LIGHTLIKE SHELL SOLITONS

OF EXTREMAL SPACE-TIME FILM

Alexander A. Chernitskii

A. Friedmann Laboratory for Theoretical Physics

Moika 48, St.-Petersburg, Russia.

AAChernitskii@mail.ru

October 17, 2019

Abstract

New exact solution class of Born – Infeld type nonlinear scalar fieldmodel is obtained. The variational principle of this model has a specificform which is characteristic for extremal four-dimensional hypersurface orhyper film in five-dimensional space-time. Obtained solutions are singularsolitons propagating with speed of light and having energy, momentum,and angular momentum which can be calculated for explicit conditions.The soliton singularity here is a moving two-dimensional surface or shell,where the model action density becomes zero. The lightlike soliton canhave a set of tubelike shells with the appropriate cavities. A twisted light-like soliton is considered. It is notable that its energy is proportional toits angular momentum in high-frequency approximation. A case with onetubelike cavity is considered. In this case the soliton shell is diffeomorphicto cylindrical surface with cuts by multifilar helix. The shell transversesize of the appropriate finite energy soliton can be converging to zero atinfinity. The ideal gas of such lightlike solitons with minimal twist param-eter is considered in a finite volume. Explicit conditions provide that theangular momentum of each soliton in the volume equals Planck constant.The equilibrium energy spectral density for the solitons is obtained. It hasthe form of Planck distribution in some approximation. A beam of twistedlightlike solitons is considered. The representation of arbitrary polariza-tion for beam with twisted lightlike solitons is discussed. It is shown thatthis beam provides the effect of mechanical angular momentum transferto absorbent by circularly polarized beam. This effect well known forphoton beam. Thus the soliton solution which have determinate likenesswith photon is obtained in particular.

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Contents

1 Introduction 2

2 Extremal space-time film 3

3 Energy-momentum and angular momentum 6

4 General lightlike soliton 7

5 Twisted lightlike soliton 13

6 Relation to photons 23

7 Conclusions 32

1 Introduction

A nonlinear space-time scalar field model considered here is known for a longtime sufficiently. This model is related to well known Born – Infeld nonlinearelectrodynamics [1], and it is sometimes called Born – Infeld type scalar fieldmodel.

This model is attractive because it has relatively simple and geometricallyclear form. It can be considered as a relativistic generalization of the minimalsurface or minimal thin film model in three-dimensional space.

In this generalization we have an extremal four-dimensional film in five-dimensional space-time. But the model equation appears as differential one forscalar field in four-dimensional space-time.

On the other hand, this model can provide the necessary effects which arerequired for a realistic filed model.

In particular, the model under consideration has a static spherically symmet-ric solution, which is identical to zero four-vector component of electromagneticpotential for dyon solution of Born – Infeld electrodynamics [2]. This staticsolution of the scalar model gives the appropriate moving soliton solution withthe aid of Lorentz transform.

As it was shown in the cited work, in the case of nonlinear electrodynam-ics there are the conformity between long-range interaction of solitons and twoknown long-range interactions of physical particles, that is electromagnetic andgravitational ones. But the methods which was used for the investigation of soli-ton long-range interaction are independent of the field model. The appropriateinstruments are integral conservation laws and characteristic equation.

2

These methods applying to the scalar model under consideration give theresults, which are similar to ones for nonlinear electrodynamics. These results indetail must be matter for another article. Here we briefly discuss the obtainingof Lorentz force for interacting scalar solitons in the next section.

An essential difference of the scalar field model from the nonlinear electro-dynamics is obviously caused by the different tensor character of the fields. Inparticular, a weak single-component scalar wave can not provide the transversepolarization of electromagnetic wave.

But at the present time we consider the light as photon beam but not aweak electromagnetic wave with constant amplitude. The photon beam couldbe represented by an appropriate scalar soliton beam. In this case an essen-tial space-time nonhomogeneous of soliton solution may provide the necessarysymmetry properties for the beam.

Thus at the present work we consider the model of extremal space-time film.We obtain its exact soliton solutions propagating with the speed of light that islightlike solitons.

Then we investigate in detail a lightlike soliton solution having a rotationabout the direction of propagation that is twisted lightlike soliton.

We consider the ideal gas of such twisted lightlike solitons. Using explicitassumptions we obtain Planck distribution formula in some approximation.

At last we consider a beam with the twisted lightlike solitons. We show thatthis beam can represent photon one. In this case we have, in particular, thepolarization property and the effect of mechanical angular momentum transferto absorbent by circularly polarized beam.

2 Extremal space-time film

Let us consider the following action which has the world volume form:

𝒜 =

∫𝑉

√|M| (d𝑥)

4, (2.1a)

where M + det(M𝜇𝜈), (d𝑥)4 + d𝑥0d𝑥1d𝑥2d𝑥3, 𝑉 is space-time volume,

M𝜇𝜈 = m𝜇𝜈 + 𝜒2 𝜕Φ

𝜕𝑥𝜇

𝜕Φ

𝜕𝑥𝜈, (2.1b)

m𝜇𝜈 are components of metric tensor for flat four-dimensional space-time, Φ isscalar real field function, 𝜒 is dimensional constant. The Greek indices takevalues {0, 1, 2, 3}.

The variational principle 𝛿𝒜 = 0 with action (2.1) corresponds to extremalfour-dimensional film Φ({𝑥𝜇}) in five-dimensional space-time {Φ, 𝑥0, 𝑥1, 𝑥2, 𝑥3}.

Determinant M in (2.1) can be represented in the form

M = m

(1 + 𝜒2 m𝜇𝜈 𝜕Φ

𝜕𝑥𝜇

𝜕Φ

𝜕𝑥𝜈

), (2.2)

3

where m + det(m𝜇𝜈).Taking into account (2.2) we can write model action (2.1) in the form

𝒜 =

∫𝑉

ℒ d𝑉 , (2.3a)

where d𝑉 +√|m| (d𝑥)

4is four-dimensional volume element,

ℒ +

√1 + 𝜒2 m𝜇𝜈

𝜕Φ

𝜕𝑥𝜇

𝜕Φ

𝜕𝑥𝜈

. (2.3b)

Variational principle with action (2.3) gives the following model equation:

1√|m|

𝜕

𝜕𝑥𝜇

√|m|Υ𝜇 = 0 , (2.4a)

where

Υ𝜇 +Φ𝜇

ℒ, Φ𝜇 = m𝜇𝜈 Φ𝜈 , (2.4b)

Φ𝜈 +𝜕Φ

𝜕𝑥𝜈. (2.4c)

We have the following evident relations from (2.4c):

𝜕Φ𝜇

𝜕𝑥𝜈− 𝜕Φ𝜈

𝜕𝑥𝜇= 0 . (2.5)

Inversion for relations (2.4b) gives

Φ𝜇 =Υ𝜇

ℒ, (2.6a)

where

ℒ +√|1 − 𝜒2 m𝜇𝜈 Υ𝜇 Υ𝜈 | . (2.6b)

For the case when the field invariant Φ𝜌 Φ𝜌 relatively small (𝜒2 |Φ𝜌 Φ𝜌| ≪ 1)we can represent the action density ℒ with two first terms in formal power seriesof 𝜒:

ℒ = 1 +𝜒2

2m𝜇𝜈 𝜕Φ

𝜕𝑥𝜇

𝜕Φ

𝜕𝑥𝜈+ 𝒪

(𝜒4)𝜒→0

. (2.7)

The appropriate linearized equation has the form

1√|m|

𝜕

𝜕𝑥𝜈

(√|m|m𝜇𝜈 𝜕Φ

𝜕𝑥𝜇

)+ 𝒪

(𝜒2)𝜒→0

= 0 . (2.8)

Also let us write the linearized relation (2.4b):

Υ𝜇 = Φ𝜇 + 𝒪(𝜒2)𝜒→0

. (2.9)

4

Nonlinear differential equation of second order (2.4) for function Φ can berepresented in the form of the first order differential equation system for four-vectors Φ𝜇 or Υ𝜇. In this case we have differential field equations (2.4a) and(2.5). In addition we must consider algebraical relations (2.4b) or (2.6).

As can be seen, the model action (2.3) is susceptible to the choice of met-ric signature. Here we will use the signature {+,−,−,−}. We introduce thefollowing designation for Minkowski metric:

−m00 = 1 , −m

0𝑖 = 0 , −m𝑖𝑗 = −𝛿𝑖𝑗 (2.10)

where 𝛿𝑖𝑗 is Kronecker symbol. The Latin indices take values {1, 2, 3}.The signature metric in use (2.10) allows the same spherically symmetric

solution of the model that was obtained by M. Born and L. Infeld in theirclassical work for nonlinear electrodynamics:

Υ𝑟 =𝑞

𝑟2,

𝜕Φ

𝜕𝑟=

𝑞√𝑟4 + 𝑟4

, (2.11)

where 𝑞 is constant, 𝑟 is radial spherical coordinate, 𝑟 +√|𝑞 𝜒|.

It is evident that solution (2.11) give birth to the class of soliton solutionswith Lorentz transformations. Such solutions in this model also can be con-sidered as point charged particles because their long-range interactions haveelectromagnetic character.

Indeed to investigate the interactions we can use the method based on inte-gral conservation law of momentum (for Born – Infeld nonlinear electrodynamicssee [2]). Let us consider the long-range interaction of an appropriate to (2.11)moving soliton-particle with the rest one (2.11). In this case the method givespure electrical interaction between the particles. Then we can transform theobtained law of particle movement with electrical force to another moving ref-erence frame. In this case Lorentz transform of the force gives its magneticcomponent. The Lorentz transform of the force was presented by A. Einsteinin last section of his classical work on special relativity [3].

It should be noted that using the metric signature {−,+,+,+} for action(2.3) leads to minus sign before the term 𝑟4 in (2.11). This gives infinity value ofΦ𝑟 on the sphere. If we want to consider the solution (2.11) with this signature,we must change the sign before 𝜒2 in (2.1b) and (2.3b).

Instead relations between four-vectors {Φ𝜇} and {Υ𝜈} (2.4b) and (2.6) wecan consider relation between quadruples of components {Φ0,Υ1,Υ2,Υ3} and{Υ0,Φ1,Φ2,Φ3} in Minkowski metric (2.10). This representation can be prefer-able for some problems.

The appropriate solution of equations (2.4b) gives the following relations:

Υ0 =

√1 + 𝜒2 Υ𝑗 Υ𝑗√

1 + 𝜒2 Φ20

Φ0 , Φ𝑖 =

√1 + 𝜒2 Φ2

0√1 + 𝜒2 Υ𝑗 Υ𝑗

Υ𝑖 . (2.12)

Comparison relations (2.12) and (2.4b) gives the following expression for the

5

action density ℒ in Cartesian coordinates:

ℒ =

√1 + 𝜒2 Φ2

0√1 + 𝜒2 Υ𝑗 Υ𝑗

. (2.13)

Now let us write the field equation (2.4) in Cartesian coordinates with metric(2.10). After differentiation Υ𝜇 (2.4b) in (2.4a) and multiplication the equationby ℒ3 we obtain (

−m𝜇𝜈 ℒ2 − 𝜒2 Φ𝜇 Φ𝜈

) 𝜕2 Φ

𝜕𝑥𝜇 𝜕𝑥𝜈= 0 . (2.14)

As we see, obtained equation does not include radicals.It is evident that the model under consideration keep invariance for space-

time rotation and scale transformation. Thus any solution give birth to theappropriate class of solutions with the following transform:

Φ({𝑥𝜇}) → 𝑎Φ({𝐿𝜇.𝜈 𝑥

𝜈/𝑎}) , (2.15)

where 𝐿𝜇.𝜈 are components of space-time rotation matrix, 𝑎 is scale parameter.

3 Energy-momentum and angular momentum

Customary method gives the following canonical energy-momentum density ten-sor of the model in Cartesian coordinates

−→𝑇𝜇𝜈 =

1

4𝜋

(Φ𝜇 Φ𝜈

ℒ− −m

𝜇𝜈

𝜒2ℒ)

. (3.1)

As we see, the canonical tensor is symmetrical.To use finite integral characteristics of solutions in infinite space-time we

introduce regularized energy-momentum density tensor with the following for-mula:

→𝑇𝜇𝜈 =−→𝑇𝜇𝜈 −

∞→𝑇𝜇𝜈 . (3.2)

where∞→𝑇𝜇𝜈 is regularizing symmetrical energy-momentum density tensor which

can be defined depending on class of solutions under consideration. Here wewill use constant regularizing tensor

∞→𝑇𝜇𝜈 = − 1

4𝜋 𝜒2 −m𝜇𝜈 . (3.3)

We have conservation law for regularized energy-momentum density tensorin Cartesian coordinates

𝜕→𝑇𝜇𝜈

𝜕𝑥𝜈= 0 . (3.4)

Let us define angular momentum density tensor by customary way. We havethe following appropriate conservation law:

𝜕 ∘𝑀𝜇𝜈𝜌

𝜕𝑥𝜌= 0 , (3.5)

6

where∘𝑀𝜇𝜈𝜌 + 𝑥𝜇→𝑇 𝜈𝜌 − 𝑥𝜈 →𝑇𝜇𝜌 . (3.6)

We introduce the following special designations for energy, momentum vec-tor, and angular momentum vector densities: ℰ , 𝒫 , 𝒥 . Let us write the appro-priate expressions taking into account relations (2.3b), (2.4b), and (2.13):

ℰ +→𝑇 00 =1

4𝜋

(Φ0 Φ0

ℒ− ℒ− 1

𝜒2

)(3.7a)

=1

4𝜋 𝜒2

(𝜒2 (Φ𝑖 Φ𝑖) − 1

ℒ+ 1

)(3.7b)

=1

4𝜋 𝜒2

(𝜒4 (Υ𝑖 Υ𝑖) Φ2

0 − 1√1 + 𝜒2 (Υ𝑖 Υ𝑖)

√1 + 𝜒2 Φ2

0

+ 1

), (3.7c)

𝒫𝑖 +→𝑇 0𝑖 = →𝑇 𝑖0 =1

4𝜋

Φ0 Φ𝑖

ℒ=

1

4𝜋Φ0 Υ𝑖 =

1

4𝜋Φ𝑖 Υ0 , (3.7d)

𝒥𝑖 + 𝜖𝑖𝑗𝑘 𝑥𝑗 𝒫𝑘 , (3.7e)

where 𝜖𝑖𝑗𝑘 is Levi-Civita symbol (𝜖123 = 1).Let us define energy, momentum, and angular momentum of field in a three-

dimensional volume 𝑉 :

E𝑉 +∫𝑉

ℰ d𝑉 , PPP𝑉 +∫𝑉

𝒫 d𝑉 , JJJ𝑉 +∫𝑉

𝒥 d𝑉 . (3.8)

4 General lightlike soliton

Let us consider solutions in a form of wave propagating along 𝑥3 axis of Cartesiancoordinate system with the speed of light. Let this solution be have sometransverse and longitudinal field distributions. Thus we can write

Φ = Φ(𝜃, 𝑥1, 𝑥2

), (4.1a)

𝜃 = 𝜔 𝑥0 − 𝑘3 𝑥3 , 𝑘3 = ±𝜔 , 𝜔 > 0 . (4.1b)

Substitution (4.1) to field equation (2.14) gives the following equation:(1 − 𝜒2

(𝜕Φ

𝜕𝑥2

)2)𝜕2Φ

(𝜕𝑥1)2+ 2𝜒2 𝜕Φ

𝜕𝑥1

𝜕Φ

𝜕𝑥2

𝜕Φ2

𝜕𝑥1𝜕𝑥2

+

(1 − 𝜒2

(𝜕Φ

𝜕𝑥1

)2)𝜕2Φ

(𝜕𝑥2)2= 0 . (4.2)

As we see this equation does not include derivatives on phase of wave 𝜃 (4.1b).Equation (4.2) is elliptical for the following condition:

1 − 𝜒2(Φ2

1 + Φ22

)> 0 . (4.3)

7

The similar in form (4.2) equations were considered. About this topic seethe paper by R. Ferraro [4] and references therein.

In particular, the similar in form but different in type equation was consid-ered by Barbashov and Chernikov [5]. The monograph by Whitham [6] containsrelatively simple way for obtaining the Barbashov – Chernikov solution with thehelp of hodograph transformation (see, for example, [7]).

Here we use in outline the Whitham method but for the elliptic (for condition(4.3)) equation (4.2). The qualitative difference between hyperbolic and ellipticequations causes the appropriate difference in the solution way.

Let us introduce new independent variables

𝜉 = 𝑥1 + 𝚤 𝑥2 , *𝜉 = 𝑥1 − 𝚤 𝑥2 , (4.4)

where 𝚤2 = −1.Also we will use cylindrical coordinates {𝜌, 𝜙, 𝑥3}. We have the following

evident relations:

𝜉 = 𝜌 e𝚤 𝜙 , *𝜉 = 𝜌 e−𝚤 𝜙 , (4.5a)

𝜌 =√

𝜉 *𝜉 , 𝜙 = −𝚤 ln(𝜉⧸√

𝜉 *𝜉). (4.5b)

Using new variables (4.4) we obtain from (4.2) the following equation:(1 − 2𝜒2 𝜕Φ

𝜕𝜉

𝜕Φ

𝜕*𝜉

)𝜕2Φ

𝜕𝜉 𝜕*𝜉

+ 𝜒2

(𝜕Φ

𝜕𝜉

)2𝜕2Φ

(𝜕*𝜉)2+ 𝜒2

(𝜕Φ

𝜕*𝜉

)2𝜕2Φ

(𝜕𝜉)2= 0 . (4.6)

Equation (4.6) is hyperbolic for the following condition:

1 − 4𝜒2(Φ2

𝜉 + Φ2*𝜉

)> 0 . (4.7)

As noted in section 2 the field model under consideration is invariant byspace-time rotation and scale transformation. But equation (4.2) does not con-tain derivatives with respect to coordinates {𝑥0, 𝑥3}. Because this here we havespace-time rotation and scale invariance in the planes {𝑥1, 𝑥2} and {𝑥0, 𝑥3} withmutually independent parameters. Thus equation (4.2) is invariant with respectto rotation about 𝑥3 axis and scale transformation in {𝑥1, 𝑥2} plane.

As applied to equation (4.6), taking into account relations (4.5) and (2.15),these two types of invariance are provided by the following general substitution:

Φ(𝜉, *𝜉) →√𝑜 *𝑜 Φ (𝜉/𝑜, *𝜉/*𝑜) , (4.8a)

where 𝑜 is arbitrary complex constant with respect to coordinates {𝜉, *𝜉}, *𝑜 iscomplex conjugate to 𝑜 quantity. The constant 𝑜 will be called the scale-rotationparameter of solution in the plane {𝑥1, 𝑥2}.

The complex constant 𝑜 can be written in the form

𝑜 = 𝜌 e𝚤 𝜙 , (4.8b)

8

where 𝜌 and 𝜙 are real constants with respect to coordinates {𝑥1, 𝑥2}. But ingeneral case these constants can be depend on phase of soliton 𝜃 (4.1b):

𝜌 = 𝜌(𝜃) , 𝜙 = 𝜙(𝜃) . (4.8c)

Because this we will call 𝑜(𝜃) the scale-rotation function of the soliton. It isevident that the function 𝜌(𝜃) defines the phase dependence of transversal scaleand the function 𝜙(𝜃) defines the phase dependence of rotation about 𝑥3 axis.

Thus if we have a solution Φ(𝜉, *𝜉) to equation (4.6) then by means of invari-ant substitution (4.8) we obtain wave propagating along 𝑥3 axis and preservingits transversal form. Longitudinal form of the wave defined by scale-rotationphase function 𝑜(𝜃) is also preserved.

As result we have wave packet propagating with speed of lite and preservingits shape. It can be called the lightlike soliton.

Equation (4.6) is equivalent to the following first order system:

𝜕Φ𝜉

𝜕*𝜉− 𝜕Φ*𝜉

𝜕𝜉= 0 , (4.9a)(

1 − 2𝜒2 Φ𝜉 Φ*𝜉

) 𝜕Φ𝜉

𝜕*𝜉+ 𝜒2 Φ2

𝜉

𝜕Φ*𝜉

𝜕*𝜉+ 𝜒2 Φ2

*𝜉

𝜕Φ𝜉

𝜕𝜉= 0 , (4.9b)

where

Φ𝜉 +𝜕Φ

𝜕𝜉, Φ*𝜉 +

𝜕Φ

𝜕*𝜉. (4.10)

By interchanging the roles of the dependent and independent variables in(4.9) we obtain the linear system

𝜕𝜉

𝜕Φ*𝜉− 𝜕*𝜉

𝜕Φ𝜉= 0 , (4.11a)

(1 − 2𝜒2 Φ𝜉 Φ*𝜉

) 𝜕𝜉

𝜕Φ*𝜉− 𝜒2 Φ2

𝜉

𝜕𝜉

𝜕Φ𝜉− 𝜒2 Φ2

*𝜉

𝜕*𝜉

𝜕Φ*𝜉= 0 , (4.11b)

which is equivalent to single equation

(1 − 2𝜒2 Φ𝜉 Φ*𝜉

) 𝜕2𝜉

𝜕Φ𝜉𝜕Φ*𝜉− 𝜒2 Φ2

𝜉

𝜕2𝜉

(𝜕Φ𝜉)2− 𝜒2 Φ2

*𝜉

𝜕2𝜉

(𝜕Φ*𝜉)2

− 2𝜒2 Φ𝜉𝜕𝜉

𝜕Φ𝜉− 2𝜒2 Φ*𝜉

𝜕𝜉

𝜕Φ*𝜉= 0 . (4.12)

Let us introduce new independent variables

𝜂 = 𝚤

√1 − 4𝜒2 Φ𝜉 Φ*𝜉 − 1

2𝜒Φ*𝜉, (4.13a)

*𝜂 = −𝚤

√1 − 4𝜒2 Φ𝜉 Φ*𝜉 − 1

2𝜒Φ𝜉. (4.13b)

9

Inversion for relations (4.13) gives

Φ𝜉 =𝚤 𝜂

𝜒 (1 + 𝜂*𝜂), (4.14a)

Φ*𝜉 = − 𝚤 *𝜂

𝜒 (1 + 𝜂*𝜂). (4.14b)

Substituting (4.13) and (4.14) into (4.11), we obtain

𝜂2𝜕𝜉

𝜕𝜂+

𝜕*𝜉

𝜕𝜂= 0 , (4.15a)

𝜕𝜉

𝜕*𝜂+ *𝜂2

𝜕*𝜉

𝜕*𝜂= 0 . (4.15b)

Sequential elimination each of the dependent variables *𝜉(𝜂, *𝜂) and 𝜉(𝜂, *𝜂)from system (4.15) gives two simple equations

𝜕2𝜉

𝜕𝜂 𝜕*𝜂= 0 ,

𝜕2*𝜉

𝜕𝜂 𝜕*𝜂= 0 , (4.16)

solutions of which have the form

𝜉 = 𝜉1(𝜂) + 𝜉3(*𝜂) , *𝜉 = 𝜉2(*𝜂) + 𝜉4(𝜂) , (4.17)

where 𝜉1(𝜂), 𝜉2(*𝜂), 𝜉3(*𝜂), 𝜉4(𝜂) are arbitrary functions.Substitution (4.17) to (4.15) gives

𝜂2 𝜉′1 + 𝜉′4 = 0 , 𝜉′3 + *𝜂2 𝜉′2 = 0 . (4.18)

Taking into consideration (4.17) and 4.18), we can write the following rela-tions for general solution of the system (4.15):

d𝜉 = d𝜉1 − *𝜂2 𝜉′2 d*𝜂 = d𝜉1 − *𝜂2 d𝜉2 , (4.19a)

d*𝜉 = d𝜉2 − 𝜂2 𝜉′1 d𝜂 = d𝜉2 − 𝜂2 d𝜉1 . (4.19b)

Thus the general solution of the system (4.15) contains only two arbitrary func-tions 𝜉1(𝜂) and 𝜉2(*𝜂).

Using (4.14) and (4.19), we obtain

𝜕Φ

𝜕𝜂= Φ𝜉

𝜕𝜉

𝜕𝜂+ Φ*𝜉

𝜕*𝜉

𝜕𝜂=

𝚤

𝜒𝜂 𝜉′1 , (4.20a)

𝜕Φ

𝜕*𝜂= Φ𝜉

𝜕𝜉

𝜕*𝜂+ Φ*𝜉

𝜕*𝜉

𝜕*𝜂= − 𝚤

𝜒*𝜂 𝜉′2 . (4.20b)

From (4.20) we have

dΦ =𝚤

𝜒

(𝜂 𝜉′1 d𝜂 − *𝜂 𝜉′2 d*𝜂

)=

𝚤

𝜒

(𝜂 d𝜉1 − *𝜂 d𝜉2

). (4.21)

10

Here the variables 𝜂 and *𝜂 in last expression must be considered as inversefunctions for 𝜉1(𝜂) and 𝜉2(*𝜂) that are 𝜂 = 𝜂(𝜉1) and *𝜂 = *𝜂(𝜉2).

Let us introduce the designations

𝚤

𝜒𝜂(𝜉1)+

dΞ1

d𝜉1, − 𝚤

𝜒*𝜂(𝜉2)+

dΞ2

d𝜉2, (4.22)

where functions Ξ1(𝜉1) and Ξ2(𝜉2) are arbitrary because of arbitrariness of thefunctions 𝜉1(𝜂) and 𝜉2(*𝜂).

Then, using (4.21) and (4.22), we have the general solution of equation (4.6)in the form

Φ = Ξ1(𝜉1) + Ξ2(𝜉2) . (4.23a)

Here arbitrariness of functions Ξ1(𝜉1) and Ξ2(𝜉2) is restricted by reality of fieldfunction Φ. The connection between variables {𝜉1, 𝜉2} and {𝜉, *𝜉} is defined byrelations (

d𝜉d*𝜉

)=

(1 𝜒2

(Ξ′2

)2𝜒2(Ξ′1

)21

)(d𝜉1d𝜉2

), (4.23b)

which are obtained from (4.19) with (4.22).Relations (4.23b) can be inverted on the assumption of nonsingularity of the

transition matrix:

1 − 𝜒4(Ξ′1

)2 (Ξ′2

)2 = 0 , (4.24a)(d𝜉1d𝜉2

)=

1

1 − 𝜒4(Ξ′1

)2 (Ξ′2

)2(

1 −𝜒2(Ξ′2

)2−𝜒2

(Ξ′1

)21

)(d𝜉d*𝜉

). (4.24b)

Obtained solution (4.23) can be checked directly. Substitution (4.23a) toequation (4.6) and using (4.24) reduce to identity.

It can be checked also that equation (4.6) is hyperbolic with the solution(4.23) for condition (4.24a).

One could say that relations (4.23b) define transformation of independentvariables {𝜉, *𝜉} to {𝜉1, 𝜉2} for equation (4.6). But the definition of transforma-tion with differential relations is not complete. Direct connection between thevariables can be obtained by path integration of relations (4.23b) in nonsingulararea, that is for condition (4.24a). At the same time we must define an initialcorrespondence between the variables {𝜉, *𝜉} and {𝜉1, 𝜉2}.

We can consider the simplest case by taking 𝜒 = 0 in (4.23b). In this casewe can put (

𝜉1𝜉2

)=

(𝜉*𝜉

), (4.25)

and expression (4.23a) is evident solution of appropriate to (4.6) linear equationwhen 𝜒 = 0.

In general case let us consider relation (4.25) as asymptotic for 𝜌 → ∞.Then we can designate

𝜉1 = 𝜉 , 𝜉2 = *𝜉 . (4.26)

11

It is useful to introduce polar coordinates {𝜌, 𝜙} for variables {𝜉, *𝜉} byanalogy with (4.5) for variables {𝜉, *𝜉}:

𝜉 = 𝜌 e𝚤 𝜙 , *𝜉 = 𝜌 e−𝚤 𝜙 , (4.27a)

𝜌 =

√𝜉 *𝜉 , 𝜙 = −𝚤 ln

(𝜉

⧸√𝜉 *𝜉

). (4.27b)

General solution in the form of lightlike soliton depending on phase 𝜃 (4.1b)can be obtained from (4.23) by invariant substitution (4.8).

It is notable that the action density for obtained solution does not con-tain radical. Substitution solution (4.8) with (4.23a) into (2.3b) with (4.4) and(4.24b) gives expression

ℒ =

1 − 𝜒2 Ξ′

1 Ξ′2

1 + 𝜒2 Ξ′1 Ξ′

2

. (4.28)

As we see, explicit dependence on phase 𝜃, which we have in (4.8), here is absent.Let us obtain the energy, momentum, and angular momentum densities for

lightlike soliton. For this purpose we substitute solution (4.23a) with scale-rotation transformation (4.8) to formulas (3.7).

Using relations (4.4) and (4.24b), we obtain the expressions for energy, mo-mentum, and angular momentum densities with some common functions, whichwill be designate as 𝑓ℰ

𝑖 . Then we have

ℰ = 𝑓ℰ0 + 𝜔2

((𝜙′)2 𝑓ℰ

1 +(𝜌′)2 (

𝑓ℰ2 /𝜌

2 + 𝑓ℰ3 /𝜌 + 𝑓ℰ

4

)+ 𝜌′ 𝜙′ (𝑓ℰ

5 /𝜌 + 𝑓ℰ6

) ), (4.29a)

𝒫3 = 𝑘 𝜔((

𝜙′)2 𝑓ℰ1 +

(𝜌′)2 (

𝑓ℰ2 /𝜌

2 + 𝑓ℰ3 /𝜌 + 𝑓ℰ

4

)+ 𝜌′ 𝜙′ (𝑓ℰ

5 /𝜌 + 𝑓ℰ6

) ), (4.29b)

𝒥3 = 𝜔(𝜙′ 𝑓ℰ

1 + 𝜌′(𝑓ℰ5 /𝜌 + 𝑓ℰ

6

)/2)

, (4.29c)

where 𝑘 = ±𝜔 according to (4.1b).Here we write explicitly only two functions 𝑓ℰ

0 and 𝑓ℰ1 :

𝑓ℰ0 +

1

2𝜋

Ξ′1 Ξ′

2

1 + 𝜒2 Ξ′1 Ξ′

2

, 𝑓ℰ1 + − 1

4𝜋

(𝜉 e−𝚤 𝜙 Ξ′

1 − *𝜉 e𝚤 𝜙 Ξ′2

)21 − 𝜒4

(Ξ′1

)2(Ξ′2

)2 . (4.30)

These functions play main role in the area, where the scale function 𝜌(𝜃) isalmost constant: 𝜌′ → 0.

We have from (4.29) the following notable relation for the case 𝜌′ → 0:

ℰ − 𝑓ℰ0 = |𝒫3| = 𝜔 |𝜙′𝒥3| . (4.31)

The arbitrary functions 𝜌(𝜃) and 𝜙(𝜃) (4.8c) define scale and rotation in theplane {𝑥1, 𝑥2} accordingly. Using (5.2), (4.8), and (4.1b), we can show that the

12

case 𝜙′ > 0 corresponds to positive rotation by angle 𝜙 in time 𝑥0 and in 𝑥3

axis for 𝑘3 > 0.Thus for right-handed coordinate system {𝑥1, 𝑥2, 𝑥3}, the cases 𝜙′ > 0 and

𝜙′ < 0 correspond to right and left local twist of the soliton accordingly.It is interesting to consider the solitons with constant twist:

𝜙′ = const = 0 . (4.32)

Such solitons can be called uniformly twisted ones. For conciseness we will callthem the twisted solitons.

As we see in (4.31), for the case (4.32) the soliton energy density ℰ is pro-portional to its angular momentum density 𝒥 in high-frequency approximation,that is for 𝜔 |𝜙′𝒥3| ≫

𝑓ℰ0

. The appropriate proportionality relation between

soliton energy and its angular momentum is notable property of the twistedlightlike soliton.

To obtain integral characteristics of the soliton it is necessary to integratethe functions {𝑓ℰ

0 , ..., 𝑓ℰ6 } in the plane {𝑥1, 𝑥2}. Considering (4.23b) and (4.27),

we can see that the appropriate integrands have notable simple form in thevariables {𝜌, 𝜙} (4.27).

We must take into consideration that the functions {𝜉, *𝜉} and, accordingly,{Ξ, *Ξ} depended on arguments {𝜉/𝑜, *𝜉/*𝑜} after scale-rotation transformation(4.8). Thus making additional substitution {𝜉/𝑜, *𝜉/*𝑜} → {𝜉, *𝜉}, we have thefollowing integrands:

𝑓ℰ0 𝜌2 𝜌 d𝜙d𝜌 =

𝜌2

2𝜋Ξ′1 Ξ′

2

(1 − 𝜒2 Ξ′

1 Ξ′2

)𝜌d𝜙d𝜌 , (4.33a)

𝑓ℰ1 𝜌2 𝜌 d𝜙d𝜌 = − 𝜌2

4𝜋

(𝑜 𝜉 e−𝚤 𝜙 Ξ′

1 − *𝑜 *𝜉 e𝚤 𝜙 Ξ′2

)2𝜌 d𝜙d𝜌 , (4.33b)

𝑓ℰ2 𝜌2 𝜌 d𝜙d𝜌 =

𝜌2

4𝜋

(𝑜 𝜉 e−𝚤 𝜙 Ξ′

1 + *𝑜 *𝜉 e𝚤 𝜙 Ξ′2

)2𝜌d𝜙d𝜌 , (4.33c)

𝑓ℰ3 𝜌2 𝜌 d𝜙d𝜌 = − 𝜌2

2𝜋(Ξ1 + Ξ2)

(𝑜 𝜉 e−𝚤 𝜙 Ξ′

1 + *𝑜 *𝜉 e𝚤 𝜙 Ξ′2

)×(1 + 𝜒2 Ξ′

1 Ξ′2

)𝜌d𝜙d𝜌 , (4.33d)

𝑓ℰ4 𝜌2 𝜌 d𝜙d𝜌 =

𝜌2

4𝜋(Ξ1 + Ξ2)

2 (1 + 𝜒2 Ξ′

1 Ξ′2

)2𝜌d𝜙d𝜌 , (4.33e)

𝑓ℰ5 𝜌2 𝜌 d𝜙d𝜌 =

𝜌2 𝚤

2𝜋

(𝑜2 𝜉2 e−𝚤 2𝜙 (Ξ′

1)2 − *𝑜2 *𝜉2 e𝚤 2𝜙 (Ξ′2)2)𝜌d𝜙d𝜌 , (4.33f)

𝑓ℰ6 𝜌2 𝜌 d𝜙d𝜌 = −𝜌2 𝚤

2𝜋(Ξ1 + Ξ2)

(𝑜 𝜉 e−𝚤 𝜙 Ξ′

1 − *𝑜 *𝜉 e𝚤 𝜙 Ξ′2

)×(1 + 𝜒2 Ξ′

1 Ξ′2

)𝜌d𝜙d𝜌 . (4.33g)

5 Twisted lightlike soliton

For further calculations, we define the arbitrary functions Ξ1 and Ξ2. Let ustake power function with integer negative exponent. Introducing necessary mul-

13

tiplicative constants for concordance of physical dimension and for simplificationof resulting formulas, we have

Ξ1 =−𝜌𝑚+1

𝜒𝑚𝜉−𝑚 , Ξ2 =

−𝜌𝑚+1

𝜒𝑚*𝜉−𝑚 , (5.1a)

Ξ′1 = −

−𝜌𝑚+1

𝜒𝜉−(𝑚+1) , Ξ′

2 = −−𝜌𝑚+1

𝜒*𝜉−(𝑚+1) , (5.1b)

where 𝑚 is natural number, constant −𝜌 has a physical dimension of length.Then formula (4.23a) representing the solution of equation (4.6) has the form

Φ =−𝜌𝑚+1

𝜒𝑚

(𝜉−𝑚 + *𝜉−𝑚

). (5.2)

Because 𝜉 ∼ 𝜉 and *𝜉 ∼ *𝜉 at 𝜌 → ∞, we have from (5.2) with the scale-rotation transformation (4.8) the following asymptotic solution:

Φ ∼ 2 (𝜌−𝜌)𝑚+1

𝜒𝑚𝜌𝑚cos(𝑚 (𝜙− 𝜙)

)at 𝜌 → ∞ . (5.3)

In view of dependence on phase 𝜌(𝜃) and 𝜙(𝜃), the formula (5.3) describesthe propagating wave along the 𝑥3 axis. The dependence 𝜙(𝜃) in (5.3) describesalso the twist of this wave about the propagation direction.

Let us consider the twisted lightlike soliton with condition (4.32). We putfor this case

𝜙 = ± 𝜃

𝑚, (5.4)

where the signs ’+’ and ’−’ correspond to right and left twisted soliton accord-ingly.

In addition, let us consider that the scale function 𝜌(𝜃) is almost constant:𝜌′ ∼ 0. As we can see in (5.3) with (5.4), in this case 𝜔 in (4.1b) is radianfrequency of the soliton wave and 2𝜋/ |𝑘3| is the appropriate wave length.

To obtain the functions 𝜉(𝜉) and *𝜉(*𝜉), first we must integrate relations(4.23b). It is convenient to use the coordinates {𝜌, ��} (4.27) for integration ofright hand part of (4.23b).

Let us take the path of integration in the plane {𝜌, ��} with sufficiently farbeginning from coordinate origin. Let the starting point be {𝜌∞, 0}. We canintegrate by the following path: from {𝜌∞, 0} to {𝜌, 0} for �� = const and from{𝜌, 0} to {𝜌, ��} for 𝜌 = const.

Then we take that 𝜌∞ → ∞, where relations (4.25) are satisfied. Usingvariables {𝜉, *𝜉} again, as result we have

𝜉 = 𝜉 −−𝜌2(𝑚+1)

(2𝑚 + 1) *𝜉2𝑚+1, *𝜉 = *𝜉 −

−𝜌2(𝑚+1)

(2𝑚 + 1) 𝜉2𝑚+1., (5.5)

Obtained relations (5.5) give the following single-valued mapping for vari-ables:

{𝜉, *𝜉} =⇒ {𝜉, *𝜉} . (5.6a)

14

But for representation of solution (5.2) in coordinates {𝑥1, 𝑥2}, according to(4.4), we must have inverse to (5.6a) mapping

{𝜉, *𝜉} =⇒ {𝜉, *𝜉} . (5.6b)

As we can see, relations (5.5), considered as equations for variables {𝜉, *𝜉},give multi-valued mapping (5.6b). But, of curse, here we must consider onlythe realization of (5.6b) with (5.5), which leads to relations (4.25) at infinity𝜌 → ∞.

Let us substitute (5.1b) to expression for the determinant of transition ma-trix in (4.23b). Then, using the coordinates {𝜌, 𝜙} (4.27), we obtain the fol-lowing value of 𝜌 for any 𝜙, which violate the reversibility condition (4.24a) fortransformation of variables (4.23b):

𝜌 = −𝜌 . (5.7)

The action density (4.28) vanishes for the set of points appropriate to (5.7).Because this the vector components Υ𝜇 (2.4b) become infinite for Φ𝜇 = 0.

Thus we have the singular line for the plane {𝜉, *𝜉} in the form of circle withradius −𝜌. For the plane {𝜉, *𝜉}, this singular line, according to (5.5) and (5.7),is described by formula

𝜉𝑚 = −𝜌 e𝚤 𝜙(

1 − e𝚤 2𝑚𝜙

2𝑚 + 1

). (5.8)

Here the function 𝜉𝑚 = 𝜉𝑚(𝜙) represents the parametric expression for thesingular line in the complex plane of variable 𝜉.

Expression (5.8) represents epicycloid with 2𝑚 cusps. This line for 𝑚 = 1is shown on Fig. 5.1 and it for 𝑚 = 2 is shown on Fig. 5.2. These figureswas obtained also by R. Ferraro [8, 9] for mathematically similar but anotherproblem.

In the present investigation, the system (5.5) with −𝜌 = 1 for given values ofparameter 𝑚 and variables {𝜉, *𝜉} is solved numerically with respect to variables{𝜉, *𝜉} in all characteristic areas of the plane {𝑥1, 𝑥2}.

In the area of the plane {𝑥1, 𝑥2} outside of the singular line (5.8) we haveone-to-one mapping (5.6) with the condition (4.25) at infinity 𝜌 → ∞.

This mapping can keep continuity for transition through the singular line,if we resign the condition of mutual complex conjugation for variables 𝜉1 and𝜉2 (4.26). But in this case the field function Φ becomes complex-valued nearlyeverywhere in the inner area of the singular line (5.8), excepting some radiallines, where the function Φ keeps reality. These lines in the plane {𝑥1, 𝑥2} aretwo-dimensional surfaces in the three-dimensional space {𝑥1, 𝑥2, 𝑥3}.

But such transition through the singular line is forbidden in the frameworkof the obtained solution. Thus we can find a solution in the inner area of thesingular line in the plane {𝑥1, 𝑥2} independently of the solution in the outsidearea. Then we could try to satisfy any conditions for the field function and itsderivatives on the singular line. But here such conditions could appear to beforcible.

15

𝒜1ℬ1

𝑥2

𝑥1

𝑚 = 1−𝜌 = 1, 𝜒 = 1

𝜌 = 1, 𝜙 = 0

Figure 5.1: Singular line on the plane {𝑥1, 𝑥2} with parameter 𝑚 = 1.

A radical solution of the inner area problem is exclusion of this area formthe space. In this case we have a soliton with the appropriate cavity. Here wefollow this way.

Taking into account also the scale-rotation transformation (4.8), we have thefollowing condition for the space of the solution:

𝜌 >𝜉𝑚(𝜙(𝜃)

)𝜌(𝜃) , 𝜌 > −𝜌 , (5.9)

where dependence 𝜉𝑚 = 𝜉𝑚(𝜙(𝜃)

)corresponds to rotation of the singular con-

tour in the plane {𝑥1, 𝑥2} by the angle 𝜙(𝜃) (4.8c).Thus according to (5.9) we have the soliton with an inner shell. It can be

called the shell soliton.The results of numerical calculations for the function Φ (5.2) on the singular

line are shown on Fig. 5.3 for 𝑚 = 1 and on Fig. 5.4 for 𝑚 = 2. The appropriateresults for the field function Φ on the plane {𝑥1, 𝑥2} are shown on Fig. 5.5 for𝑚 = 1 and on Fig. 5.6 for 𝑚 = 2. The points {𝒜1, ℬ1} for 𝑚 = 1 and{𝒜2, ℬ2, 𝒞2, 𝒟2} for 𝑚 = 2 are corresponding on Figures 5.1 - 5.6.

As we see on Figs. 5.3 and 5.4, the field function Φ on the singular line isnearly triangle function by angle 𝜙. But there are slight deflection from straightlines.

The cusps are the derivative discontinuities for the field function along thesingular line in the plane {𝑥1, 𝑥2}. As we can see on Figs. 5.5 and 5.6, thesederivative discontinuities are absent outside of the singular line.

Now let us obtain the expressions of full energy, momentum, and angular mo-mentum for the solution under consideration in bounded three-dimensional vol-

16

𝒜2ℬ2

𝒞2

𝒟2

𝑥2

𝑥1

𝑚 = 2−𝜌 = 1, 𝜒 = 1

𝜌 = 1, 𝜙 = 0

Figure 5.2: Singular line on the plane {𝑥1, 𝑥2} with parameter 𝑚 = 2.

ume. For convenience we consider the tubular volume in coordinates {𝜌, 𝜙, 𝑥3}.Its internal radius is defined in (5.9). Let its external radius and length be des-ignated as 𝜌∞ and 𝑙𝑠 accordingly. Thus in addition to condition (5.9) we have

𝜌 6 𝜌∞ , − 𝑙𝑠26 𝑥3 6

𝑙𝑠2. (5.10)

First we calculate the integrals on right-hand parts of relations (4.33) byvariables {𝜌, 𝜙} in area {[−𝜌, 𝜌∞], [−𝜋, 𝜋]}. That corresponds to integration onleft-hand parts of relations (4.33) by variables {𝜌, 𝜙} in the outside area of thesingular line 𝜉𝑚 and bounded by the line 𝜉(𝜌∞ e−𝚤 𝜙).

Making the integration in the plane {𝑥1, 𝑥2} we can get the rotation param-eter be zero: 𝜙 = 0. Let us substitute (5.1) and (5.5) with (4.27) to right-handparts of (4.33). We change the integration by variable 𝑥3 to one by phase 𝜃(4.1b).

As result we have the following expressions for energy and absolute valuesof momentum and angular momentum:

E = P + E , (5.11a)

E =−𝜌2

𝜔 𝜒2𝒞0 ℐ0 = P

1

−𝜌2 𝜔2

𝒞0 ℐ0𝒞1 (𝒞2 ℐ1 + 𝒞3 ℐ2)

, (5.11b)

P =𝜔−𝜌4

𝜒2𝒞1 (𝒞2 ℐ1 + 𝒞3 ℐ2) , (5.11c)

J =−𝜌4

𝜒2𝒞1 𝒞2 |ℐ3| , (5.11d)

17

Φ(𝜉𝑚)

𝒜1

ℬ1ℬ1

𝑚 = 1−𝜌 = 1, 𝜒 = 1

𝜌 = 1, 𝜙 = 0

𝜙/𝜋

Figure 5.3: The field function Φ on the singular line of the plane {𝑥1, 𝑥2} for𝑚 = 1.

where E is the part of soliton energy obtained from the part 𝑓ℰ0 of energy density

ℰ (4.29a),

ℐ0 +

𝜔 𝑙𝑠/2∫−𝜔 𝑙𝑠/2

𝜌2d𝜃 , ℐ1 +

𝜔 𝑙𝑠/2∫−𝜔 𝑙𝑠/2

𝜌4 (𝜙′)2

d𝜃 ,

ℐ2 +

𝜔 𝑙𝑠/2∫−𝜔 𝑙𝑠/2

𝜌2 (𝜌′)2

d𝜃 , ℐ3 +

𝜔 𝑙𝑠/2∫−𝜔 𝑙𝑠/2

𝜌4 𝜙′ d𝜃 , (5.12)

𝒞0 +𝑚 + 1

2𝑚 (2𝑚 + 1)− (−𝜌/𝜌∞)2𝑚

2𝑚+

(−𝜌/𝜌∞)2 (2𝑚+1)

2 (2𝑚 + 1), (5.13a)

𝒞1 + ln

(𝜌∞−𝜌

)+

13

72− 1

6

(−𝜌

𝜌∞

)4− 1

72

(−𝜌

𝜌∞

)8for 𝑚 = 1 , (5.13b)

𝒞1 +1

2𝑚2

((𝑚 + 1)2 (12𝑚2 − 2𝑚− 1)

𝑚 (3𝑚 + 1) (2𝑚 + 1)2 (𝑚− 1)− (−𝜌/𝜌∞)2 (𝑚−1)

𝑚− 1

− (−𝜌/𝜌∞)4𝑚

𝑚 (2𝑚 + 1)− (−𝜌/𝜌∞)2 (3𝑚+1)

(2𝑚 + 1)2 (3𝑚 + 1)

)for 𝑚 > 2 , (5.13c)

𝒞2 + 𝑚2 , 𝒞3 + (𝑚 + 1)2 . (5.13d)

The value of 𝑥3 momentum projection is defined by the sign of wave vector

18

Φ(𝜉𝑚)

𝒜2 ℬ2ℬ2

𝒞2 𝒟2

𝑚 = 2−𝜌 = 1, 𝜒 = 1

𝜌 = 1, 𝜙 = 0

𝜙/𝜋

Figure 5.4: The field function Φ on the singular line of the plane {𝑥1, 𝑥2} for𝑚 = 2.

projection 𝑘3 (4.1b): P3 = ±P.In general case the 𝑥3 angular momentum projection is defined by integral

ℐ3 (5.12), which can be called the integral twist of the soliton with weight 𝜌4:J3 = ±J.

Let us write the appropriate to (5.11) expressions for the twisted soliton.Using condition (5.4) and formulas (5.12), we have

|ℐ3| =1

𝑚ℐ1 , ℐ1 =

1

𝑚2ℐ1 , ℐ1 +

𝜔 𝑙𝑠/2∫−𝜔 𝑙𝑠/2

𝜌4d𝜃 . (5.14)

Using (5.14) and (5.13), we obtain from (5.11) the following expressions forthe twisted soliton:

E = P(

1 +1

−𝜌2 𝜔2

𝒞0 ℐ0𝒞1 (ℐ1 + 𝒞3 ℐ2)

), (5.15a)

P = 𝜔 J1

𝑚

(1 +

𝒞3 ℐ2ℐ1

), (5.15b)

J =−𝜌4

𝜒2𝑚 𝒞1 ℐ1 , (5.15c)

For the twisted soliton let us consider the case for slowly varying scale func-tion 𝜌(𝜃), such that ℐ2 → 0 (5.12). Also we suppose that the frequency 𝜔 issufficiently high, such that −𝜌𝜔 → ∞. According to expressions (5.15), in this

19

Φ𝒜1

ℬ1𝑚 = 1−𝜌 = 1, 𝜒 = 1

𝜌 = 1, 𝜙 = 0

𝑥2

𝑥1

Figure 5.5: The field function Φ on the plane {𝑥1, 𝑥2} for 𝑚 = 1.

Φ 𝒜2ℬ2

𝒞2𝒟2

𝑚 = 2−𝜌 = 1, 𝜒 = 1

𝜌 = 1, 𝜙 = 0𝑥2

𝑥1

Figure 5.6: The field function Φ on the plane {𝑥1, 𝑥2} for 𝑚 = 2.

case we have the following relations:

E = P = �� J , (5.16a)

where�� +

𝜔

𝑚(5.16b)

is the angular velocity of the twisted soliton.Let us consider the twisted soliton with scale function in the form of Gaussian

curve:

𝜌 = exp

(− 𝜃2

2 𝜃2

), (5.17)

where 𝜃 is characteristic length of the soliton measured in radians and numeri-cally equals to a total angle of twist on the characteristic length of the soliton

20

along 𝑥3 axis. A twist angle 2𝜋/𝑚 corresponds to soliton wave-length along 𝑥3

axis.Let us consider the case of infinite space with the conditions

𝜌∞−𝜌

→ ∞ , 𝜔 𝑙𝑠 → ∞ . (5.18)

Then the calculation of the essential integrals in (5.12) and (5.14) for the func-tions (5.17) gives

ℐ0 = 𝜃√𝜋 , ℐ1 = 𝜃

√𝜋

2, ℐ2 =

1

4 𝜃

√𝜋

2. (5.19)

As we see in (5.11) and (5.13) with (5.18) and (5.19), for the case of infinitespace we have the finite values of energy, momentum, and angular momentumif 𝑚 > 2. Using (5.15) with (5.13), (5.14), (5.18), and (5.19), let us write theappropriate expressions for 𝑚 = 2:

E = P

(1 +

560√

2 𝜃2

129 (9 + 4 𝜃2)−𝜌2 𝜔2

), (5.20a)

P = 𝜔 J1

2

(1 +

9

4 𝜃2

), (5.20b)

J =387

1400

√𝜋

2

−𝜌4 𝜃

𝜒2. (5.20c)

It is evident that the case 𝜃 ≫ 1 and −𝜌 𝜔 ≫ 1 for expressions (5.20) givesrelations (5.16a).

The shell of the twisted soliton with Gaussian scale phase functions is shownon Fig. 5.7 and Fig. 5.8.

BB

BB��1��)

𝜆𝑚 = 1

𝜃 = 10

Figure 5.7: The shell of Gaussian twisted soliton for 𝑚 = 1.

It is significant that the twist parameter 𝑚 is a topological invariant fordiffeomorphism. The shell of twisted lightlike soliton is diffeomorphic to cylin-drical surface with cuts by multifilar helix, where the number of continuous cuts

21

BBBB1)

𝜆𝑚 = 2

𝜃 = 10

Figure 5.8: The shell of Gaussian twisted soliton for 𝑚 = 2.

is 2𝑚. These cuts correspond to the singular lines on the shell, which we cansee on Fig. 5.7 and Fig. 5.8.

The field function Φ of the Gaussian twisted soliton in the plane section{𝑥1, 𝑥3} for 𝑥2 = 0 is shown on Fig. 5.9 and Fig. 5.10.

𝑚 = 1

𝜃 = 10

Φ

𝑥3

𝑥1

Figure 5.9: The field function Φ of Gaussian twisted soliton for 𝑚 = 1 on theplane {𝑥1, 𝑥3} for 𝑥2 = 0.

At last we show zero level surfaces of the field function Φ for the Gaussiantwisted soliton with 𝑚 = 1 (Fig. 5.11) and 𝑚 = 2 (Fig. 5.12). The twist of thesolitons is well seen also on these figures. We have two-sheeted helical surfacewith excluded cavity for 𝑚 = 1 and we have four-sheeted one for 𝑚 = 2.

All figures 5.7 – 5.12 are appropriate to the solitons twisted on the right.Here we have considered the simplest arbitrary functions Ξ1 and Ξ2, which

give the twisted shell lightlike soliton with one cavity. For more complicated

22

𝑚 = 2

𝜃 = 10

Φ

𝑥3

𝑥1

Figure 5.10: The field function Φ of Gaussian twisted soliton for 𝑚 = 2 on theplane {𝑥1, 𝑥3} for 𝑥2 = 0.

cases we can have the appropriate solitons with a set of cavities. But we willhave the notable asymptotic relation between energy, momentum, and angularmomentum of type (5.16) for these cases, because of the appropriate relationfor densities (4.31).

6 Relation to photons

Because of notable connection (5.16) between energy, momentum, and angularmomentum of the twisted lightlike solitons, it is reasonable to consider theirrelation to photons.

For this purpose first we consider an ideal gas of these solitons in boundedthree-dimensional volume 𝑉 .

As is known, the ideal gas behaviour is characterized by zero interactionbetween the particles. But an interaction of the particles with the volume wallsprovides thermodynamic equilibrium of the ideal gas.

Let us suppose that absorptive and emissive capacities of the walls are pro-vided by soliton-particles having the following constant absolute value of angularmomentum

J𝑒 =~2, (6.1)

where ~ is Planck constant.We suppose also that each lightlike soliton can interact simultaneously with

only one soliton-particle of the wall. We assume angular momentum conserva-tion for the combination of lightlike soliton with soliton-particle in the wall inabsorption or emission event.

23

𝑚 = 1

𝜃 = 10

Φ = 0

𝑥3

𝑥1

𝑥2

Figure 5.11: Zero level surfaces of the field function Φ for the Gaussian twistedsoliton with 𝑚 = 1.

Then, because of the angular momentum conservation, absorption or emis-sion of twisted lightlike soliton is possible only when the angular momentum ofsoliton-particle in the wall is oppositely directed to the angular momentum oflightlike soliton. The soliton-particle angular momentum is reversed in absorp-tion or emission event.

Thus the absolute value of angular momentum of twisted lightlike solitonsin the volume 𝑉 must be equal to ~.

The structure of twisted lightlike solitons depends on structure and statesof emissive and absorbent soliton-particles. We must define the value of twistparameter 𝑚 and the scale phase function 𝜌(𝜃) for the twisted lightlike solitonsin the volume 𝑉 .

Let us consider the case𝑚 = 1 . (6.2a)

As we see in (5.13b), in this case the energy of the soliton is logarithmicallydivergent in infinite space. But here we consider the finite volume, where itsenergy is finite.

Strictly speaking, the obtained soliton solutions must be modified for finitevolume. But here we consider the integral characteristics of the solitons only.Thus we can consider the soliton solutions of infinite space for the finite volumein some approximation.

Let us suppose also that the scale phase function 𝜌(𝜃) is slow variable:

𝜌′ → 0 . (6.2b)

Thus, taking into account (5.11) – (5.14) and (6.2), we have the following

24

𝑚 = 2

𝜃 = 10

Φ = 0

𝑥3 𝑥1

𝑥2

Figure 5.12: Zero level surfaces of the field function Φ for the Gaussian twistedsoliton with 𝑚 = 2.

relations for the twisted lightlike solitons in the volume 𝑉 :

E = P + E , (6.3a)

P = 𝜔 ~ , (6.3b)

J = ~ , (6.3c)

where

E =

∫𝑉

𝑓ℰ0 d𝑉 , (6.3d)

𝑓ℰ0 is static part of energy density ℰ for lightlike soliton in expression (4.29a).

As we see in (4.29a), the static part of energy E is independent of the fre-quency 𝜔 of the soliton. Disregarding the dependence of E from peculiarity ofthe volume 𝑉 , we take

E = const . (6.4)

The finiteness of the volume under consideration confines the set of possiblefrequencies of the solitons. As it is known, the field in any finite volume can berepresented by the appropriate mode expansion. In the case of cuboid we havethe simple space-time Fourier components, which satisfies the periodic boundaryconditions.

In the case of arbitrary volume with cavities, the finding of volume modeslooks very complicated. Here we consider that the cavities inside the solitonshells are sufficiently small to neglect of their influence. Also we take that eachsoliton in the volume has one of its allowed frequencies.

Hereafter up to formulas (6.16) we obtain the equilibrium distribution func-tion by soliton frequencies. The appropriate derivation of formulas is similarto ones represented in classical works by S. Bose [10], A. Einstein [11, 12], andcontained in monographs (see, for example, [13]).

25

As distinct from cited works, here we use the natural energy cells insteadof finite phase space cells. Complete deduction is expounded to show that allassumptions are in the framework of real soliton dynamics only.

For simplicity let us consider the volume 𝑉 in cubic form with side 𝑙𝑣. Thenthe allowed frequencies are defined by formula

𝜔𝑖 =2𝜋

𝜆𝑖=

2𝜋

𝑙𝑣𝑛𝑖 =

2𝜋

𝑙𝑣

√𝑛21 + 𝑛2

2 + 𝑛23 , (6.5)

where {𝑛1, 𝑛2, 𝑛3} are integer numbers, excepting the case when all number arezero, 𝑖 is the index for different frequencies.

According to (6.5) we have the following minimal frequency in the volume

𝜔min =2𝜋

𝑙𝑣. (6.6)

If there are 𝑁𝑖 solitons with frequency 𝜔𝑖 in the volume 𝑉 , then the fullenergy of solitons in it is given by formula

𝑈 =

∞∑𝑖=1

𝑁𝑖 E𝑖 , (6.7a)

where E𝑖 is energy of the soliton with frequency 𝜔𝑖,

E𝑖 = 𝜔𝑖 ~ + E , (6.7b)

𝑁 =

∞∑𝑖=1

𝑁𝑖 , (6.7c)

𝑁 is a total number of solitons in the volume 𝑉 .Because there is the minimal frequency 𝜔min (6.6) for the solitons in the

volume 𝑉 , then according to (6.7) we have the following expression for theirmaximal quantity:

𝑁max =𝑈

𝜔min ~ + E. (6.8)

If we suppose that a full angular momentum as well as a full momentum ofthe soliton gas in the volume 𝑉 are zero, then the total number of solitons mustbe even. Thus their minimal quantity is 2 and we have from (6.7) the maximalvalue for frequency:

𝑁min = 2 , 𝜔max =𝑈 − 2E

2 ~. (6.9)

Among all the possible distributions by soliton frequencies {𝑁𝑖} there is apiece providing an identical total energy 𝑈 . According to general principles ofstatistical physics such distributions are considered as equally probable.

Let us introduce the size of energy cell 𝐸𝑖, which are the quantity of solitonshaving the energy E𝑖 and the corresponding frequency 𝜔𝑖. Different states in

26

the sell are defined with the set of numbers {𝑛1, 𝑛2, 𝑛3} in (6.5) for frequency𝜔𝑖 and two directions of twist (right and left).

Let us count up the number of ways to provide the part of total energy𝑈 produced by the solitons with energy E𝑖 that is 𝑁𝑖 E𝑖 (6.7a). According toknown representation we line up 𝑁𝑖 solitons (∘) and (𝐸𝑖 − 1) dividing walls (|)in random order:

∘ ∘ | ∘ ∘ ∘ | | ∘ | ∘ ∘ ∘ ∘| ∘ ∘| · · · ∘ ∘ · · · | ∘ ∘ ∘ | | ∘ . (6.10a)

Here the dividing walls (|) separate the different soliton states ({𝑛1, 𝑛2, 𝑛3} andtwist direction).

In that case the permutation number (𝑁𝑖 + 𝐸𝑖 − 1)! is a total number ofdistributions for solitons with energy E𝑖. Then we take into account that 𝑁𝑖!permutations of solitons and (𝐸𝑖−1)! permutations of dividing walls correspondto one state. As result we have the sought number of ways to provide the part𝑁𝑖 E𝑖 of total energy 𝑈 :

𝑊𝑖 =(𝑁𝑖 + 𝐸𝑖 − 1)!

𝑁𝑖! (𝐸𝑖 − 1)!. (6.10b)

We obtain the total number of ways providing the energy 𝑈 by multiplicationof the numbers 𝑊𝑖:

𝑊 =

∞∏𝑖=1

𝑊𝑖 =

∞∏𝑖=1

(𝑁𝑖 + 𝐸𝑖 − 1)!

𝑁𝑖! (𝐸𝑖 − 1)!. (6.11)

According to usual method, we take that the most probable distributionprovided with maximum number of the ways 𝑊 corresponds to equilibrium.The total number of solitons 𝑁 is not fixed here.

Let us solve the problem for maximization of number 𝑊 with fixed totalenergy 𝑈 . (6.7a). For this purpose the method of Lagrange multipliers is used.For convenience we maximize the natural logarithm of number 𝑊 . Thus theproblem for finding of the equilibrium distribution {𝑁𝑖} take the form:

𝑆 = ln𝑊 − 𝒯 −1 𝑈 , 𝑆 → max , (6.12)

where 𝒯 −1 is Lagrange multiplier, the parameter 𝒯 has a physical dimension ofenergy.

Let us consider the case when the numbers 𝑁𝑖 and 𝐸𝑖 are sufficiently great.In this case we use the Stirling formula for factorial of number. Thus for 𝑁𝑖 ≫ 1and 𝐸𝑖 ≫ 1 we have

ln𝑊 ≈∞∑𝑖=1

((𝑁𝑖 + 𝐸𝑖) ln (𝑁𝑖 + 𝐸𝑖) −𝑁𝑖 ln𝑁𝑖 − 𝐸𝑖 ln𝐸𝑖

). (6.13)

Considering the sequence of numbers 𝑁𝑖 as quasicontinuous, we have thefollowing necessary conditions for maximum of the function 𝑆:

𝜕𝑆

𝜕𝑁𝑖= 0 . (6.14)

27

From (6.14) with (6.12), (6.13), and (6.7a) we have the following equilibriumdistribution:

𝑁𝑖 =𝐸𝑖

eE𝑖/𝒯 − 1. (6.15)

Here the constant 𝒯 can be expressed through the total energy 𝑈 by using thecondition (6.7a). Thus the physical quantity 𝒯 is an energy parameter of thedistribution (6.15).

Let us use the representation of quasicontinuous soliton energy spectrum toobtain the size of energy cell 𝐸𝑖. In this case the energy cell 𝐸𝑖 is characterizedby energy gap from E𝑖 to E𝑖 + ∆E𝑖.

Having in view one-to-one correspondence between the number 𝑛 in (6.5),frequency, and energy (6.3), we can obtain the quantity of different soliton stateswith frequencies from 𝜔𝑖 to 𝜔𝑖 + ∆𝜔𝑖. A spherical layer in the space of numbers{𝑛1, 𝑛2, 𝑛3} corresponds to the frequency interval ∆𝜔𝑖. Taking into account alsothe two directions of twist and proceeding to the limit ∆𝜔𝑖 → 0, we obtain

𝐸𝜔,Δ𝜔 ≈ 2 · 4𝜋 𝑛2 ∆𝑛 ≈ 𝑙3𝑣𝜋2

𝜔2 ∆𝜔 → 𝐸𝜔 d𝜔 =𝑙3𝑣𝜋2

𝜔2 d𝜔 . (6.16a)

𝑁𝜔 =𝑙3𝑣𝜋2

𝜔2

eE𝜔/𝒯 − 1, (6.16b)

whereE𝜔 = ~𝜔 + E . (6.16c)

Then we integrate the expressions E𝜔 𝑁𝜔 and 𝑁𝜔 with substitution 𝑁𝜔 andE𝜔 from (6.16) over frequency from 𝜔 = 0 to infinity. As result we obtain thefollowing expressions for total energy and number of solitons in the volume 𝑉 :

𝑈 =𝑙3𝑣 𝒯 4 6 Li4

(e−E/𝒯 )

𝜋2 ~3+ 𝑁 E , (6.17a)

𝑁 =𝑙3𝑣 𝒯 3 2 Li3

(e−E/𝒯 )

𝜋2 ~3, (6.17b)

where Li𝑠(𝑧) is polylogarithm function.For connection between energy parameter 𝒯 of distribution {𝑁𝑖} (6.15) and

absolute temperature ∘𝑇 we take

𝒯 = 𝑘𝐵 ∘𝑇 , (6.18)

where 𝑘𝐵 is Boltzmann constant.The relation (6.18) can be validated by means of comparison between sta-

tistical determination for entropy 𝑆 and its thermodynamic one for the case ofconstant volume (𝑉 = const):

𝑆 = 𝑘𝐵 ln𝑊 , (6.19a)

d𝑆 =d𝑈∘𝑇

. (6.19b)

28

But because the equivalence of these determinations must be postulated, it isreasonable here to postulate the relation (6.18).

Let us write the equilibrium energy spectral density for the twisted lightlikesolitons in the volume 𝑉 . According to (6.16) and (6.18) we have

𝑢(𝜔, ∘𝑇 ) +E𝜔 𝑁𝜔

𝑉=

𝜔2

𝜋2

~𝜔 + Eexp ~𝜔+E

𝑘𝐵 ∘𝑇 − 1. (6.20)

For the cases of negligible static soliton energy E → 0, we have from (6.20)the following known Planck formula for photons:

𝑢(𝜔, ∘𝑇 ) =𝜔2

𝜋2

~𝜔exp ~𝜔

𝑘𝐵 ∘𝑇 − 1. (6.21)

Thus we can consider the relation between twisted lightlike solitons andphotons.

Now let us estimate the possible values of soliton parameters in the volume𝑉 using certain suppositions.

Taking into account (6.2b), we put 𝜌′ = 0 and without loss of generality𝜌 = 1.

Let the longitudinal size of the soliton in (5.12) and the external diameterof cylindrical integration domain in (5.13) be equal to the side of the consideredcubic volume:

𝑙𝑠 = 2 𝜌∞ = 𝑙𝑣 . (6.22)

Then, taking into account (6.2a), we have the following values contained in(5.12) – (5.14):

ℐ0 = ℐ1 = 𝜔 𝑙𝑣 , ℐ2 = 0 , (6.23a)

𝒞0 =1

3+

32−𝜌6

3 𝑙6𝑣− 2−𝜌2

3 𝑙2𝑣, 𝒞2 = 1 ,

𝒞1 = ln

(𝑙𝑣2−𝜌

)+

13

72−

−𝜌4

6 𝑙6𝑣−

−𝜌8

72 𝑙8𝑣. (6.23b)

Condition (6.3c) with expression (5.15c) gives relation

−𝜌4 𝜔 𝑙𝑣 𝒞1 = 𝜒2 ~ . (6.24)

Thus, by virtue of fixedness of the angular momentum of the soliton (6.3c),the radius of its shell −𝜌 depends on frequency 𝜔. But to calculate −𝜌 we musthave the value of the constant 𝜒.

Nevertheless, to make a very rough estimate, we assume that for a visiblelight frequency 𝜔 the shell radius −𝜌 has an order of values in the range from theelectron classical radius to half of soliton wave-length.

Let

𝜔 = 𝑘 ∼ 107 m−1 , (6.25a)

−𝜌 ∼ 3 ·(10−15 ÷ 10−7

)m , 𝑙𝑣 ∼ 0.1 m . (6.25b)

29

Expressions (6.23b) with (6.25b) give

𝒞0 ≈ 1

3, 𝒞1 ∼ (12 ÷ 31) , 𝒞2 = 1 . (6.25c)

Standard value of Planck constant must be multiply by the velocity of lightfor used unit of frequency (6.25a):

~ ≈ 2 · 10−7 eV · m . (6.26)

Then relation (6.24) with (6.25) and (6.26) give

𝜒 ∼(1 · 10−22 ÷ 7 · 10−7

)m3/2 · eV−1/2 , (6.27a)

𝜒−2 ∼(2 · 1012 ÷ 1 · 1044

) eV

m3∼(3 · 10−7 ÷ 2 · 1025

) J

m3, (6.27b)

where minimal value of 𝜒 in (6.27a) and maximal value of 𝜒−2 in (6.27b) cor-respond to minimal value of −𝜌 in (6.25b).

According to formula (5.11b) and taking into account (6.23), (6.25), and(6.27), we have the following values for the static part of soliton energy:

E ≈−𝜌2 𝑙𝑣3𝜒2

∼(6 · 10−3 ÷ 2 · 1013

)eV , (6.28)

where minimal value of E corresponds to minimal value of 𝜒−2 in (6.27b) andmaximal value of −𝜌 in (6.25b).

Thus to provide the condition E ≪ 𝜔 ~, the diameter of soliton shell 2−𝜌 mustbe closer to the soliton wavelength than to the electron classical diameter.

Expressing −𝜌 from (6.24) and substituting it to formula for E in (6.28), weobtain from (6.3) the following formula for soliton energy:

E ≈ ~𝜔 +1

3𝜒

√~ 𝑙𝑣𝜔 𝒞1

. (6.29)

This dependence is shown on Fig. 6.1 for the explicit values of parameters. Ofcourse, it can be considered only for a qualitative analysis.

As we see on Fig. 6.1, the distinction of soliton energy function from thelinear one ~𝜔 (dashed line) can be noticeable in low-frequency region.

The question arises as to whether there is a static part of energy for realphotons. The appropriate experimental check may be possible with the helpof extrinsic photoeffect. If the photon energy not exactly equals to ~𝜔, thenthe frequency dependence of photoelectron energy may have a weak nonlinearitynear photoemission threshold. The substances with low photoemission thresholdis preferable for such experiments.

Let us next consider all values of twist parameter 𝑚 for lightlike solitons.For 𝑚 = 1 we have the known expression for photon energy in the case ~𝜔 ≫ E.

Thus for 𝑚 > 2 here we could be considered a fractional photon with thefollowing energy expression, according to (5.16):

E =~𝜔𝑚

+ E . (6.30)

30

𝜒 = 7 · 10−7 m3/2 · eV−1/2

~ = 2 · 10−7 eV · m

𝑙 = 0.1 m

𝒞1 = 12

E, eV

𝜔, m−1

Figure 6.1: Dependence of soliton energy from soliton frequency.

But we must pay attention once again to the fact that the twisted lightlikesolitons with 𝑚 > 2 have qualitative distinction from ones with 𝑚 = 1 in thepart of energy representation.

The energy of longitudinal limited and twisted lightlike soliton with 𝑚 = 1logarithmically diverges in infinite space, but for 𝑚 > 2 its energy is finite. Inthis point of view the solitons with 𝑚 = 1 more closely resemble the plane waveswith constant amplitude, the energy of which also diverges in infinite space.

Let us consider the representation of polarization property of light by twistedlightlike solitons.

A beam of these solitons with right or left twist has a necessary symmetry ofright or left circularly polarized light wave accordingly. This beam, in particular,provides the Sadovskii effect [14], which is a mechanical angular momentumtransfer to absorbent by circularly polarized electromagnetic wave. This effecthas the experimental verification [15, 16], including one for electromagneticcentimeter waves [17].

As it is known, the plane circularly polarized electromagnetic wave withconstant amplitude does not have angular momentum. Thus this wave does notprovide the Sadovskii effect. But the twisted lightlike solitons as well as photonshave angular momentum and provide this effect.

Elliptical polarization and, as limiting case, linear one of the soliton beamcould be provided by a coherent combining of solitons twisted to the right andto the left.

This representation for elliptical polarization conforms to one for the beam

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of photons, which have two helicity states only.Peculiarity of the value 𝑚 = 1 for twist parameter becomes apparent here.

According to solution symmetry for this case (see Fig. 5.5), the coherent com-bining of equal quantities of such right and left twisted solitons can give a beamhaving a crystal like symmetry with axes of the first order. This case can beinterpreted as a linear polarization.

But for the case of solitons with higher twist parameters we have for the sameconditions the appropriate crystal like symmetry with axes of 𝑚 > 2 order (seeFig. 5.6). This case can not be interpreted as a linear polarization.

Thus the lightlike solitons with twist parameter 𝑚 = 1 can be considered asusual photons in some approximation. But the solitons of higher twist 𝑚 > 2have qualitative differences from the solitons of the lowest twist 𝑚 = 1.

7 Conclusions

Thus we have considered the field model for extremal space-time film, which issometimes called Born – Infeld type scalar field model.

We have obtained new exact solution class for this model that is lightlikesolitons. We have considered an appropriate significant subclass that is twistedlightlike solitons. It is notable that its energy is proportional to its angularmomentum in high-frequency approximation.

The soliton under consideration has a singularity which is a moving two-dimensional surface or shell, where the model action density becomes zero. Thelightlike soliton can have a set of tubelike shells with the appropriate cavities.

A relatively simple twisted lightlike soliton with one cavity has consideredin details. This soliton is characterized, in particular, by a twist parameter 𝑚which is a natural number. The energy of this soliton in infinite space can befinite for 𝑚 > 2, but for 𝑚 = 1 its energy is logarithmically divergent. For thecase 𝑚 = 1 we have the asymptotic relation between soliton energy, momentum,and angular momentum, which is characteristic for photon.

Then we have investigated relations of the twisted lightlike solitons with𝑚 = 1 to photons. The model of ideal gas of the twisted lightlike solitons in abounded volume has considered for this purpose. Planck formula for the solitonenergy spectral density in the volume has obtained with explicit assumptions insome approximation.

An experimental check for a conformity of exact formula for the twistedsoliton energy with known formula for photon is proposed.

A beam of twisted lightlike solitons have considered. We have shown that thisbeam provides the effect of mechanical angular momentum transfer to absorbentby circularly polarized beam. This effect well known for photon beam.

It has been found that a twisted lightlike soliton beam with 𝑚 = 1 canprovide polarization as well as photon beam.

Thus we have a correspondence between photon and lightlike twisted solitonwith the minimal value of twist parameter.

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