos{\phi}\left[C_{5}\Omega^{pq}_{(a)}D_{p}U_{q}^{(a)}{+}2C_{6}\Omega^{qm(b)}D_{p}U_{m}^{(a)}\left(U_{q(a)}U^{p}_{(b)}{+}U_{q(b)}U^{p}_{(a)}\right)\right]\right\}$$

presents the source associated with the color aether contribution.

The last variation procedure in our context relates to the variation with respect to the metric. The gravitational field is obtained to be described by the following set of equations:

$$R_{pq}{-}\frac{1}{2}Rg_{pq}-\Lambda g_{pq}=\lambda U^{(a)}_{p}U_{(a)q}+T^{(\rm U)}_{pq}+\kappa T^{(\rm YM)}_{pq}+\kappa T^{(\rm A)}_{pq}\,.$$

(24)

The effective stress-energy tensor of the color vector fields $T^{(\rm U)}_{pq}$ is of the following form:

$$T^{(\rm U)}_{pq}=\frac{1}{2}g_{pq}{\cal K}^{ijmn}_{(a)(b)}D_{i}U^{(a)}_{m}D_{j}U^{(b)}_{n}{+}D^{m}\left[U_{(a)(p}{\cal J}_{q)m}^{(a)}{-}{\cal J}_{m(p}^{(a)}U_{q)(a)}{-}{\cal J}_{(pq)}^{(a)}U_{(a)m}\right]{+}$$

(25)

$$+\cos{\phi}\left[C_{1}\left(D_{m}U_{p}^{(a)}D^{m}U_{q(a)}-D_{p}U^{m(a)}D_{q}U_{m(a)}\right)+C_{4}U^{m}_{(a)}D_{m}U^{(a)}_{p}U^{n}_{(b)}D_{n}U^{(b)}_{q}\right]+$$

$$+4C_{5}\sin{\phi}\left[\frac{1}{4}g_{pq}D_{j}U_{n(a)}\Omega^{jn(a)}-D_{j}U^{(a)}_{(p}\Omega^{j}_{q)(a)}\right]+$$

$$+4C_{6}\sin{\phi}\left[2D^{j}U^{m}_{(a)}U^{((b)}_{j}U^{(a))}_{(p}\Omega_{q)m(b)}+2D^{j}U_{(a)(p}\Omega^{m}_{q)(b)}U_{m}^{((a)}U_{j}^{(b))}+\right.$$

$$\left.+D_{i}U^{m(a)}\epsilon_{jsm(p}D^{j}U^{(b)}_{q)}U^{i}_{((b)}U^{s}_{(a))}-g_{pq}D_{j}U^{m(b)}\Omega_{sm}^{(a)}U^{(j}_{(a)}U^{s)}_{(b)}\right]\,.$$

Here the parentheses $(pq)$ denote the symmetrization with respect to the coordinate indices $p$ and $q$, i.e., $a_{(p}b_{q)}=\frac{1}{2}(a_{p}b_{q}+a_{q}b_{p})$. The symbol $T^{(\rm YM)}_{pq}$ relates to the effective stress-energy tensor of the nonlinear Yang-Mills field

$$T^{(\rm YM)}_{pq}={\cal L}^{\prime}({\cal I})\left[\frac{1}{4}g_{pq}F^{(a)}_{mn}F_{(a)}^{mn}-F^{(a)}_{pm}F_{(a)qn}g^{mn}\right]+g_{pq}\left[{\cal L}-{\cal I}{\cal L}^{\prime}({\cal I})\right]\,.$$

(26)

The last term in (24)

$$T^{(\rm A)}_{pq}=\Psi^{2}_{0}\left[\nabla_{p}\phi\nabla_{q}\phi+\frac{1}{2}g_{pq}\left(V(\phi)-\nabla_{n}\phi\nabla\phi^{n}\right)\right]$$

(27)

describes the canonic effective stress-energy tensor of the pure axion field.

In this Section we established the new self-consistent SU(N) symmetric model of nonlinear interaction between gauge, vector, pseudoscalar and gravitational fields. The set of the model master equations is derived for arbitrary spacetime platform. This model as a whole deserves detailed analysis and has a lot of potential applications. But what we plan to do now? Our purpose is to make the first step and to analyze the solutions to the equation for the axion field (22) with the sources, which correspond to the decay of the color aether in the early Universe. We assume that this story can be described in the framework of Bianchi-I spacetime model.

First of all, the Yang-Mills fields become quasi-Maxwellian, i.e.,

$$F^{(a)}_{mn}=\nu q^{(a)}F_{mn}\,,\quad F_{mn}=\nabla_{m}A_{n}-\nabla_{n}A_{m}\,.$$

(28)

The terms $D_{k}U^{j(a)}$ and $\Omega^{pq(a)}$ convert, respectively, into

$$D_{k}U^{j(a)}=q^{(a)}\nabla_{k}U^{j}\,,\quad\Omega^{pq(a)}=q^{(a)}\Omega^{pq}\,,\quad\Omega^{pq}=\frac{1}{2}\epsilon^{pqmn}\nabla_{m}U_{n}\,.$$

(29)

The extended Yang-Mills equations (14) take now the Maxwell-type form

$$\nabla_{k}\left[{\cal L}^{\prime}({\cal I})\left(F^{ik}+\sin{\phi}\ {}^{*}\!F^{ik}\right)\right]=0\,,$$

(30)

and the equation (11) reads

$$\nabla_{k}{}^{*}\!F^{ik}=0\,.$$

(31)

With the proposed ansatz we can transform the axion field equation (22) as follows

$$\nabla^{k}\nabla_{k}\phi{+}m^{2}_{A}\sin{\phi}={-}\frac{\nu^{2}\cos{\phi}}{4\Psi^{2}_{0}}\ {\cal L}^{\prime}({\cal I})\ {}^{*}\!F_{mn}F^{mn}{+}\frac{\cos{\phi}}{\kappa\Psi^{2}_{0}}\left[2(C_{5}{+}2C_{6})\ \omega^{p}DU_{p}{-}C_{5}\omega^{pq}\omega^{*}_{pq}\right]{+}$$

(32)

$$+\frac{\sin{\phi}}{2\kappa\Psi^{2}_{0}}\left[(C_{1}{+}C_{3})\sigma_{mn}\sigma^{mn}+\frac{1}{3}\Theta^{2}(C_{1}{+}3C_{2}{+}C_{3}){+}(C_{1}{-}C_{3})\omega_{mn}\omega^{mn}{+}(C_{1}{+}C_{4}){\cal D}U_{m}{\cal D}U^{m}\right]\,.$$

Here we use the notations standard for the Einstein-aether theory, namely, the convective derivative ${\cal D}$, the scalar of the medium flow expansion $\Theta$, the projector $\Delta^{p}_{m}$, the skew-symmetric vortex tensor $\omega_{mn}$ and its dual $\omega^{*pq}$, the symmetric traceless shear tensor $\sigma_{mn}$, and the (pseudo) four-vector of the rotation velocity $\omega^{p}$:

$${\cal D}\equiv U^{p}\nabla_{p}\,,\quad\Theta\equiv\nabla_{p}U^{p}\,,\quad\omega_{mn}\equiv\frac{1}{2}\Delta^{p}_{m}\Delta^{q}_{n}(\nabla_{p}U_{q}-\nabla_{q}U_{p})\,,\quad\Delta^{p}_{m}\equiv\delta^{p}_{m}-U^{p}U_{m}\,,$$

(33)

$$\sigma_{mn}\equiv\frac{1}{2}\Delta^{p}_{m}\Delta^{q}_{n}(\nabla_{p}U_{q}+\nabla_{q}U_{p})-\frac{1}{3}\Delta_{mn}\Theta\,,\quad\omega^{*pq}\equiv\frac{1}{2}\epsilon^{pqmn}\omega_{mn}\,,\quad\omega^{p}\equiv\omega^{*pq}U_{q}=\Omega^{pq}U_{q}\equiv\Omega^{p}\,.$$

The equation (32) admits the equilibrium-type solutions $\phi=2\pi n$ ($n$ is an integer), introduced and advocated in E1 ; E2 ; E3 ; E4 , if and only if

$${\cal L}^{\prime}({\cal I}){}^{*}\!F_{mn}F^{mn}=\frac{4}{\kappa\nu^{2}}\left[2(C_{5}{+}2C_{6})\ \omega^{p}DU_{p}{-}C_{5}\omega^{pq}\omega^{*}_{pq}\right]\,,$$

(34)

otherwise, the axion production takes place inevitably in the course of the color aether decay.

Since now ${\cal J}^{im}_{(a)}=q_{(a)}{\cal J}^{im}$ and ${\cal I}^{m}_{(a)}=q_{(a)}{\cal I}^{m}$, the balance equations (17) take the form

$$\nabla_{i}{\cal J}^{im}=\lambda\ U^{m}+{\cal I}^{m}\,,$$

(35)

where the corresponding reduced quantities are

$${\cal J}^{im}=\cos{\phi}\left[C_{1}\nabla^{i}U^{m}{+}C_{2}g^{im}\Theta{+}C_{3}\nabla^{m}U^{i}+C_{4}U^{i}{\cal D}U^{m}\right]+$$

(36)

$$+\sin{\phi}\left[(2C_{6}-C_{5})U_{s}\epsilon^{imsn}{\cal D}U_{n}-8C_{6}U^{i}\Omega^{m}-2C_{5}\epsilon^{impq}\Omega_{pq}\right]\,,$$

$${\cal I}^{l}=C_{4}\cos{\phi}\nabla^{l}U^{n}\ {\cal D}U_{n}+4C_{6}\sin{\phi}\left[\Omega^{lm}DU_{m}-\Omega^{m}\nabla^{l}U_{m}\right]\,.$$

(37)

The new coupling constant $C_{5}$ happens to be hidden in (19).

Using the ansatz of parallel fields we have to replace $\lambda U^{(a)}_{p}U_{(a)q}$ with $\lambda U_{p}U_{q}$ in (24), and to delete the group indices $(a)$ in the stress-energy tensor of the Yang-Mills field (26). The stress-energy tensor (25) converts now into

$$T^{(\rm U)}_{pq}=\nabla^{m}\left[U_{(p}{\cal J}_{q)m}{-}{\cal J}_{m(p}U_{q)}{-}{\cal J}_{(pq)}U_{m}\right]{+}$$

(38)

$$+\cos{\phi}\left[C_{1}\left(\nabla_{m}U_{p}\nabla^{m}U_{q}-\nabla_{p}U^{m}\nabla_{q}U_{m}\right)+C_{4}{\cal D}U_{p}{\cal D}U_{q}\right]+$$

$$+\frac{1}{2}g_{pq}\cos{\phi}\left[C_{1}\nabla^{j}U^{m}\nabla_{j}U_{m}+C_{2}\Theta^{2}+C_{3}\nabla_{j}U_{m}\nabla^{m}U^{j}+C_{4}{\cal D}U^{m}{\cal D}U_{m}\right]-$$

$$-2C_{5}\sin{\phi}\nabla_{j}U_{(p}\epsilon^{j}_{\ q)mn}\nabla^{m}U^{n}+$$

$$+4C_{6}\sin{\phi}\left[{\cal D}U^{m}U_{(p}\epsilon_{q)mls}\nabla^{l}U^{s}+{\cal D}U_{(p}\epsilon_{q)mls}U^{m}\nabla^{l}U^{s}+{\cal D}U^{m}U^{s}\epsilon_{jsm(p}\nabla^{j}U_{q)}\right]\,.$$

Next simplification of the total set of master equations appears, when we choose the spacetime platform, which corresponds to the homogeneous Bianchi-I cosmological model.

Taking into account the symmetry of the reduced model, we can state that the equations (31) admit the solution $F_{12}=const$, and the equations (30) can be reduced to one equation

$$\frac{d}{dt}\left[{\cal L}^{\prime}({\cal I})\left(abcF^{30}-F_{12}\sin{\phi}\right)\right]=0\,.$$

(42)

The solution to (42) is

$$F^{30}(t)=F^{30}(t_{0})\frac{a(t_{0})b(t_{0})c(t_{0})}{a(t)b(t)c(t)}\cdot\frac{{\cal L}^{\prime}({\cal I})(t_{0})}{{\cal L}^{\prime}({\cal I})(t)}+\frac{F_{12}}{a(t)b(t)c(t)}\left[\sin{\phi(t)}-\sin{\phi(t_{0})}\cdot\frac{{\cal L}^{\prime}({\cal I})(t_{0})}{{\cal L}^{\prime}({\cal I})(t)}\right]\,,$$

(43)

where $t_{0}$ is some specific time moment. For illustration we assume the following initial data: $F^{30}(t_{0})=0$ and $\phi(t_{0})=\pi n$, so that below we work with the exact solution

$$F^{30}(t)=\frac{F_{12}}{abc}\sin{\phi(t)}\,,$$

(44)

which does not include the nonlinear function ${\cal L}({\cal I})$. For this elegant solution the quadratic terms

$$\frac{1}{4}F^{(a)}_{mn}F^{mn}_{(a)}=\frac{\nu^{2}F^{2}_{12}}{2a^{2}b^{2}}\cos^{2}{\phi}\,,\quad\frac{1}{4}{}^{*}\!F^{(a)}_{mn}F^{mn}_{(a)}=\frac{\nu^{2}F^{2}_{12}}{a^{2}b^{2}}\sin{\phi}\,,\quad{\cal I}=\frac{\nu^{2}F^{2}_{12}}{2a^{2}b^{2}}(1+\sin^{2}{\phi})$$

(45)

also are rather simple.