**1. Introduction**

50 Hydrodynamics – Advanced Topics

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.

Mandelshtam L.I., Leontovich M.A. (1937). To the sound absorption theory in liquids, *The* 

Martynov G.A. (2001). Hydrodynamic theory of sound wave propagation. *Theoretical and* 

Maximov G.A. (2006). On the variational principle for dissipative hydrodynamics. *Preprint 006-2006, Moscow Engineering Physics Institute*, Moscow. (in Russian) Maximov G.A. (2008). Generalized variational principle for dissipative hydrodynamics and

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Onsager L. (1931b). Reciprocal relations in irreversible process II. *Physical Review*, Vol. 38,

Shliomis M.I. (1966). Hydrodynamics of a fluid with intrinsic rotation*, The Journal of Experimental and Theoretical Physics*, Vol.51, No.7, pp.258-265, ISSN 0044-4510 (in Russian). Sorokin V.S. (1943). On internal friction of liquids and gases possessed hidden angular

Zhdanov V.M., Roldugin V.I. (1998). Non-equilibrium thermodynamics and kinetic theory of rarefied gases. *Physics-Uspekh,*. Vol.41, No.4, pp. 349-381, ISSN 0042-1294.

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its application to the Biot's equations for multicomponent, multiphase media with temperature gradient, In: *New Research in Acoustics*, B.N. Weis, (Ed.), 21-61, Nova

its application to the Biot's theory for the description of a fluid shear relaxation,

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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 external conditions.

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

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

are well known in the context of quantum mechanics. Ablowitz, Kaup, Newell, Segur (AKNS) (Ablowitz et al., 1973) have found that many physically meaningful nonlinear models can be

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics 53

In the traditional scheme of the IST method, the spectral parameter Λ of the auxiliary

fundamental role in the development of analytical theory (Zakharov, 1980). The nonlinear evolution equations that arise in the approach of variable spectral parameter, Λ�

contain, as a rule, some coefficients explicitly dependent on time. The IST method with variable spectral parameter makes it possible to construct not only the well-known models for nonlinear autonomous physical systems, but also discover many novel integrable and

In this work, we clarify our algorithm based on the Lax pair generalization and reveal generic properties of nonautonomous solitons. We consider the generalized nonautonomous NLSE and KdV models with varying dispersion and nonlinearity from the point of view of their exact integrability. It should be stressed that to test the validity of our predictions, the experimental arrangement should be inspected to be as close as possible to the optimal map of parameters, at which the problem proves to be exactly integrable (Serkin & Hasegawa, 2000a;b; 2002). Notice, that when Serkin and Hasegawa formulated their concept of solitons in nonautonomous systems (Serkin & Hasegawa, 2000a;b; 2002), known today as nonautonomous solitons and SH-theorems (Serkin & Hasegawa, 2000a;b; 2002) published for the first time in (Serkin & Hasegawa, 2000a;b; 2002), they emphasized that "the methodology developed provides for a systematic way to find an infinite number of the novel stable bright and dark "soliton islands" in a "sea of solitary waves" with varying dispersion, nonlinearity, and gain or absorption" (Belyaeva et al., 2011; Serkin et al., 2010a;b). The concept of nonautonomous solitons, the generalized Lax pair and generalized AKNS methods described in details in this Chapter can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter-waves and offer many opportunities for further scientific studies. As an illustrative example, we show that important mathematical analogies between different physical systems open the possibility to study optical rogue waves and ocean rogue waves in parallel and, due to the evident complexity of experiments with rogue waves in open oceans, this method offers remarkable possibilities in studies nonlinear hydrodynamic problems by performing experiments in the nonlinear

*<sup>t</sup>* = 0, and this fact plays a

*<sup>t</sup>* �= 0,

linear problem is assumed to be a time independent constant Λ�

physically significant nonlinear nonautonomous equations.

optical systems with nonautonomous solitons and optical rogue waves.

**dispersive models with external potentials**

Balachandran, 2004).

**2. Lax operator method and exact integrability of nonautonomous nonlinear and**

The classification of dynamic systems into autonomous and nonautonomous is commonly used in science to characterize different physical situations in which, respectively, external time-dependent driving force is being present or absent. The mathematical treatment of nonautonomous system of equations is much more complicated then of traditional autonomous ones. As a typical illustration we may mention both a simple pendulum whose length changes with time and parametrically driven nonlinear Duffing oscillator (Nayfeh &

In the framework of the IST method, the nonlinear integrable equation arises as the

*ψ<sup>x</sup>* = F*ψ*(*x*, *t*), *ψ<sup>t</sup>* = G*ψ*(*x*, *t*). (1)

compatibility condition of the system of the linear matrix differential equations

solved by the IST method.

the independent variable and has not appeared explicitly in the nonlinear evolution equation. A not uncommon situation is one in which a system is subjected to some form of external time-dependent force. Such situations could include repeated stress testing of a soliton in nonuniform media with time-dependent density gradients.

Historically, the study of soliton propagation through density gradients began with the pioneering work of Tappert and Zabusky (Tappert & Zabusky, 1971). As early as in 1976 Chen and Liu (Chen, 1976; 1978) substantially extended the concept of classical solitons to the accelerated motion of a soliton in a linearly inhomogeneous plasma. It was discovered that for the nonlinear Schrödinger equation model (NLSE) with a linear external potential, the inverse scattering transform (IST) method can be generalized by allowing the time-varying eigenvalue (TVE), and as a consequence of this, the solitons with time-varying velocities (but with time invariant amplitudes) have been predicted (Chen, 1976; 1978). At the same time Calogero and Degaspieris (Calogero, 1976; 1982) introduced a general class of soliton solutions for the nonautonomous Korteweg-de Vries (KdV) models with varying nonlinearity and dispersion. It was shown that the basic property of solitons, to interact elastically, was also preserved, but the novel phenomenon was demonstrated, namely the fact that each soliton generally moves with variable speed as a particle acted by an external force rather than as a free particle (Calogero, 1976; 1982). In particular, to appreciate the significance of this analogy, Calogero and Degaspieris introduced the terms boomeron and trappon instead of classical KdV solitons (Calogero, 1976; 1982). Some analytical approaches for the soliton solutions of the NLSE in the nonuniform medium were developed by Gupta and Ray (Gupta, 1981), Herrera (Herrera, 1984), and Balakrishnan (Balakrishnan, 1985). More recently, different aspects of soliton dynamics described by the nonautonomous NLSE models were investigated in (Serkin & Hasegawa, 2000a;b; 2002; Serkin et al., 2004; 2007; 2001a;b). In these works, the "ideal" soliton-like interaction scenarios among solitons have been studied within the generalized nonautonomous NLSE models with varying dispersion, nonlinearity and dissipation or gain. One important step was performed recently by Serkin, Hasegawa and Belyaeva in the Lax pair construction for the nonautonomous nonlinear Schrödinger equation models (Serkin et al., 2007). Exact soliton solutions for the nonautonomous NLSE models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. In particular, solitons in nonautonomous physical systems exist only under certain conditions and varying in time nonlinearity and dispersion cannot be chosen independently; they satisfy the exact integrability conditions. The law of soliton adaptation to an external potential has come as a surprise and this law is being today the object of much concentrated attention in the field. The interested reader can find many important results and citations, for example, in the papers published recently by Zhao et al. (He et al., 2009; Luo et al., 2009; Zhao et al., 2009; 2008), Shin (Shin, 2008) and (Kharif et al., 2009; Porsezian et al., 2007; Yan, 2010).

How can we determine whether a given nonlinear evolution equation is integrable or not? The ingenious method to answer this question was discovered by Gardner, Green, Kruskal and Miura (GGKM) (Gardner et al., 1967). Following this work, Lax (Lax, 1968) formulated a general principle for associating of nonlinear evolution equations with linear operators, so that the eigenvalues of the linear operator are integrals of the nonlinear equation. Lax developed the method of inverse scattering transform (IST) based on an abstract formulation of evolution equations and certain properties of operators in a Hilbert space, some of which 2 Will-be-set-by-IN-TECH

the independent variable and has not appeared explicitly in the nonlinear evolution equation. A not uncommon situation is one in which a system is subjected to some form of external time-dependent force. Such situations could include repeated stress testing of a soliton in

Historically, the study of soliton propagation through density gradients began with the pioneering work of Tappert and Zabusky (Tappert & Zabusky, 1971). As early as in 1976 Chen and Liu (Chen, 1976; 1978) substantially extended the concept of classical solitons to the accelerated motion of a soliton in a linearly inhomogeneous plasma. It was discovered that for the nonlinear Schrödinger equation model (NLSE) with a linear external potential, the inverse scattering transform (IST) method can be generalized by allowing the time-varying eigenvalue (TVE), and as a consequence of this, the solitons with time-varying velocities (but with time invariant amplitudes) have been predicted (Chen, 1976; 1978). At the same time Calogero and Degaspieris (Calogero, 1976; 1982) introduced a general class of soliton solutions for the nonautonomous Korteweg-de Vries (KdV) models with varying nonlinearity and dispersion. It was shown that the basic property of solitons, to interact elastically, was also preserved, but the novel phenomenon was demonstrated, namely the fact that each soliton generally moves with variable speed as a particle acted by an external force rather than as a free particle (Calogero, 1976; 1982). In particular, to appreciate the significance of this analogy, Calogero and Degaspieris introduced the terms boomeron and trappon instead of classical KdV solitons (Calogero, 1976; 1982). Some analytical approaches for the soliton solutions of the NLSE in the nonuniform medium were developed by Gupta and Ray (Gupta, 1981), Herrera (Herrera, 1984), and Balakrishnan (Balakrishnan, 1985). More recently, different aspects of soliton dynamics described by the nonautonomous NLSE models were investigated in (Serkin & Hasegawa, 2000a;b; 2002; Serkin et al., 2004; 2007; 2001a;b). In these works, the "ideal" soliton-like interaction scenarios among solitons have been studied within the generalized nonautonomous NLSE models with varying dispersion, nonlinearity and dissipation or gain. One important step was performed recently by Serkin, Hasegawa and Belyaeva in the Lax pair construction for the nonautonomous nonlinear Schrödinger equation models (Serkin et al., 2007). Exact soliton solutions for the nonautonomous NLSE models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. In particular, solitons in nonautonomous physical systems exist only under certain conditions and varying in time nonlinearity and dispersion cannot be chosen independently; they satisfy the exact integrability conditions. The law of soliton adaptation to an external potential has come as a surprise and this law is being today the object of much concentrated attention in the field. The interested reader can find many important results and citations, for example, in the papers published recently by Zhao et al. (He et al., 2009; Luo et al., 2009; Zhao et al., 2009; 2008), Shin (Shin, 2008) and (Kharif et al.,

How can we determine whether a given nonlinear evolution equation is integrable or not? The ingenious method to answer this question was discovered by Gardner, Green, Kruskal and Miura (GGKM) (Gardner et al., 1967). Following this work, Lax (Lax, 1968) formulated a general principle for associating of nonlinear evolution equations with linear operators, so that the eigenvalues of the linear operator are integrals of the nonlinear equation. Lax developed the method of inverse scattering transform (IST) based on an abstract formulation of evolution equations and certain properties of operators in a Hilbert space, some of which

nonuniform media with time-dependent density gradients.

2009; Porsezian et al., 2007; Yan, 2010).

are well known in the context of quantum mechanics. Ablowitz, Kaup, Newell, Segur (AKNS) (Ablowitz et al., 1973) have found that many physically meaningful nonlinear models can be solved by the IST method.

In the traditional scheme of the IST method, the spectral parameter Λ of the auxiliary linear problem is assumed to be a time independent constant Λ� *<sup>t</sup>* = 0, and this fact plays a fundamental role in the development of analytical theory (Zakharov, 1980). The nonlinear evolution equations that arise in the approach of variable spectral parameter, Λ� *<sup>t</sup>* �= 0, contain, as a rule, some coefficients explicitly dependent on time. The IST method with variable spectral parameter makes it possible to construct not only the well-known models for nonlinear autonomous physical systems, but also discover many novel integrable and physically significant nonlinear nonautonomous equations.

In this work, we clarify our algorithm based on the Lax pair generalization and reveal generic properties of nonautonomous solitons. We consider the generalized nonautonomous NLSE and KdV models with varying dispersion and nonlinearity from the point of view of their exact integrability. It should be stressed that to test the validity of our predictions, the experimental arrangement should be inspected to be as close as possible to the optimal map of parameters, at which the problem proves to be exactly integrable (Serkin & Hasegawa, 2000a;b; 2002). Notice, that when Serkin and Hasegawa formulated their concept of solitons in nonautonomous systems (Serkin & Hasegawa, 2000a;b; 2002), known today as nonautonomous solitons and SH-theorems (Serkin & Hasegawa, 2000a;b; 2002) published for the first time in (Serkin & Hasegawa, 2000a;b; 2002), they emphasized that "the methodology developed provides for a systematic way to find an infinite number of the novel stable bright and dark "soliton islands" in a "sea of solitary waves" with varying dispersion, nonlinearity, and gain or absorption" (Belyaeva et al., 2011; Serkin et al., 2010a;b). The concept of nonautonomous solitons, the generalized Lax pair and generalized AKNS methods described in details in this Chapter can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter-waves and offer many opportunities for further scientific studies. As an illustrative example, we show that important mathematical analogies between different physical systems open the possibility to study optical rogue waves and ocean rogue waves in parallel and, due to the evident complexity of experiments with rogue waves in open oceans, this method offers remarkable possibilities in studies nonlinear hydrodynamic problems by performing experiments in the nonlinear optical systems with nonautonomous solitons and optical rogue waves.
