**6. References**

48 Hydrodynamics – Advanced Topics

*I* γ

ω

( ) <sup>2</sup> 2 ... [] ( ) ... <sup>2</sup> *<sup>t</sup> t t <sup>I</sup> dt e u sh t t*

( ) <sup>2</sup> <sup>2</sup> ... [ ] sin ( ) ... <sup>2</sup> *<sup>t</sup> t t <sup>I</sup> dt e u t t*

δ

Substitution of the explicit expressions (47b) or (47c) in the equation (44c) gives the generalisation of the Navier – Stokes equation for a solid medium with local relaxation of angular momentum. As was mentioned above under special condition (49) and in the limiting case (52) this equation reduces exactly to the form of Navier – Stokes equation. Thus, it is shown that relaxation of angular momentum of material points consisting a continuum can be considered as physical reason for appearance of terms with shear viscosity in Navier-Stokes equation. Without dissipation additional degree of freedom dealt

The first part of the chapter presents an original formulation of the generalized variational principle (GVP) for dissipative hydrodynamics (continuum mechanics) as a direct combination of Hamilton's and Onsager's variational principles. The GVP for dissipative continuum mechanics is formulated as Hamilton's variational principle in terms of two independent field variables i.e. the mean mass and the heat displacement fields. It is important to mention that existence of two independent fields gives us opportunity to consider a closed mechanical system and hence to formulate variational principle. Dissipation plays only a role of energy transfer between the mean mass and the heat displacement fields. A system of equations for these fields is derived from the extreme condition for action with a Lagrangian density in the form of the difference between the kinetic and the free energies minus the time integral of the dissipation function. All mentioned potential functions are considered as a general positively determined quadratic

δ

*I*

*I* is used. For the case of resonant relaxation <sup>2</sup> *I* >

*I*

= − ′ ′ <sup>∇</sup> <sup>−</sup> (47c)

(47b)

zeros are real and have the following asymptotics for small momentum of inertia 0 *I* → :

The first zero does not depend on momentum of inertia *I* and the second root goes to

negative real part, which decreases with increase of *I* . The last case corresponds to the

= − ′ ′ <sup>∇</sup> <sup>−</sup>

γ /(4 ) σ

γ /(4 ) σ

In the time representation the solution of the equation (50) can be written in the form

γ

γ

with angular momentum leads to the well known Cosserat continuum.

 − − ′ −∞

σ

 − − ′ −∞

σ

γ≈ − 2 *<sup>i</sup>* γ /(4 ) σ

> γ /(4 ) σ

≈ − (52)

the zeros coincide and have the value

the zeros are complex conjugated with

both

determine two modes of angular momentum relaxation. Under condition <sup>2</sup> *I* <

<sup>1</sup> *i*

infinity when 0 *I* → . Under condition <sup>2</sup> *I* =

resonant relaxation of angular momentum.

the corresponding expression has the form

here the notation <sup>2</sup> ... 4 = −

≈ − , and under the condition <sup>2</sup> *<sup>I</sup>* <sup>&</sup>gt;

ϕ

γ

ϕ

<sup>1</sup> *i* 2

ω

σ

γ

**4. Conclusion** 

ω

Berdichevsky V.L. (2009). *Variational principles of continuum mechanics*, Springer-Verlag, ISBN 978-3-540-88466-8, Berlin.

Biot M. (1970). *Variational principles in heat transfer*. Oxford, University Press.

Deresiewicz H. (1957). Plane wave in a thermoplastic solids. *The Journal of the Acoustcal Society of America*, Vol.29, pp.204-209, ISSN 0001-4966.

Erofeev V.I. (1998). *Wave processes in solids with microstructure,* Moscow State University, Moscow.

Glensdorf P., Prigogine I., (1971). *Thermodynamic Theory of Structure, Stability, and Fluctuations*, Wiley, New York.

Gyarmati I. (1970). *Non-equilibrium thermodynamics. Field theory and variational principles*. Berlin, Springer-Verlag.

Kunin I.A. (1975). *Theory of elastic media with micro structure* , Nauka, Moscow.

Landau L.D., Lifshitz E.M. (1986). *Theoretical physics. Vol.6. Hydrodynamics*, Nauka, Moscow.

**0**

**3**

*Mexico*

T. L. Belyaeva1 and V. N. Serkin<sup>2</sup>

<sup>1</sup>*Universidad Autónoma del Estado de México* <sup>2</sup>*Benemerita Universidad Autónoma de Puebla*

**Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics**

Nonlinear science is believed by many outstanding scientists to be the most deeply important frontier for understanding Nature (Christiansen et al., 2000; Krumhansl, 1991). The interpenetration of main ideas and methods being used in different fields of science and technology has become today one of the decisive factors in the progress of science as a whole. Among the most spectacular examples of such an interchange of ideas and theoretical methods for analysis of various physical phenomena is the problem of solitary wave formation in nonautonomous and inhomogeneous dispersive and nonlinear systems. These models are used in a variety of fields of modern nonlinear science from hydrodynamics and plasma physics to nonlinear optics and matter waves in Bose-Einstein condensates. The purpose of this Chapter is to show the progress that is being made in the field of the exactly integrable nonautonomous and inhomogeneous nonlinear evolution equations possessing the exact soliton solutions. These kinds of solitons in nonlinear nonautonomous systems are well known today as nonautonomous solitons. Most of the problems considered in the present Chapter are motivated by their practical significance, especially the hydrodynamics applications and studies of possible scenarios of generations and controlling of monster **(**rogue) waves by the action of different nonautonomous and inhomogeneous

Zabusky and Kruskal (Zabusky & Kruskal, 1965) introduced for the first time the soliton concept to characterize nonlinear solitary waves that do not disperse and preserve their identity during propagation and after a collision. The Greek ending "on" is generally used to describe elementary particles and this word was introduced to emphasize the most remarkable feature of these solitary waves. This means that the energy can propagate in the localized form and that the solitary waves emerge from the interaction completely preserved in form and speed with only a phase shift. Because of these defining features, the classical soliton is being considered as the ideal natural data bit. It should be emphasized that today, the optical soliton in fibers presents a beautiful example in which an abstract mathematical concept has produced a large impact on the real world of high technologies (Agrawal, 2001;

Solitons arise in any physical system possessing both nonlinearity and dispersion, diffraction or diffusion (in time or/and space). The classical soliton concept was developed for nonlinear and dispersive systems that have been autonomous; namely, time has only played the role of

Akhmediev, 1997; 2008; Dianov et al., 1989; Hasegawa, 1995; 2003; Taylor, 1992).

**1. Introduction**

external conditions.

Landau L.D., Lifshitz E.M. (1972). *Theoretical physics. Vol.7. Theory of elasticity,* Nauka, Moscow.

Landau L.D., Lifshitz E.M. (1964). *Theoretical physics. Vol.5. Statistical physics.* Nauka, Moscow.

Lykov A.V. (1967). *Theory of heat conduction,* Moscow, Vysshaya Shkola.

