## Abstract

The new Vavilov-Cherenkov radiation theory which is based on the relativistic generalization of the Landau theory for superfluid threshold velocity and Abraham theory of the electromagnetic field (EMF) in medium is represented. The new exact solution of the Cauchy problem in unbounded space is obtained for the n-dimensional Euler-Helmholtz (EH) equation in the case of a nonzero-divergence velocity field for an ideal compressible medium. The solution obtained describes the inertial vortex motion and coincides with the exact solution to the n-dimensional Hopf equation which simulates turbulence without pressure. Due to the introduction of a fairly large external friction or by introducing an arbitrary small effective volume viscosity, a new analytic solution of the Cauchy problem for the three-dimensional Navier-Stokes (NS) equation is obtained for compressible flows. This gives the positive solution to the Clay problem (www.clamath.org) generalization on the compressible NS equation. This solution also gives the possibility to obtain a new class of regular solutions to the n-dimensional modification of the Kuramoto-Sivashinsky equation, which is ordinarily used for the description of the nonlinear propagation of fronts in active media. The example for potential application of the new exact solution to the Hopf equation is considered in the connection of nonlinear geometrical optics with weak nonlinear medium at the nonlocality of the small action radii.

### Keywords

- hydrodynamics
- compressibility
- viscosity
- turbulence
- vorticity
- EMF waves
- Abraham theory
- photon in medium

## 1. Introduction

The main subject of the nonlinear optic theory is a nonlinear activity of a medium where electromagnetic field (EMF) is propagated.

In this connection, the analogy between electromagnetic and hydrodynamic phenomena, which was noted yet by Helmholtz and Maxwell [1], is considered. In more recent papers, also different types of this analogy are used [2, 3, 4] and give possibility to open new ways for the solution of some nonlinear hydrodynamic problems on the basis of this analogy.

However up to now, there are only a few examples of the direct mathematical correspondence between hydrodynamics and EMF theory, which gives resolution of the EMF problems on the basis of hydrodynamics [5, 6].

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 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 for the vortices in quantum superfluid He-3A [8, 9, 10, 11].

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 all observations and experiment data.

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 field of nonlinear optics.

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 velocity [13, 14, 15]:

In (1),

In contraposition to the classic VCR theory [16, 17, 18], the new VCR theory in [13, 14, 15] and (1) admits the conditions for effective and direct VCR realization even for high-frequency transverse waves of EMF in isotropic plasma when

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, 20, 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 electric wave E is propagating along axis x):

On the other side, the problem of the propagation of a flame front (generated by a self-sustained exothermal chemical reaction) may be considered on the basis of the simplified version of the Sivashinsky equation ∇ [21]:

In the one-dimensional case, (3) is the same as (2) if

In Eq. (3), the function

(for the inertial motion of compressible medium with velocity

The common solution of 1D, 2D, and 3D equations (4) in Euler variables is first time obtained in [22, 23, 24, 25, 26]. On the basis of this solution, we give the positive answer to the generalization of the Clay problem [27] on the case of compressible medium motion with nonzero divergence of velocity field [23, 24, 25, 26]. The existence and smoothness of this solution for all time may take place only for super threshold friction

### 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

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 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:

In (5)

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

For (5), the real nonzero photon rest mass

In the new VCR quantum theory [13, 14, 15], the energy

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

where

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

In the nonrelativistic limit when

Thus for the possibility of arising VCR photon with positive energy

where the value

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

In the classic VCR theory in (1) and (8), the value

Let us introduce the values

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

Thus, the classic VCR theory gives good correspondence with experiment [28] only in the determination of angle

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, 33, 34, 35, 36].

### 1.2 Exact solution of hydrodynamic equations

Fundamental turbulence problem was unsolved during many years by virtue of the absence of analytical, time-dependent, smooth-at-all-time solutions of the nonlinear hydrodynamic equations. A few exact solutions are known in hydrodynamics, but none of these solutions is time-dependent and defined in unbounded space or in space with periodic boundary conditions [38, 39, 40].

The importance of this problem is determined by stability and predictability problems in all fields of science where solutions and methods of hydrodynamics are used. In this connection in 2000, the problem of the existence of smooth time-dependent hydrodynamic solutions was stated as one of the seven Millennium Prize Problems (MPPs) by the Clay Institute of Mathematics [27]. MPPs relate only to incompressible flows “since it is well known that the behavior of compressible flows is abominable” [41].

Here we show that even for a compressible case, it is possible to obtain exact analytical, time-dependent, smooth-at-all-time solutions of Hopf equation (4) (which gives also new class solution also for vortex typ. 2D and 3D Euler equation) when any viscosity of super threshold friction is taken into account [22, 23, 24, 25, 26].

With the aim to introduce effective volume viscosity (in addition to external friction in (4)), let us consider the n-dimensional Hopf equation (4) in the moving with velocity

In (9)

Thus, the initial equation (4) (for the case

As shown in Appendix, in the case of an arbitrary dimensionality of the space (*n* = 1, 2, 3, etc.), Eq. (10) has the following exact solution (see also [22, 23, 24, 25, 26]):

where

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 the two-dimensional Hopf equation (4) (or when *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 if we have a solution in Euler variables as in (11).

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

After averaging over the random field

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 the determinant of the matrix

If, on the other side, we neglect the viscosity forces when *n*th-order algebraic equation (and successive minimization of the expression obtained, which depends on the spatial coordinates, with respect to these coordinates):

where

In the two-dimensional case, the condition in the form of Eq. (13) exactly coincides with the collapse condition obtained in [42] in connection with the problem of propagation of a flame front investigated on the basis of the Kuramoto-Sivashinsky Eq. (3). In this case for exact coincidence, it is necessary to replace

In the one-dimensional case, when *n* = 1, from Eq. (13) we can obtain the minimum time of appearance of the singularity

This means that the singularity (collapse) of the smooth solution can never occur when the initial velocity field is nonzero only for negative values of the spatial coordinate

Similarly, we can also determine the vortex wave burst time *n* > 1. For (13) in the two-dimensional case (when the initial velocity field is divergence-free) for the initial stream function in the form

In the example considered, this minimum time of existence of the smooth solution is implemented for the spatial variables corresponding to points on the ellipse

In accordance with (13), the necessary condition of implementation of the singularity is the condition of existence of a real positive solution to a quadratic (when *n* = 2) or cubic (when *n* = 3) equation for the time variable *t*. For example, in the case of two-dimensional flow with the initial divergence-free velocity field

For the example considered above from (14), there follows the inequality *n* = 2 there exists a real positive solution to the quadratic equation in (13) for which the minimum collapse time

On the contrary, if the initial velocity field is defined in the form of a finite function which is nonzero only in the domain

The condition of existence of a real positive solution of Eq. (13) (e.g., see (14)) is the necessary and sufficient condition of implementation of the singularity (collapse) of the solution, as distinct from the sufficient but not necessary integral criterion which was proposed in [45] (see formula (38) in [45]) and has the form:

In fact, in accordance with this criterion proposed in [45], the collapse of the solution is not possible in the case of the initial divergence-free velocity field, i.e., when *n* = 2.

On the basis of the solution (11), using (13) and the Lagrangian variables

In this case the expression (16) exactly coincides with the formula (30) given in [45] for the Lagrangian time evolution of the matrix of the first derivatives of the velocity which must satisfy the three-dimensional Hopf equation (10) (when *n* = 1, in the Lagrangian representation from (11) and (13), we obtain a particular case of the formula (16):

where

The solution (17) also coincides with the formula (14) in [45] and describes the catastrophic process of collapse of a simple wave in a finite time

Let us take into account only the external friction. For this purpose it is necessary to consider the case with *t* by the variable

for given initial conditions, the quantity

Providing (18), the solution to the n-dimensional EH equation is smooth on an unbounded interval of time *t*. The corresponding analytic vortical solution to the three-dimensional Navier–Stokes equation also remains smooth for any

Note that under the formal coincidence of the parameters *n* = 2 and in accordance with the solution of the Kuramoto-Sivashinsky equation in [42] and the regularization of this solution for all times if (18) takes place.

Moreover the example of interesting prosperity for the direct application for solution (11) (see also (12)–(18)) may be done in the connection of the results [46], where the description of light propagation in a nonlinear medium on the basis of the Burgers-Hopf equation is done.

Indeed, in [46], the model of light propagation in weak nonlinear 3D Coul-Coul’s medium with small action radii of nonlocality is represented. In [46], it was stated that in the geometric optic approach, this model is integrated and described by the Veselov-Novikov equation which has a 1D reduction in the form of the Burgers-Hopf equation. The last equation is considered in connection with nonlinear geometrical optics when 1D reduction is made for the case when the refractive index has no dependence on one of the space coordinates. It is important when the property of nonlinear wave finite-time breakdown for Burgers-Hopf solutions is considered in the application to the case of nonlinear geometrical optics. These solutions are useful for modeling of dielectrics which have impurities which induced sharp variations of the refractive index. Indeed, in the points of breakdown, the curvature of the light rays obtained discontinues property as it takes place at the boundary between different media [46].

In [46], the only hodograph method is used for the Burgers-Hopf (or Hopf equation which is obtained from the Burgers’ equation in the limit of zero viscosity) equation solution in this connection. Thus the direct analytical description of the 1D–3D solutions to the Hopf equation in the form (11) gives the new possibility also for the nonlinear optic problem which is considered in [46]. For example, according to this solution, it is possible to obtain the important effect of avoidance of finite-time singularities when viscosity or friction forces are taken into account (when condition (18) takes place for the case of external friction).

## 2. Conclusions

Here we represent some examples where hydrodynamic methods and solutions may be useful for different problems in nonlinear optics. In these examples, the medium itself has the first degree of importance in realization of all mentioned phenomena. Indeed, the main future of the Vavilov-Cherenkov radiation is that the medium is the source of this radiation instead of any kinds of bremsstrahlung radiations by moving charged particles. The VCR theory presented here for the first time takes into account the real mechanism of VCR by the medium itself, excited by a sufficiently fast electron. It can also be shown only from the microscopic theory, but not from the macroscopic one stated in [16]. The first step in this direction was made in [47] also on the basis of the Abraham theory where it is proposed that the Vavilov-Cherenkov radiation is emitted by the medium in a nonequilibrium polarization state which is arising due to the parametric resonance interaction of the medium with a fast-charged particle.

The second example, which is represented here, also gives new perspectives on the basis of the new exact solution (in the Euler variables) for n-dimensional Hopf equation because this equation is known as the possible model for weak nonlinear optic problems [46]. The importance of the new solution is connected with its Euler form in dependence from space variables, which are not represented in the solution of the Burgers-Hopf equation well known before (see [45] and others).

The Appendix presents a procedure for deriving the exact solution of the 3D Hopf equation.

The Hopf equation in the n-dimensional space (n = 1..3) is as follows:

When the external friction coefficient tends to zero in Eq. (4),

In the unbounded space, the general Cauchy problem solution for Eq. (19) under arbitrary smooth initial conditions

Eq. (19) may be represented in an implicit form as follows:

In (20),

In (21), the matrix

To infer (21), it is necessary to use the following delta-function property that is true for any smooth function

In (23), the values

To prove (23), it is necessary to use Taylor series decomposition wrt

Using variable substitution in the argument of the right-hand side of (25) (of the type

When in (23),

The solution of Eq. (26) is as follows:

This can be verified substituting (27) into (26) and taking into account that the general implicit solution of the equation (19) can be represented as

Let us use a known property of the delta function that for any smooth function

From (28) and (27), identical holding of the equality (21) follows.

Taking into account (21), from (20), we get an exact general (for any smooth initial velocity fields) solution of the Cauchy problem for Eq. (19) as

where

That is why, sign of

We can check that the very (29) under condition (30) exactly satisfies Eq. (19) by direct substitution of (29) in (19). The solution (29) describes not only potential but also vortex solutions of Eq. (19) in two- and three-dimensional cases for any smooth initial velocity field

The solution (29) of Eq. (19) allows getting an exact solution of Eq. (10) if in (29) to make a substitution:

The solution (29) also can be described as an exact solution of Eq. (4) for

## A.1 The direct validation of the solution

To verify the solution (29) satisfies Eq. (19), let us substitute (29) in Eq. (19). Then we get from (19):

where

To transform sub-integral expression in (32), the following identities shall be used:

The identity (33) is obtained from the relationship (obtained by differentiating the delta function having argument as a given function of

The validity of the identities (34) and (35) is proved by the direct checking. In the one-dimensional case, when

Taking into account (33)–(35), from (32), we get

where the sub-integral expression in the second term of the left-hand side of (36) is as follows:

To transform (37), it is necessary to use the following identities:

In (39), as it is noted above,

The identity (38) is a consequence of the noted above property of the delta function (see discussion before the formula (28)).

To infer the identity (39), it is necessary to consider in the argument of the delta function a Taylor series decomposition of the function

After the application of the identity (39) to the expression (37), defining the form of the second term in (36), from (36), we get

Equality (40) holds identically due to the identical equality to zero of the expression in the brackets in the sub-integral expression in (40).

Thus, we have proved that (29) exactly satisfies the Hopf equation (19) for any smooth initial velocity fields on the finite-time interval under condition

## A.2 The validation of identities (34) and (35)

In the two-dimensional case, the elements of the inverse matrix

Here, (42) corresponds to the formula (13) for n = 2.

Using (41), it is possible to show that the following equality holds (in the left-hand side of (43), summation is assumed on the repeating indices from 1 to 2):

From (42), it follows that the right-hand side of (43) exactly matches

To prove the identity (35), let us introduce

Using (41), one gets from (44)

The identities (45) and (46) confirm the truth of the identity (35) in the two-dimensional case.

Similarly, the identity (35) is proved in the three-dimensional case. For that, we need the following representation of the entries of the inverse matrix

From (44), in the three-dimensional case, we get on the basis of (47) that all three components of the vector

For example, in the expression for *B* _{1} the sum of terms proportional to the first degree of time has the form

Proof of the identity (34) also is possible in the 3D case on the basis of (47) and (13) but is related to the cumbersome transformations.