1.1 New theory of the Vavilov-Cherenkov radiation (VCR)

The Vavilov-Cherenkov radiation (VCR) phenomenon has justly become an inherent part of modern physics. The VCR in a refractive medium was experimentally discovered by Cherenkov and Vavilov [28] more than half a century ago. This was also the time when Tamm and Frank [16, 17] developed the electromagnetic macroscopical theory of this phenomenon, which, as well as the VCR discovery, was marked later by a Nobel Prize. The Tamm-Frank theory appeared to be very similar to the Heaviside theory, which had been forgotten for a century [29].

The Heaviside-Tamm-Frank (HTF) theory demonstrated that the cylindrically symmetrical EMF, created in a medium by an electron, which moves rectilinearly with the constant velocity V0, does not exponentially reduce only in the case of the super threshold electron velocity V0≥c=n. According to the HTF theory, this field must be identical to the VCR field, observed in the experiment [28].

However, such direct identification is not in agreement with the basic microscopical conception that VCR photons are radiated by a medium and not by an electron itself [16, 30]. The latter can serve only for the initiation of such radiation

However up to now, there are only a few examples of the direct mathematical correspondence between hydrodynamics and EMF theory, which gives resolution

Thus in [5] there is an exact mathematical correspondence between the solutions for the point electric dipole potential and velocity potential obtaining for the rigid

Moreover, in [6], it was proved that the hydrodynamic equations do not allow the existence of a solution in the form of a single isolated vortex on sphere, but allow the exact solution in the form of two antipodal point vortices (which have the same value but different signs of circulation and located on the sphere on the maximal possible distance from each other). This result gives the first theoretical base for the proposition that DMM also cannot exist in the single form, but they must be included in the structure of point magnetic dipole, which is confirmed by

Here we consider some examples of the application of hydrodynamic methods for the problems of EMF interaction with medium which may be important in the

In Part 1 of the chapter, we give the example for demonstration of the new mechanism of the Vavilov-Cherenkov radiation (VCR), which is obtained only on the basis of relativistic generalization to the Landau theory of superfluid threshold velocity [12]. In analogy with the Landau criterion its relativistic

generalization is deduced for the determination of threshold conversion of medium Bose-condensed excitation into Cherenkov's photon. Thus, the VCR arises only due to the reaction of medium on the electric charge moving with super threshold

In (1), с is the light speed in vacuum and n is the medium refractive index. In contraposition to the classic VCR theory [16–18], the new VCR theory in [13–15] and (1) admits the conditions for effective and direct VCR realization even for high-frequency transverse waves of EMF in isotropic plasma when n < 1 in (1). This is possible in the new VCR theory only because it is based on the Abraham theory for EMF in a medium where photons have nonzero real mass of rest, which determines necessary (in energy balance equation) energy difference for the medium when the medium emits photon VCR only for the

In the second part of this chapter, we consider a new exact solution of nonlinear hydrodynamic equations. This gives corresponding possibility of its application to the problems of nonlinear EMF and other wave propagation in active and dissipative medium, where the Kuramoto-Sivashinsky equation [19–21] is used, giving the generalization of the Korteweg-de Vries (KdV) equation. Indeed, in nonlinear optic the KdF equation may describe the EMF wave propagation (for the case when

<sup>n</sup><sup>2</sup> � <sup>1</sup> <sup>p</sup> , n <sup>&</sup>gt; <sup>1</sup>; n<sup>∗</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 � n<sup>2</sup>

<sup>∂</sup>x<sup>3</sup> <sup>¼</sup> <sup>0</sup> (2)

� � <sup>p</sup> <sup>=</sup>n, n <sup>&</sup>lt; 1 (1)

of the EMF problems on the basis of hydrodynamics [5, 6].

Nonlinear Optics ‐ Novel Results in Theory and Applications

for the vortices in quantum superfluid He-3A [8–11].

all observations and experiment data.

<sup>V</sup><sup>0</sup> <sup>&</sup>gt; Vth <sup>¼</sup> <sup>c</sup>=n∗; n<sup>∗</sup> <sup>¼</sup> <sup>n</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

electric wave E is propagating along axis x):

∂E ∂t

<sup>þ</sup> <sup>σ</sup><sup>E</sup> <sup>∂</sup><sup>E</sup> ∂x þ σ<sup>0</sup> ∂3 E

field of nonlinear optics.

velocity [13–15]:

condition (1).

54

sphere moving with constant speed in the ideal incompressible fluid.

In [6] an exact correspondence is established between the mathematical description of the single vortex velocity on the sphere and the Dirac magnetic monopole (DMM) [7] vector potential. Similar analogy with DMM was noted also by the medium. The phenomenological quantum theory of the VCR, developed by Ginzburg [18] on the basis of the Minkowski EMF theory in medium, still does not take into consideration the changes of the radiating medium energy state, which might be necessary for the VCR realization. As we show the latter, this is so because, in contrast to the Abraham EMF theory, for the momentum of photon in the Minkowski EMF theory, the corresponding photon mass of rest in medium always has only exact imaginary (with zero real part) value and cannot be taken into account in the energy balance equation for the VCR.

Thus, the classic theory of the VCR phenomenon leaves a question of the energy mechanism of the VCR effect open. Indeed, to elaborate this mechanism, we need to find out the necessary possible changes of the energy state of the medium itself, which ensure the VCR effect realization.

The suggested theory is based on directly using the Abraham momentum of photon:

$$
\overrightarrow{p}\_A = \frac{\varepsilon\_{ph}}{cn} \overrightarrow{k} \text{ , } n > 1; \overrightarrow{p}\_A = \frac{\varepsilon\_{ph}n}{c} \overrightarrow{k} \text{ , } n < 1; \ \overrightarrow{k} = \frac{\overrightarrow{V}\_{ph}}{\left| \overrightarrow{V}\_{ph} \right|} \tag{5}
$$

For example, in the case n > 1 in (7), we have Vph ¼ c=n and in the right-hand side

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory…

<sup>c</sup> cos <sup>θ</sup> � ffiffiffiffiffiffiffiffi

<sup>2</sup> is the only kinetic energy of excitation (in [12] these are vorton

c V0n<sup>∗</sup>

Thus for the possibility of arising VCR photon with positive energy εph > 0, it is

where the value n∗ð Þ n > 1 for any cases of n > 1 or n < 1 as it shown in (1). From the condition cos j j θ ≤ 1 in (8), the value of threshold velocity in (1) is obtained. The conditions (8) and (1) give the necessary condition for arising VCR, and from (8) it is possible to obtain the maximal angle of the VCR cone of rays. The classic VCR theory gives good correspondence to experiment only in the determination of position for the maximum of intensity in the VCR cone of rays, but not to the maximal angle of this cone. In [13, 14] it is shown that the new VCR theory gives a better agreement with the experiment [28] than classical VCR theory when

According to [28] the VCR effect is observed in the whole region of angles

corresponds to the VCR induced by Ra. Thus, <sup>I</sup>ð Þ¼ <sup>θ</sup> 0 when <sup>θ</sup> <sup>&</sup>gt; <sup>θ</sup>A,B max. In the [31] the same result was also obtained for VCR realization through the direct use of high-

In the classic VCR theory in (1) and (8), the value n<sup>∗</sup> must be replaced with the

when (8) is used for evaluation of parameter <sup>β</sup> <sup>¼</sup> <sup>V</sup>0=<sup>c</sup> and the analogy values <sup>β</sup>A; <sup>β</sup><sup>B</sup>

For example, when the medium where the VCR arising is water (H2O), where

corresponding to the inequality β ¼ V0=c < 1 of the relativity theory because from the classic VCR theory, <sup>β</sup><sup>A</sup> <sup>¼</sup> <sup>1</sup>:1177; <sup>β</sup><sup>B</sup> <sup>¼</sup> <sup>1</sup>:0064 may be obtained. The same results obtained for all other media are considered in the experiment [28, 31] (see [13, 14]). Thus, the classic VCR theory gives good correspondence with experiment [28]

<sup>∗</sup> which correspond to θA,B max of experiment [28]

<sup>0</sup> , but not of the angle θA,B max. In this connection

max <sup>¼</sup> <sup>0</sup>:6691; cos <sup>θ</sup><sup>B</sup>

<sup>∗</sup> ¼ 0:6049 which are smaller than 1, as they need

<sup>0</sup> <sup>≤</sup> <sup>θ</sup> <sup>≤</sup> <sup>θ</sup>A,B max with the maximum of radiation intensity <sup>I</sup>ð Þ<sup>θ</sup> at the angle

<sup>∗</sup> ; β<sup>B</sup>

from the relativity theory. For the classic VCR theory, the result is not

the classic VCR theory tied only with interference maximum at <sup>θ</sup> <sup>¼</sup> <sup>θ</sup>A,B

not consider at all the energetic base for threshold arising of this coherent VCR. Actually, this is clearer for the case of plasma with n < 1, where the classic VCR theory total excludes the possibility of the VCR in the form of transverse

<sup>0</sup> < θA,B max. Here Index A corresponds to gamma rays of ThC″

The left-hand side of (7) is always negative (it is zero only for the case when the

In the nonrelativistic limit when V0≪c; Vph≪c from (7) for ε<sup>p</sup> > 0, the Landau

! V ! 0

necessary to have in the right-hand side of (7) the negative value of A < 0 or

cos θ >

<sup>n</sup>2�<sup>1</sup> <sup>p</sup> <sup>n</sup> .

� � <sup>&</sup>lt; <sup>0</sup>; <sup>ε</sup><sup>V</sup> <sup>¼</sup> <sup>ε</sup><sup>p</sup> <sup>1</sup> �

! <sup>0</sup> ¼ V ! 1).

> ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>V</sup><sup>2</sup> p c2

ffi <sup>ε</sup>pV<sup>2</sup> p <sup>2</sup>c<sup>2</sup> .

(8)

, and the Index B

max ¼ 0:7431 from

<sup>0</sup> and does

� � q

of (7) <sup>A</sup> <sup>¼</sup> <sup>1</sup> � <sup>V</sup>

Then <sup>ε</sup><sup>V</sup> <sup>¼</sup> Vpp

inequality:

<sup>θ</sup> <sup>¼</sup> <sup>θ</sup>A,B

energy electron beam.

value n for the case with n > 1. Let us introduce the values β<sup>A</sup>

<sup>n</sup> <sup>¼</sup> <sup>1</sup>:333, <sup>n</sup><sup>∗</sup> <sup>¼</sup> <sup>2</sup>:247, and for the values cos <sup>θ</sup><sup>A</sup>

<sup>∗</sup> <sup>¼</sup> <sup>0</sup>:6718; <sup>β</sup><sup>B</sup>

only in the determination of angle θA,B

for the classic VCR theory.

(8), we obtain β<sup>A</sup>

57

elementary excitations).

! <sup>0</sup>V ! ph � �

DOI: http://dx.doi.org/10.5772/intechopen.80813

criterion [12] may be obtained: ε<sup>V</sup> � p

<sup>c</sup><sup>2</sup> � mphc<sup>2</sup>

initial and finite velocity of the electron are the same V

describing the threshold edge of the VCR cone of rays.

<sup>ε</sup>ph <sup>¼</sup> <sup>1</sup> � <sup>V</sup><sup>0</sup>

In (5) εph is the photon energy and V ! ph its velocity in medium.

For the Minkowski EMF theory, the momentum of photon in medium with n > 1 has the form: p ! <sup>M</sup> <sup>¼</sup> <sup>ε</sup>phn <sup>c</sup> k !

For (5), the real nonzero photon rest mass mph is determined from the known relativistic equation m<sup>2</sup> phc<sup>2</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup> ph <sup>c</sup><sup>2</sup> � <sup>p</sup><sup>2</sup> <sup>A</sup>, and from (5), we have

$$m\_{ph} = \frac{\varepsilon\_{ph}}{c^2 n} \sqrt{n^2 - 1}, n > 1;\\ m\_{ph} = \frac{\varepsilon\_{ph}}{c^2} \sqrt{1 - n^2}, n < 1 \tag{6}$$

In the new VCR quantum theory [13–15], the energy <sup>Δ</sup>Em <sup>¼</sup> mphc<sup>2</sup> may correspond to the energy of a medium long-wave Bose excitation which can transform into the VCR photon only when the super threshold condition (1) takes place. Thus, the value ΔEm must be taken into account in the energy balance equation for VCR realization possibility (when medium must lose this energy when the VCR photon is arising from it), and this new VCR theory is provided in [13, 14]. In [15] we also give examples where it is easy to obtain experimental and observational evidence of the difference between Abraham's and Minkowski's EMF theories when the VCR may be observed during the electron beam transfer through the medium which is the light of intense laser or when high-energy cosmic rays go through the relict background radiation.

To obtain a relativistic generalization of the Landau criterion [12] for the VCR realization, it is necessary to use the energy balance equation for the VCR (including in it the value of medium energy loss <sup>Δ</sup>Em <sup>¼</sup> mphc2, where mph may be taken from (6)) in the coordinate system moving with the initial electron velocity V ! 0 [13, 14]:

$$m\_{\epsilon}c^{2}\left[\mathbf{1}-\Gamma\_{0}\Gamma\_{1}\left(\mathbf{1}-\frac{\left(\vec{V}\_{0}\vec{V}\_{1}\right)}{c^{2}}\right)\right]=\varepsilon\_{ph}\Gamma\_{0}\left[\mathbf{1}-\frac{\left(\vec{V}\_{0}\vec{V}\_{ph}\right)}{c^{2}}-\frac{m\_{ph}c^{2}}{\varepsilon\_{ph}}\right] \tag{7}$$

where V ! <sup>1</sup> is the velocity of electron after VCR photon arising. In (7) Γ<sup>α</sup> ¼ 1= ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>V</sup><sup>2</sup> α c2 q , where <sup>α</sup> <sup>¼</sup> 0 or <sup>α</sup> <sup>¼</sup> 1 and mphc<sup>2</sup>=εph <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>V</sup><sup>2</sup> ph c2 q according to (6). Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory… DOI: http://dx.doi.org/10.5772/intechopen.80813

For example, in the case n > 1 in (7), we have Vph ¼ c=n and in the right-hand side

$$\text{of (7)}\ A = \mathbf{1} - \frac{\left(\overrightarrow{V}\_0 \overrightarrow{V}\_{pl}\right)}{c^l} - \frac{m\_{plt}}{\varepsilon\_{pl}} = \mathbf{1} - \frac{V\_0}{c} \cos\theta - \frac{\sqrt{n^2 - 1}}{n}.$$

$$\text{Thus 1.4.6, bound yields } \hat{\mathbf{n}} \text{ for (7)}\ \hat{\mathbf{n}} \text{ to } \hat{\mathbf{n}} \text{ a.e.} \text{ (-1.1.1)}$$

by the medium. The phenomenological quantum theory of the VCR, developed by Ginzburg [18] on the basis of the Minkowski EMF theory in medium, still does not take into consideration the changes of the radiating medium energy state, which might be necessary for the VCR realization. As we show the latter, this is so because, in contrast to the Abraham EMF theory, for the momentum of photon in the Minkowski EMF theory, the corresponding photon mass of rest in medium always has only exact imaginary (with zero real part) value and cannot be taken into

Thus, the classic theory of the VCR phenomenon leaves a question of the energy mechanism of the VCR effect open. Indeed, to elaborate this mechanism, we need to find out the necessary possible changes of the energy state of the medium itself,

The suggested theory is based on directly using the Abraham momentum of

! <sup>¼</sup> <sup>ε</sup>phn c k !

For the Minkowski EMF theory, the momentum of photon in medium with n > 1

For (5), the real nonzero photon rest mass mph is determined from the known

In the new VCR quantum theory [13–15], the energy <sup>Δ</sup>Em <sup>¼</sup> mphc<sup>2</sup> may correspond to the energy of a medium long-wave Bose excitation which can transform into the VCR photon only when the super threshold condition (1) takes place. Thus, the value ΔEm must be taken into account in the energy balance equation for VCR realization possibility (when medium must lose this energy when the VCR photon is arising from it), and this new VCR theory is provided in [13, 14]. In [15] we also give examples where it is easy to obtain experimental and observational evidence of the difference between Abraham's and Minkowski's EMF theories when the VCR may be observed during the electron beam transfer through the medium which is the light of intense laser or when high-energy cosmic rays go through the relict

To obtain a relativistic generalization of the Landau criterion [12] for the VCR realization, it is necessary to use the energy balance equation for the VCR (including in it the value of medium energy loss <sup>Δ</sup>Em <sup>¼</sup> mphc2, where mph may be taken from (6)) in the coordinate system moving with the initial electron velocity V

5 ¼ εphΓ<sup>0</sup> 1 �

<sup>1</sup> is the velocity of electron after VCR photon arising. In (7)

2 4

V ! <sup>0</sup>V ! ph � �

<sup>c</sup><sup>2</sup> � mphc<sup>2</sup> εph

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>V</sup><sup>2</sup> ph c2

q

3

<sup>A</sup>, and from (5), we have

c2

ffiffiffiffiffiffiffiffiffiffiffiffiffi

!

<sup>n</sup><sup>2</sup> � <sup>1</sup> <sup>p</sup> , n <sup>&</sup>gt; <sup>1</sup>; mph <sup>¼</sup> <sup>ε</sup>ph

, n < 1; k ! <sup>¼</sup> <sup>V</sup> ! ph V ! ph � � �

ph its velocity in medium.

� � �

<sup>1</sup> � <sup>n</sup><sup>2</sup> <sup>p</sup> , n <sup>&</sup>lt; 1 (6)

(5)

! 0

5 (7)

according to (6).

, n > 1; pA

account in the energy balance equation for the VCR.

Nonlinear Optics ‐ Novel Results in Theory and Applications

which ensure the VCR effect realization.

p ! <sup>A</sup> <sup>¼</sup> <sup>ε</sup>ph cn k !

!

relativistic equation m<sup>2</sup>

background radiation.

[13, 14]:

Γ<sup>α</sup> ¼ 1=

56

mec

where V !

q

2 4

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>V</sup><sup>2</sup> α c2

<sup>2</sup> <sup>1</sup> � <sup>Γ</sup>0Γ<sup>1</sup> <sup>1</sup> �

0 @

V ! <sup>0</sup>V ! 1 � �

c2

1 A

, where <sup>α</sup> <sup>¼</sup> 0 or <sup>α</sup> <sup>¼</sup> 1 and mphc<sup>2</sup>=εph <sup>¼</sup>

3

In (5) εph is the photon energy and V

mph <sup>¼</sup> <sup>ε</sup>ph c<sup>2</sup>n

phc<sup>2</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup> ph <sup>c</sup><sup>2</sup> � <sup>p</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>M</sup> <sup>¼</sup> <sup>ε</sup>phn <sup>c</sup> k !

photon:

has the form: p

The left-hand side of (7) is always negative (it is zero only for the case when the initial and finite velocity of the electron are the same V ! <sup>0</sup> ¼ V ! 1).

In the nonrelativistic limit when V0≪c; Vph≪c from (7) for ε<sup>p</sup> > 0, the Landau criterion [12] may be obtained: ε<sup>V</sup> � p ! V ! 0 � � <sup>&</sup>lt; <sup>0</sup>; <sup>ε</sup><sup>V</sup> <sup>¼</sup> <sup>ε</sup><sup>p</sup> <sup>1</sup> � ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>V</sup><sup>2</sup> p c2 � � q ffi <sup>ε</sup>pV<sup>2</sup> p <sup>2</sup>c<sup>2</sup> .

Then <sup>ε</sup><sup>V</sup> <sup>¼</sup> Vpp <sup>2</sup> is the only kinetic energy of excitation (in [12] these are vorton elementary excitations).

Thus for the possibility of arising VCR photon with positive energy εph > 0, it is necessary to have in the right-hand side of (7) the negative value of A < 0 or inequality:

$$\cos \theta > \frac{c}{V\_0 n\_\*} \tag{8}$$

where the value n∗ð Þ n > 1 for any cases of n > 1 or n < 1 as it shown in (1). From the condition cos j j θ ≤ 1 in (8), the value of threshold velocity in (1) is obtained.

The conditions (8) and (1) give the necessary condition for arising VCR, and from (8) it is possible to obtain the maximal angle of the VCR cone of rays. The classic VCR theory gives good correspondence to experiment only in the determination of position for the maximum of intensity in the VCR cone of rays, but not to the maximal angle of this cone. In [13, 14] it is shown that the new VCR theory gives a better agreement with the experiment [28] than classical VCR theory when describing the threshold edge of the VCR cone of rays.

According to [28] the VCR effect is observed in the whole region of angles <sup>0</sup> <sup>≤</sup> <sup>θ</sup> <sup>≤</sup> <sup>θ</sup>A,B max with the maximum of radiation intensity <sup>I</sup>ð Þ<sup>θ</sup> at the angle <sup>θ</sup> <sup>¼</sup> <sup>θ</sup>A,B <sup>0</sup> < θA,B max. Here Index A corresponds to gamma rays of ThC″ , and the Index B corresponds to the VCR induced by Ra. Thus, <sup>I</sup>ð Þ¼ <sup>θ</sup> 0 when <sup>θ</sup> <sup>&</sup>gt; <sup>θ</sup>A,B max. In the [31] the same result was also obtained for VCR realization through the direct use of highenergy electron beam.

In the classic VCR theory in (1) and (8), the value n<sup>∗</sup> must be replaced with the value n for the case with n > 1.

Let us introduce the values β<sup>A</sup> <sup>∗</sup> ; β<sup>B</sup> <sup>∗</sup> which correspond to θA,B max of experiment [28] when (8) is used for evaluation of parameter <sup>β</sup> <sup>¼</sup> <sup>V</sup>0=<sup>c</sup> and the analogy values <sup>β</sup>A; <sup>β</sup><sup>B</sup> for the classic VCR theory.

For example, when the medium where the VCR arising is water (H2O), where <sup>n</sup> <sup>¼</sup> <sup>1</sup>:333, <sup>n</sup><sup>∗</sup> <sup>¼</sup> <sup>2</sup>:247, and for the values cos <sup>θ</sup><sup>A</sup> max <sup>¼</sup> <sup>0</sup>:6691; cos <sup>θ</sup><sup>B</sup> max ¼ 0:7431 from (8), we obtain β<sup>A</sup> <sup>∗</sup> <sup>¼</sup> <sup>0</sup>:6718; <sup>β</sup><sup>B</sup> <sup>∗</sup> ¼ 0:6049 which are smaller than 1, as they need from the relativity theory. For the classic VCR theory, the result is not corresponding to the inequality β ¼ V0=c < 1 of the relativity theory because from the classic VCR theory, <sup>β</sup><sup>A</sup> <sup>¼</sup> <sup>1</sup>:1177; <sup>β</sup><sup>B</sup> <sup>¼</sup> <sup>1</sup>:0064 may be obtained. The same results obtained for all other media are considered in the experiment [28, 31] (see [13, 14]).

Thus, the classic VCR theory gives good correspondence with experiment [28] only in the determination of angle θA,B <sup>0</sup> , but not of the angle θA,B max. In this connection the classic VCR theory tied only with interference maximum at <sup>θ</sup> <sup>¼</sup> <sup>θ</sup>A,B <sup>0</sup> and does not consider at all the energetic base for threshold arising of this coherent VCR. Actually, this is clearer for the case of plasma with n < 1, where the classic VCR theory total excludes the possibility of the VCR in the form of transverse

high-frequency EMF waves. The present new VCR theory gives this possibility due to the transformation of a longitudinal Bose-condensed plasmon into transverse VCR photon, during the scattering of a plasmon on the relativistic electron [14, 37]. satisfies Eq. (10) only at such times for which the determinant of the matrix A^ is

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory…

In the case of the potential initial velocity field, the solution (11) is potential for all successive instants of time, corresponding to a zero-vortex field. On the contrary, in the case of nonzero initial vortex field, the solution also determines the evolution of velocity with a nonzero vortex field. In [42] the potential solution to

in the Lagrangian representation which also exactly follows from (11) for n = 2. It is important to understand that here in (11) we have a solution in Euler variables, which is firstly obtained in [22] for n = 2 and n = 3. From the solution of (10) or (4) in Lagrangian variables, it is unreal to obtain a solution of (4) or (10) in Euler variables. From the other side, it is easy to obtain a solution in Lagrangian variables

, and the solution (11) coincides exactly with the solutions

� � � � with the use of the Dirac delta function (see

x ! � ξ ! �tu<sup>0</sup> ! ξ � � � �! <sup>2</sup>

2 6 4

As distinct from (11), the average solution (12) of Eq. (10) is already arbitrarily smooth on any unbounded time interval and not only providing the positiveness of

smooth solution (11) is defined, as was already noted, only under the condition det A^ > 0 [22–26] (see Appendix). This condition corresponds to a bounded time interval 0 ≤ t < t0, where the minimum limiting time t<sup>0</sup> of existence of the solution can be determined from the solution to the following nth-order algebraic equation (and successive minimization of the expression obtained, which depends on the

4νt

!

3 7

<sup>5</sup> (12)

ðÞ¼ t 0 in (11), the

<sup>3</sup> det <sup>U</sup>^ <sup>0</sup> <sup>¼</sup> <sup>0</sup>, n <sup>¼</sup> <sup>3</sup>

<sup>∂</sup>xm , and

(13)

For example, in the one-dimensional case (n = 1) in (11), we have

!; t

1 2 ffiffiffiffiffiffiffi πν<sup>t</sup> � � <sup>p</sup> <sup>n</sup> exp �

representation for the implicit solution of Eq. (10) in the form

density), from (11) we can obtain the exact solution in the form:

If, on the other side, we neglect the viscosity forces when B

ðÞ�<sup>t</sup> t u! <sup>x</sup>

det A^ � � � �

spatial coordinates, with respect to these coordinates):

¼ 0, n ¼ 1

<sup>2</sup> det <sup>U</sup>^ <sup>012</sup> <sup>¼</sup> <sup>0</sup>, n <sup>¼</sup> <sup>2</sup>

where det <sup>U</sup>^ <sup>0</sup> is the determinant of the 3 � 3 matrix <sup>U</sup>0nm <sup>¼</sup> <sup>∂</sup>u0<sup>n</sup>

dimensional case for the variables ð Þ <sup>x</sup>1; <sup>x</sup><sup>2</sup> . In this case det <sup>U</sup>^ <sup>013</sup>, det <sup>U</sup>^ <sup>023</sup> are the determinants of the matrices in the two-dimensional case for the variables ð Þ x1; x<sup>3</sup>

<sup>2</sup> det <sup>U</sup>^ <sup>012</sup> <sup>þ</sup> det <sup>U</sup>^ <sup>013</sup> <sup>þ</sup> det <sup>U</sup>^ <sup>023</sup> � � <sup>þ</sup> <sup>t</sup>

<sup>∂</sup>x<sup>1</sup> is the determinant of a similar matrix in the two-

du01ð Þ x<sup>1</sup> dx<sup>1</sup>

! þt

! <sup>þ</sup><sup>t</sup>

obtained in [43, 44]. The solution (11) can be obtained if we use the integral

After averaging over the random field Bið Þt (with the Gaussian probability

!

¼ 0 in (12)) was obtained only

positive for any values of the spatial coordinates det A^ > 0.

the two-dimensional Hopf equation (4) (or when B

DOI: http://dx.doi.org/10.5772/intechopen.80813

if we have a solution in Euler variables as in (11).

det <sup>A</sup>^ <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>t</sup> du<sup>01</sup>

Appendix or [22, 23]).

ð dn ξu0<sup>i</sup> ξ � �!

the determinant of the matrix A^ .

h i ui ¼

det A t ^ ðÞ¼ <sup>1</sup> <sup>þ</sup> <sup>t</sup>

det <sup>U</sup>^ <sup>012</sup> <sup>¼</sup> <sup>∂</sup>u<sup>01</sup>

59

det A t ^ ðÞ¼ <sup>1</sup> <sup>þ</sup> tdivu<sup>0</sup>

det A t ^ ðÞ¼ <sup>1</sup> <sup>þ</sup> tdivu<sup>0</sup>

∂x<sup>1</sup> ∂u<sup>02</sup> <sup>∂</sup>x<sup>2</sup> � <sup>∂</sup>u<sup>01</sup> ∂x<sup>2</sup> ∂u<sup>02</sup>

and ð Þ x2; x<sup>3</sup> , respectively.

uk x !; t � � <sup>¼</sup> <sup>u</sup>0<sup>k</sup> <sup>x</sup>

dξ<sup>1</sup>

! � <sup>B</sup> !

Moreover in this new VCR theory, the VCR phenomenon has the same nature as for numerous physical systems where dissipative instability is realized when corresponding excitations in a medium become energetically favorable at some super threshold conditions [12, 32–36].
