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 [2–4] 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].

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

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

In the one-dimensional case, (3) is the same as (2) if <sup>E</sup> <sup>¼</sup> <sup>∂</sup><sup>f</sup> <sup>=</sup>∂x; Us ¼ �<sup>σ</sup> and if

In Eq. (3), the function x<sup>3</sup> ¼ f xð Þ <sup>1</sup>; x2; t determines the flame front which represents the interface between a combustible matter (x<sup>3</sup> > 0) and the combustion products (x<sup>3</sup> < 0); Us and γ<sup>0</sup> are constant positive quantities which characterize the front velocity and the combustion intensity, respectively. For γ<sup>0</sup> ¼ 0 Eq. (3) coincides with the Hamilton-Jacobi equation for a free nonrelativistic particle. In the two-dimensional case (more exactly, in its modification with account for the external friction with the coefficient μ when μ ¼ �γ0), the exact solution of the n-

(for the inertial motion of compressible medium with velocity ui) gives also the

The common solution of 1D, 2D, and 3D equations (4) in Euler variables is first time obtained in [22–26]. 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 [23–26]. The existence and smoothness of this solution for all time may take place only for super threshold friction μ > μth ¼ 1=t<sup>0</sup> (here t<sup>0</sup> is the minimal finite time of singularity realization for solution of the Hopf equation (4)) or for any finite volume viscosity [22–26]. This gives the possibility to obtain also exact solutions in nonlinear optic when equations of Kuramoto-Sivashinsky type are used for EMF wave propagation in

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

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

¼ γ<sup>0</sup> f (3)

¼ �μui; i, k ¼ 1, ::, n (4)

!¼ �Usr ! f.

the simplified version of the Sivashinsky equation ∇ [21]:

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

we replace (for the case <sup>γ</sup><sup>0</sup> <sup>&</sup>lt; 0) <sup>σ</sup>0∂<sup>3</sup>E=∂x<sup>3</sup> ! �γ0E.

dimensional Hopf equation modification with μ 6¼ 0

∂ui ∂t þ uk

nonlinear medium.

55

∂ui ∂xk

exact solution of Eq. (3) when the velocity of compressible medium u

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

must be identical to the VCR field, observed in the experiment [28].

∂f ∂t � 1 2 Us r ! f <sup>2</sup>

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 [8–11].

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 [13–15]:

$$V\_0 > V\_{th} = c/n\_\*; n\_\* = n + \sqrt{n^2 - 1}, n > 1; n\_\* = \left(1 + \sqrt{1 - n^2}\right)/n, n < 1 \tag{1}$$

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 condition (1).

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 electric wave E is propagating along axis x):

$$\frac{\partial E}{\partial t} + \sigma E \frac{\partial E}{\partial \mathbf{x}} + \sigma\_0 \frac{\partial^3 E}{\partial \mathbf{x}^3} = \mathbf{0} \tag{2}$$

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory… DOI: http://dx.doi.org/10.5772/intechopen.80813

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]:

$$\frac{\partial f}{\partial t} - \frac{1}{2} U\_s \left( \vec{\nabla} f \right)^2 = \gamma\_0 f \tag{3}$$

In the one-dimensional case, (3) is the same as (2) if <sup>E</sup> <sup>¼</sup> <sup>∂</sup><sup>f</sup> <sup>=</sup>∂x; Us ¼ �<sup>σ</sup> and if we replace (for the case γ<sup>0</sup> < 0) σ0∂<sup>3</sup> <sup>E</sup>=∂x<sup>3</sup> ! �γ0E.

In Eq. (3), the function x<sup>3</sup> ¼ f xð Þ <sup>1</sup>; x2; t determines the flame front which represents the interface between a combustible matter (x<sup>3</sup> > 0) and the combustion products (x<sup>3</sup> < 0); Us and γ<sup>0</sup> are constant positive quantities which characterize the front velocity and the combustion intensity, respectively. For γ<sup>0</sup> ¼ 0 Eq. (3) coincides with the Hamilton-Jacobi equation for a free nonrelativistic particle. In the two-dimensional case (more exactly, in its modification with account for the external friction with the coefficient μ when μ ¼ �γ0), the exact solution of the ndimensional Hopf equation modification with μ 6¼ 0

$$\frac{\partial u\_i}{\partial t} + u\_k \frac{\partial u\_i}{\partial \mathbf{x}\_k} = -\mu u\_i; i, k = 1, \dots, n \tag{4}$$

(for the inertial motion of compressible medium with velocity ui) gives also the

exact solution of Eq. (3) when the velocity of compressible medium u !¼ �Usr ! f.

The common solution of 1D, 2D, and 3D equations (4) in Euler variables is first time obtained in [22–26]. 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 [23–26]. The existence and smoothness of this solution for all time may take place only for super threshold friction μ > μth ¼ 1=t<sup>0</sup> (here t<sup>0</sup> is the minimal finite time of singularity realization for solution of the Hopf equation (4)) or for any finite volume viscosity [22–26]. This gives the possibility to obtain also exact solutions in nonlinear optic when equations of Kuramoto-Sivashinsky type are used for EMF wave propagation in nonlinear medium.
