A. Exact solution of n-D Hopf equation (n = 1, 2, 3)

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:

$$\frac{\partial u\_i}{\partial t} + u\_l \frac{\partial u\_i}{\partial \mathbf{x}\_l} = \mathbf{0} \tag{19}$$

When the external friction coefficient tends to zero in Eq. (4), μ ! 0, Eq. (4) also coincides with the Hopf equation (19).

In the unbounded space, the general Cauchy problem solution for Eq. (19) under arbitrary smooth initial conditions u ! <sup>0</sup> x !� � may be obtained as follows (see also in [22, 23]):

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

$$u\_i(\overrightarrow{\boldsymbol{x}},t) = u\_{0i}(\overrightarrow{\boldsymbol{x}} - t \,\overrightarrow{\boldsymbol{u}} \,\, (\overrightarrow{\boldsymbol{x}},t)) = \int d^n \xi u\_{0i}\left(\overrightarrow{\boldsymbol{\xi}}\right) \delta\left(\overrightarrow{\boldsymbol{\xi}} - \overrightarrow{\boldsymbol{x}} + t \,\, \overrightarrow{\boldsymbol{u}} \,\, (\overrightarrow{\boldsymbol{x}},t)\right) \tag{20}$$

In (20), δ is the Dirac delta function. Using known (see farther) properties of the delta function, it is possible to express the delta function in (20) with the help of an identity true for the very velocity field meeting Eq. (19):

$$\delta\left(\overrightarrow{\xi} - \overrightarrow{\varkappa} + t\ u \,\, \overrightarrow{u}\,\left(\overrightarrow{\varkappa}, t\right)\right) \equiv \delta\left(\overrightarrow{\xi} - \overrightarrow{\varkappa} + t\overline{u\_0}\,\left(\overrightarrow{\xi}\right)\right) |\det \hat{A}|\,\tag{21}$$

In (21), the matrix A^ depends only on the initial velocity field and is as follows:

$$\hat{A} \equiv A\_{km} = \delta\_{km} + t \frac{\partial u\_{0k}\left(\stackrel{\rightarrow}{\xi}\right)}{\partial \xi\_m} \tag{22}$$

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

$$\delta\left(\overrightarrow{\Phi}\left(\overrightarrow{\xi}\right)\right) = \frac{\delta\left(\overrightarrow{\xi} - \overrightarrow{\xi\_0}\right)}{\left|\det\left(\frac{\partial\Phi\_k}{\partial\xi\_m}\right)\_{\overrightarrow{\xi} = \overrightarrow{\xi\_0}}\right|}\tag{23}$$

In (23), the values ξ<sup>0</sup> ! are defined from the solution of the equation

$$\overrightarrow{\Phi} \left( \overrightarrow{\xi\_0} \right) = \mathbf{0} \tag{24}$$

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 t≥0 if the

Note that under the formal coincidence of the parameters μ ¼ �γ<sup>0</sup> (see the Sivashinsky equation (3) in Introduction), the equality τðÞ¼ t b tð Þ takes place providing the implementation of the singularity (13) when n = 2 and in

accordance with the solution of the Kuramoto-Sivashinsky equation in [42] and

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

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

In [46], the only hodograph method is used for the Burgers-Hopf (or Hopf

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

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

the regularization of this solution for all times if (18) takes place.

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

condition (18) is satisfied [22–26].

Nonlinear Optics ‐ Novel Results in Theory and Applications

Burgers-Hopf equation is done.

media [46].

external friction).

2. Conclusions

62

medium with a fast-charged particle.

To prove (23), it is necessary to use Taylor series decomposition wrt ξ ! near ξ ! ¼ ξ<sup>0</sup> ! for the argument of the delta function Φ ! ξ � �! when in the limit ξ ! ! ξ<sup>0</sup> ! taking into account (24), we get

$$\delta \left( \Phi\_k \left( \stackrel{\rightarrow}{\xi}\_0 \right) + \left( \frac{\partial \Phi\_k}{\partial \xi\_m} \right)\_{\stackrel{\rightarrow}{\xi} = \stackrel{\rightarrow}{\xi}\_0} \left( \xi\_m - \xi\_{0m} \right) + O \left( \stackrel{\rightarrow}{\xi} - \stackrel{\rightarrow}{\xi}\_0 \right)^2 \right) = \delta \left( \left( \frac{\partial \Phi\_k}{\partial \xi\_m} \right)\_{\stackrel{\rightarrow}{\xi} = \stackrel{\rightarrow}{\xi}\_0} \left( \xi\_m - \xi\_{0m} \right) \right) \tag{25}$$

Using variable substitution in the argument of the right-hand side of (25) (of the type A x ^!¼<sup>y</sup> ! and taking into account that d x!¼ d y! det <sup>A</sup>^ j j [48]), we get from the righthand side of (25) the right-hand side of (23).

When in (23), Φ ! ξ � �! �ξ ! � x ! þ tu<sup>0</sup> ! ξ � �! and det <sup>∂</sup>Φ<sup>k</sup> <sup>∂</sup>ξ<sup>m</sup> ¼ det Akm where Akm is from (22); then Eq. (24) is reduced to the following equation:

$$
\overrightarrow{\xi\_0} - \overrightarrow{x} + t\overrightarrow{u\_0} \left(\overrightarrow{\xi\_0}\right) = \mathbf{0} \tag{26}
$$

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

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

The solution (29) of Eq. (19) allows getting an exact solution of Eq. (10) if in (29)

The solution (29) also can be described as an exact solution of Eq. (4) for μ > 0

<sup>t</sup> ! <sup>1</sup> � exp ð Þ �t<sup>μ</sup> μ

To verify the solution (29) satisfies Eq. (19), let us substitute (29) in Eq. (19).

u0<sup>i</sup> ξ � �!

A�<sup>1</sup>

∂δ ξ!

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

¼ �A�<sup>1</sup> km

The identity (33) is obtained from the relationship (obtained by differentiating

The validity of the identities (34) and (35) is proved by the direct checking. In

ously follows directly from (34) and (35). Further, in Item 3, the proof of the identities (34) and (35) of the two- and three-dimensional cases is given.

<sup>∂</sup><sup>t</sup> � <sup>∂</sup>u0<sup>m</sup> ∂ξk

A�<sup>1</sup>

!� � that was not known earlier for the solutions of

ð Þt that yields Eq. (10) representation as in (11).

� <sup>u</sup>0iu0<sup>m</sup> det <sup>A</sup>^ <sup>∂</sup>δ ξ!

det A^ ξ

� x ! þ tu<sup>0</sup> ! ξ � � � �!

∂ξk

km det <sup>A</sup>^ � � � <sup>0</sup> (35)

! )

<sup>∂</sup>xl Alk after multiplying it both sides by the inverse

km ¼ δlmи and δlm is the unity matrix or the Kronecker

<sup>d</sup>ξ<sup>1</sup> <sup>¼</sup> det A; ^ <sup>A</sup>^ �<sup>1</sup>

� �! <sup>∂</sup>δ ξ!

� x ! þtu<sup>0</sup> ! ξ � � � �!

> �x !þtu<sup>0</sup> ! ξ ! � � � � <sup>∂</sup>xm .

km det <sup>A</sup>^ (34)

<sup>¼</sup> det <sup>A</sup>^ � ��<sup>1</sup>

∂xm

(31)

3 5

(32)

(33)

, it obvi-

! x

!!x ! � B !

A.1 The direct validation of the solution

<sup>∂</sup><sup>t</sup> δ ξ!

ξ1F ¼ 0

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

� � � � !

∂ det A^

∂ ∂ξk

the delta function having argument as a given function of ξ

smooth initial velocity field u<sup>0</sup>

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

to make a substitution: x

if in (29) to substitute:

Then we get from (19):

� �! ∂ det A^

det A^ ξ<sup>1</sup> � � ! δ ξ<sup>1</sup> ! � x ! þ tu<sup>0</sup> ! ξ<sup>1</sup>

∂δ ξ!

<sup>∂</sup>ξ<sup>k</sup> ¼ � <sup>∂</sup>δ ξ!

km (where AlkA�<sup>1</sup>

� x ! þ tu<sup>0</sup> ! ξ � � � �!

> �x !þ tu<sup>0</sup> ! ξ ! � � � �

the one-dimensional case, when <sup>A</sup>^ <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>t</sup> du<sup>01</sup>

∂xm

ξ u0<sup>i</sup> ξ

þ ð dn ξ ð dn

� � !

where

F � u0<sup>m</sup> ξ<sup>1</sup>

used:

∂δ ξ! �x !þ tu<sup>0</sup> ! ξ ! � � � �

matrix A�<sup>1</sup>

delta).

65

2 4

ð dn

Eq. (19) [22–26].

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

$$
\overrightarrow{\xi\_0} = \overrightarrow{\varkappa} - t\overrightarrow{u}\left(\overrightarrow{\varkappa}, t\right) \tag{27}
$$

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 u ! x !; t � � <sup>¼</sup> <sup>u</sup><sup>0</sup> ! x ! � t u!ð Þ x; t � � that is used in (20).

Let us use a known property of the delta function that for any smooth function f ! x !� �, the following equality <sup>f</sup> ! x !� �<sup>δ</sup> <sup>x</sup> ! � x ! 0 � � <sup>¼</sup> <sup>f</sup> ! x ! 0 � �<sup>δ</sup> <sup>x</sup> ! � x<sup>0</sup> ! � � holds. That is why, in the general case, it is possible to multiply both sides of (23) by det <sup>∂</sup>Φ<sup>k</sup> <sup>ξ</sup> !� � ∂ξ<sup>m</sup> � � � � � � � � getting the following:

$$\delta\left(\overrightarrow{\xi} - \overrightarrow{\xi\_0}\right) = \delta\left(\overrightarrow{\Phi}\left(\overrightarrow{\xi}\right)\right) \left| \det \frac{\partial \Phi\_k\left(\overrightarrow{\xi}\right)}{\partial \xi\_m} \right| \tag{28}$$

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

$$u\_i(\overrightarrow{\boldsymbol{x}},t) = \int d^n \xi u\_{0i}(\overrightarrow{\boldsymbol{\xi}}) \delta(\overrightarrow{\boldsymbol{\xi}} - \overrightarrow{\boldsymbol{x}} + t\overrightarrow{u\_0}(\overrightarrow{\boldsymbol{\xi}})) \det \hat{A},\tag{29}$$

where det <sup>A</sup>^ <sup>¼</sup> det <sup>δ</sup>mk <sup>þ</sup> <sup>t</sup> ∂u0<sup>m</sup> ξ !� � ∂ξk � �. That solution of Eq. (19) is considered under the following condition:

$$\det \hat{A} > 0 \tag{30}$$

That is why, sign of det A^ is absent in (29). The condition (30) provides smoothness of the solution only on the finite-time interval defined above from (13). Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory… DOI: http://dx.doi.org/10.5772/intechopen.80813

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 u<sup>0</sup> ! x !� � that was not known earlier for the solutions of Eq. (19) [22–26].

The solution (29) of Eq. (19) allows getting an exact solution of Eq. (10) if in (29) to make a substitution: x !!x ! � B ! ð Þt that yields Eq. (10) representation as in (11).

The solution (29) also can be described as an exact solution of Eq. (4) for μ > 0 if in (29) to substitute:

$$t \to \frac{1 - \exp\left(-t\mu\right)}{\mu} \tag{31}$$
