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It is usually believed that solitons always expand due to absorption, and vice versa, they can be significantly compressed due to amplification. Contrary to this usual opinion, we demonstrate that the higher-order nonlinearity opens the possibility to observe unusual and contrasting effects, namely, the soliton self-spreading effect arising due to the soliton amplification and vise versa, the soliton self-compression effect appearing in the absorption regime of the soliton propagation. The detailed contrasting scenarios of the soliton self-spreading or self-compression owing to amplification or absorption are considered in the framework of the simplest cubic-quintic nonlinear Schrodinger equation model.

Using the complimentary interplay between direct computer experiments and the generalized Karpman–Solov’ev adiabatic soliton perturbation theory, we find remarkable empirical formulae describing completely, qualitatively and quantitatively, the impact of distributed losses and gain on the interaction scenarios of the NLSE solitons.

We analyze the most important ideas of the Karpman-Maslov-Solov’ev soliton perturbation theory, and based on it, we solve analytically the long-standing problem of the Schrodinger soliton interactions in the systems with distributed losses or gain. Our analytical results do give a quite good qualitative and quantitative check of the numerical results known so far. Unexpected and nontrivial result consists in the fact that the relative soliton separation depends on the accumulated (integrated) losses or gain.

We present the generalized Tappert transformation for the higher-order nonlinear Schrodinger equation model of femtosecond nonlinear optics, in which the gravitational-like potential is introduced with the aim of the effective parameterization of the Raman soliton self-scattering effect. By means of the nonlinear change of variables and nontrivial reversible gauge transformation the higher-order forced nonlinear Schrodinger equation model is turning into the “free” higher-order nonlinear Schrodinger equation without external potential, but with varying in time dispersion and nonlinearity. The proposed generalized Tappert transformations allow one to find the direct relationships between the soliton solutions of the forced nonisospectral equations and the corresponding isospectral equations without external potential, but with varying in time dispersion and nonlinearity coefficients.

Based on the symmetry reduction procedures, we introduce nonlinear solitonic analogues of coherent and squeezed states both for self-focusing and self-defocusing nonautonomous NLSE models with harmonic oscillator confining potentials. We demonstrate that chirping of the de Broglie wave is a key physical condition for the existence of squeezed states, and it is precisely this wave effect that opens the way to the fundamental extension of the squeezed state concept to different nonlinear physical systems. We show that a subtle interplay between nonlinearity and space dimensionality can result in a rich variety of nonlinear squeezed states. In one dimensional symmetry, nonlinear squeezed states are formed if and only if the nonlinear response of one-dimensional graded-index external potential (waveguide) is inversely proportional to the “hidden” squeezing parameter. And vice versa, in the case of three-dimensional symmetry, the nonlinearity is required to be proportional to the “hidden” squeezing parameter. This remarkable finding carries the germ of the idea of the 3-D soliton bullet formation and the self-focusing collapse suppression in the experimental setup arranged so that periodic variations of the nonlinearity and the maximum peaks of nonlinear squeezed states are being opposite in phases, and as the result, these two processes are alternating to each other. By means of direct computer experiments, we demonstrate the stability of soliton-like coherent and squeezed states, and reveal remarkable and complete analogies with their canonical linear progenitors.

In the framework of the nonautonomous exactly integrable higher-order equations belonging to the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy, the method for systematic construction of the bound soliton states (localized breathers on the zero background) is presented, and general and specific conditions for their appearance are formulated. We show that both all parameters of solitons forming the breather, and the varying dispersion and nonlinearities should be appropriately chosen in accordance with the exact integrability conditions of the models. Novel breather solutions of the higher-order nonlinear evolution equations with vanishing boundary conditions generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. We demonstrate that novel soliton breather solutions substantially extend the breather concept known so far. The revealed specific conditions for the breather formation can be generalized and applied to the evolution equations of the next, more and more higher orders of the AKNS hierarchy.

We apply the method for systematic construction of the two-soliton (and in principal, N-soliton) breather solutions on the zero background to the Lakshmanan–Porsezian–Daniel model with varying dispersion and nonlinearity, gain or loss, and derive the specific conditions for the appearance of the so-called standing and moving soliton breathers. We show that both all parameters of constituent solitons forming the breather and the varying dispersion and nonlinearities should be appropriately chosen in accordance with the exact integrability conditions of the Lakshmanan–Porsezian–Daniel equation with vanishing boundary conditions. We compare the details of periodic breather dynamics for the complex modified KdV, Hirota, and Lakshmanan–Porsezian–Daniel models with the classical Satsuma–Yajima breather behavior. We derive simple formulas for the soliton breather periods and use them to understand the possibilities to control breather dynamics, and, in particular, to realize the so-called regimes of “artificial breathing”, when periods of soliton breathing in space and time are controlled by choosing appropriate complex spectral parameters defining the amplitudes and velocities of constituent solitons. Of fundamental importance is the remarkable theoretical fact that the higher-order nonlinear and dispersion effects included into (or removed from) the new higher-order equations of the AKNS hierarchy, as a rule, do not destroy soliton breathers discovered here and do not transform them into dispersive waves. On the contrary, we demonstrate that novel and novel soliton breathers can appear in the framework of the next, more and more higher orders of the AKNS and NLSE hierarchies.

We reveal novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg–de Vries equation with variable coefficients and the loss (or gain) term (vc cmKdV). Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. This finding allows us to systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.

Discovered by Busch and Anglin, the most important and puzzling nonlinear effect for solitons in external potentials lies in the fact that in a harmonic oscillator potential, a dark soliton oscillates with a frequency that is smaller by a factor of 2 in comparison with the ground state oscillations frequency. We review the main features of the Busch-Anglin effect and show that the physical nature of the self-defocusing nonlinearity can radically alter the Busch-Anglin dynamics. To bridge the gap between the matter-wave and optical dark solitons, we consider both the impact of linear and nonlinear absorption and the crucial role of nonlocality (or nonstationarity) of the nonlinear response in the triggering of the Busch-Anglin effect. We conclude that nonlocality (or nonstationarity) brings into existence the self-induced Busch-Anglin effect. Both linear and nonlinear losses cannot radically affect the period of dark solitons oscillations. We establish that from the physical point of view, it is best to redefine the specific dark soliton angle between the asymptotic phases of non-vanishing boundary conditions in accordance with Faddeev and Takhtajan's mathematical arrangement based on the phase shift across a dark soliton “core” controlling the soliton “blackness”. Among other things, we show that the effects analogous to the self-induced Busch-Anglin effect exist for bright solitons as well. Namely, triggered by nonlocal (or time-delayed) self-focusing nonlinearity, the bright solitons oscillate in the parabolic trap with a shifted equilibrium position and a frequency that exactly coincides with the oscillator frequency.

We consider optimal control methods for dark solitons and their higher-order states based on the completely integrable nonautonomous nonlinear Schrodinger equation (NLSE) models with varying gravitational-like and harmonic oscillator potentials, and in the presence of gain or absorption. In view of the fact that the time-dependence of the dark soliton phase (chirp) represents a major difference between bright and dark solitons, our main goal here is to gain a deeper understanding at a fundamental level based on the Lax pairs of how this time-dependent phase appears in the framework of the nonautonomous NLSE models with varying dispersion and nonlinearity. Specifically, we consider dynamics of dark solitons and their higher-order states in a variety of different optical scenarios. The main features of dark solitons are supplied by numerical calculations performed both for the exactly integrable and nonintegrable cases. We reveal nontrivial dynamics of black and gray solitons under the optimal control conditions. Since the experimental realization of dark solitons is possible only if a finite background pedestal is used, we consider dynamics of dark solitons and their higher-order states superimposed on the ground state of nonlinear harmonic oscillator. Direct numerical experiments performed in the framework of the Gross-Pitaevskii (GP) model in a broad range of parameters reveal many fundamental features of the dark solitons dynamics in confining harmonic oscillator potential, including the background BEC cloud spreading near the Feshbach resonance (when repulsive nonlinearity has a dispersive form). One of the most important effects is related to the “saturation” of the maximum value of self-compression near the Feshbach resonance. This effect exists both for the central black soliton and for the gray solitons formed during the higher-order dark soliton oscillations in the parabolic trap. The soliton becomes black at the turning points in the harmonic potential and gray at the trap center. This makes possible to observe the nontrivial dynamics of the periodic transformation of the black soliton to the gray one.

We reveal remarkable, but obviously only formal analogies and parallels between the soliton behavior in external gravitational-like potentials and the cosmic dark energy and antigravitation effects. The introduced generalized and exactly integrable higher-order nonlinear Schrodinger equation model for the wave function of a soliton in the attractive gravitational-like potential implies the existence of many fundamental nonlinear phenomena in completely different systems, independent of the physical origin of the nonlinearity and the higher-order linear and nonlinear dispersion. We show that the proposed model is governed by nontrivial Galilean-like invariance and, under the exact integrability conditions given by the Lax pair, demonstrates important formal nonlinear-optical analogies with enigmatic cosmic dark energy and antigravitation effects. Guided by interesting, but only formal analogy between the linear Doppler red shift and the nonlinear soliton self-frequency red shift effects, we reveal nontrivial features of the soliton propagation scenario in the attractive gravitation-like potential. Namely, the soliton attraction in the gravitation-like potential is changed to strong repulsion similar to the Newtonian particle dynamics in the repulsive antigravitational field. The wave-function red-shift effect (which is accumulated linearly with the propagation distance) is conserved under the transformation from attractive to repulsive scenarios in the ordinary space–time domain.

In view of the fact that the nonlinear Schrodinger equation model has a wide range of applicability, from nonlinear femtosecond optics, Bose–Einstein condensates and plasmas, to ocean extreme (monster) waves and hurricanes, we conclude that existing mathematical analogies open advantageous possibility to study optical, matter-wave and hydrodynamic nonlinear waves and solitons in parallel and due to the evident complexity of experiments with extreme nonlinear waves in oceans, offer remarkable potentialities in studies of these so complex systems by performing “optic hydrodynamics” experiments in the nonlinear fiber-optics systems. Specifically, we present the results of computer experiments of soliton interactions with different external gravitational-like potentials including that with varying levels of inclinations. We demonstrate novel effective mechanism of the supercontinuum generation both in nonlinear optics and in hydrodynamics based on the higher order N-soliton wave packets decay induced by external potentials of the hyperbolic tangent profiles with varying inclinations similar to the real ocean coast line profiles.

Inspirited by many remarkable but, obviously, only formal analogies appearing within the framework of the nonlinear Schrodinger equation model from nonlinear femtosecond optics, Bose–Einstein condensates and plasmas, to ocean extreme waves and hurricanes, we conclude that these important mathematical parallels offer novel possibilities to perform “optic hydrodynamics” experiments in the nonlinear fiber-optics systems. Specifically, we present the results of the wide-range computer experiments of the soliton dynamics in the external gravitational-like potentials with embedded barriers and wells. We show that the external gravitational-like potential adds significant new physics both to the nonlinear soliton tunneling through classically forbidden potential barriers and to the solitonic analog of the quantum-mechanical Ramsauer–Townsend effect. The main special feature of a soliton in the external gravitational-like potential lies in the fact that its velocity is no longer constant, but depends linearly on time. This soliton acceleration changes dramatically the wave-particle duality of a soliton giving rise, by virtue of the Galilean symmetry, the appearance of its own solitonic analog of the “accelerating de Broglie wavelength”. Guided by this constructive (but obviously only formal) analogy, we reveal the hidden role of the soliton self-interaction (binding) energy both for the soliton nonlinear tunneling and the solitonic nonlinear analog of the Ramsauer–Townsend effect in the effective gravitational-like external potentials with superimposed barriers and wells.

We present three exactly integrable nonisospectral nonlinear evolution models with varying in time linear potentials: the forced complex modified Korteweg-de Vries (cmKdV), the forced Hirota, and the forced Sasa-Satsuma equations. In all three models we have added the time-varying linear potential to effectively simulate two effects of tremendous significance for soliton management and fiber optics communications: the sliding filter method of the noise separation from a soliton and the Raman soliton self-scattering effect. We find nonlinear gauge transformations between introduced nonisospectral models arranged in such a way that their soliton solutions, accelerating in the linear potential nonautonomous solitons, can be directly obtained without resolving the nonisospectral inverse scattering transform problem.

Our direct computer experiments reveal the main features of the induced modulational instability in the sign-reversal dispersion traps and offer a clearer view of how nonlinear analogies to the “deja vu” phenomena have arisen. We demonstrate the crucial role of the higher-order dispersion in the origin of the imperfect Fermi–Pasta–Ulam recurrence and partial “deja vu” phenomena. Induced modulational instability can be used to generate the steady-state soliton trains in the sign-reversal parabolic dispersion traps with exponentially decreasing dispersion.

We clarify an analytical approach in the theory of modulational instability developed in the framework of the nonautonomous nonlinear Schrodinger equation model taking into account three mathematical models of the Raman self-scattering effect and arbitrary orders of the even group velocity dispersion. We reveal important analogies and significant distinctions between the higher-order induced modulational instability and the internal dynamics of the higher-order N-soliton bound states. We demonstrate that periodic interference splitting onto two or more well-separated wave components and the numbers of split rogue wave-like peaks are determined by the total number of solitons accumulated in the initial wave profile. Our direct computer experiments reveal the crucial role of the sliding pump effect at the nonlinear stage of the higher-order induced modulational instability giving rise to the soliton self-cleaning effect.

The induced modulational instability is studied in the framework of the nonautonomous nonlinear Schrodinger equation (NLSE) model with varying dispersion and nonlinearity. The Hasegawa algorithm of the analysis of the modulational instability (MI) is generalized taking into account both varying dispersion and nonlinearity. Main features of the induced MI are revealed using direct computational experiments. In particular, the induced modulational instability is considered from a point of view of dynamical memory behavior, and the main restrictions leading to imperfect restoration of the information signal are found. Our direct computer experiments offer a clearer view of how nonlinear analogies to the “deja vu” phenomena have arisen. It is shown that in the framework of the nonautonomous NLSE model, the “deja vu” phenomena must be accompanied by the imperfect spectral energy restoration into its fundamental Fourier mode, the variation of the Fermi–Pasta–Ulam recurrence period along the propagation distance, the excitation of the higher-order modulation instability, and the rogue waves formation.

The generalized nonautonomous complex modified Korteweg–de Vries (mKdV) equation with external potentials is introduced in the framework of the nonisospectral generalization of the Inverse Scattering Transform method with associated spectral parameter varying in accordance with the Riccati equation. Nonautonomous accelerating and self-compressing mKdV solitons exist only if the varying dispersion, nonlinearity, and the external linear, parabolic, and cubic potentials satisfy to the conditions of the exact integrability of the introduced model.

The generalized nonautonomous nonlinear Schrodinger equation is introduced in the framework of the nonisospectral generalization of the Inverse Scattering Transform method with associated spectral parameter varying in accordance with the Riccati equation. Nonautonomous solitons of the introduced model conserve the soliton main feature to interact elastically both in the linear, parabolic, and cubic external potentials, and are controlled only if the varying dispersion and nonlinearity satisfy to the conditions of the exact integrability. Novel features of nonautonomous solitons arising due to the dependence between the soliton amplitudes and their velocities consist in the decay of the soliton bound states.

We demonstrate the possibility to generalize the Karpman and Solov’ev theory of interaction forces among solitons taking into account an external potential as a small perturbation. Closely following to the Karpman and Solov’ev adiabatic perturbation algorithm, we obtain analytical solutions completely describing the dynamics of two solitons trapped in the harmonic oscillator potential. Period of oscillations of two in-phase solitons decreases both with increasing the oscillator frequency and decreasing the initial separation between them, whereas two out-of-phase solitons can be found in the bound states of two practically noninteracting solitons. Analytical relations demonstrate a quite good qualitative and quantitative agreement with the numerical experiments. The main discrepancy between the generalized Karpman and Solov’ev perturbation theory and the results of direct computer experiments is explained by periodic perturbations of the soliton pair by the noise-like dispersive waves emitted by overlapped neighboring solitons.

We consider the dynamics of the simplest Schrodinger's cat composed by a superposition of two canonical coherent states of the Schrodinger quantum harmonic oscillator. We demonstrate that the interchange of the “alive” and “dead” constituents in the whole Schrodinger's cat wave function opens the possibility to reveal one interesting, but obviously, only formal analogy with the Pauli exclusion principle. We show analytically and computationally that the symmetric cat states will occupy the ground state of the harmonic oscillator, whereas the antisymmetric cat states, vice versa, will never occupy the ground state when these states closely approach in space each other. The antisymmetric overlapping in space and time “alive” and “dead” Schrodinger cat states simply jump onto the first excited energy level of the quantum harmonic oscillator. This effect could be considered as the formal analogy of the Pauli exclusion principle applied to the antisymmetric “jumping off” Schrodinger cats. As the name suggests, the main feature of the “jumping off” Schrodinger cat consists in the transformation of the probability distribution (both in ordinary coordinate space and in the momentum space as well) in such a way that antisymmetric “alive” and “dead” cats are found to be in the first excited energy state of the quantum harmonic oscillator. We conclude that the wave and particle-like analogies in the dynamics of the symmetric and antisymmetric Schrodinger's cat states confirm at least formally the famous Pauli statements that “a symmetric solution can never develop into an antisymmetric one, and vice versa”, and that “for particles with antisymmetric states it can, therefore, never happen that two particles find themselves in the same state”

We present a formal extension of the Pauli exclusion principle to the arbitrary-order jumping up and down excited Schrodinger’s cats. We show that the excited symmetric Schrodinger’s cat of the n-order will occupy the same n-order excited eigenstate of the harmonic oscillator when the “alive” and “dead” cat’s constituents closely approach in space and time each other. The antisymmetric n-order cat states, vice versa, will never occupy the same excited n-order eigenstate. For example, the respective probabilities for antisymmetric cat states of n = 1, 2, and 3 orders to jump up onto the corresponding energy levels with n = 2, 3, and 4, and, at the same time, jump down onto the levels with n = 0, 1, and 2, are found to be in the ratios: 2/3:1/3, 3/5:2/5, and 4/7:3/7, with the asymptotic saturated ratio of 1/2:1/2 for the cat states distributed over the higher-energy levels. Moreover, we demonstrate the remarkable analogies between linear and nonlinear dynamics of the higher-order Schrodinger’s cats revealed in the framework of the nonlinear Schrodinger equation model with harmonic oscillator potential.

We consider how the higher-energy excited Schrodinger's cats can be transformed into the even and odd winking states, the main feature of which consists in the periodic jumping of the probability density distribution from one to other regions of space. In particular, the winking states can be formed among closely displaced ground and arbitrary-order coherent states involved into the Schrodinger's “alive” and “dead” cat dynamics. From a pedagogical standpoint it is worth mentioning that the general analytical solution for arbitrary-order winking states can be obtained practically without any calculations. We use only the method of mathematical induction and the “often forgotten” quantum-mechanical Husimi and Taniuti symmetry transformation. We show that the Husimi and Taniuti gauge invariance establishes a one to one correspondence between the harmonic oscillator eigenstates and the higher-energy coherent states of the same order in such a way that their space and time-dependent phase factors are identical for all orders of coherent states. By way of illustration, we exemplify the main logical-operation properties of winking states up to the fifth-order.

Nonlinear analogues of coherent states arise in the framework of the nonlinear Schrodinger equation models with confining harmonic oscillator potentials. We clarify the profound physical linkage between the quantum-mechanical Schrodinger coherent states and their nonlinear solitonic analogues. Guided by remarkable, but obviously only formal analogy between the soliton negative self-action energy and the nuclear binding energy, we reveal how the nonlinear ground and coherent states could be built up from the parton-like solitonic constituents when the absolute value of the soliton binding energy increases. The enhancement of the soliton binding energy contribution in the total conserved energy of the nonlinear ground and coherent states radically changes their internal structures and allows one to apply the formal analogies from the parton model of nucleons.

A quantum superposition of two displaced higher-order coherent states can be considered as the higher-energy excited Schrodinger's cat state. These linear self-localized and robust wave packets elastically interact and preserve their internal higher-energy Hermitian-Gaussian structures after repeated collisions. We demonstrate the transformation of the higher-energy excited Schrodinger's cat states into the winking states. As the name suggests, the main novel feature of winking state consists in the periodic jumping of the probability distribution from one region of space to another. The winking states of the Schrodinger harmonic oscillator are formed when the lower-energy coherent state, for example, the displaced ground state (playing the role of the “alive” cat state), closely approaches in space (and at the same time) the first excited coherent state, which has the meaning of the “dead” cat. We combine analytical and computational studies of winking states formed among two closely displaced ground and first-order excited states involved into the Schrodinger's “alive and dead” cat dynamics.

Nonlinear solitonic analogues of Schrodinger's cat states arise in the framework of the nonlinear Schrodinger equation (NLSE) model with confining harmonic oscillator potential. In-phase or out-of-phase displaced canonical(linear) Schrodinger's coherent states (well known as the “male” or “female” Schrodinger's cats) periodically oscillate and interfere in the crossing central point of confining harmonic oscillator potential. They are self-localized and robust linear solitary waves that do not disperse and preserve their identity after repeated collisions, and, in this sense, they are analogous to the well-known nonlinear bound states of the NLSE solitons. We discuss the main parallels and distinctions between canonical Schrodinger's cat states and their nonlinear solitonic analogues. Our primary aim is to reveal the main features of nonlinear solitonic analogues of Schrodinger's cat states restricted by the quantum-mechanical normalization condition and interpretation. The fulfillment of this condition means that, in general terms, the hypothesis of the possibility to develop nonlinear quantum mechanics can be tested directly by computational experiments with generalized NLSE models. We clarify how the strong short-range nonlinear forces lead to considerable decreasing (or increasing) of the oscillation periods between even and odd soliton-like Schrodinger's coherent states, when they closely approach each other. We demonstrate the possibility to realize the coherent superposition of practically noninteracting displaced solitonic Schrodinger's coherent states with opposite phases. Our analytical results do give a quite good qualitative and quantitative check of the numerical results known so far.

Formal mathematical analogies existing within the framework of the nonlinear Schrodinger equation (NLSE) model open remarkable possibilities to perform novel experiments in nonlinear-optical hydrodynamics that would never have been possible in the sea hydrodynamics. We demonstrate that inherent phase jump of a dark soliton is a key physical condition giving rise to the formal nonlinear-optical analogy to the Moses sea parting effect and sequential appearance of the out-of-phase NLSE solitons formed from the rogue wave-like structures on the wings of the spreading spatial dip profile including the case of varying water depth.

We demonstrate both the existence of the self-induced sub-barrier transparency in soliton tunneling through classically forbidden localized potential barriers and controllable ‘shooting out’ the high-energy solitons from arbitrary -soliton superpositions.

Based on the formal analogy between soliton tunneling through a classically forbidden potential barrier and –particle tunneling and decay we introduce a ‘toy solitonic model of a nucleus’ and show that the analog of the Geiger–Nuttall law can be discovered for Schrodinger solitons. We reveal and explain the analogy with virtual particle–antiparticle pair production and annihilation mechanisms and show how the nonlinear particle-like wave packet is being ‘dressed in a coat of virtual pairs’ and transformed into the soliton. In computational experiments with different soliton parameters and for different profiles of the barriers, including the charged sphere Coulomb potential, we demonstrate how the scaling symmetry breaking reveals the hidden role of the soliton self-interaction (binding) energy in the soliton scattering on external potentials and its dramatic impact on the wave-particle duality of solitons.

A generalized saturable nonlinear Schrodinger equation was derived for the quasi-spin model of DNA that was subjected to the action of external agents such as a protein and a bath of phonons, by using the Generalized Coherent States approach (GCS). The GCS approach was applied for averaging the quasi-spin Hamiltonian for DNA macromolecule proposed by Takeno and Homma. In weakly saturating approximation, several important collectively formed stretching and unfolding traveling structures have been obtained for angle deviations that subsequently made crucial influence in the formation of compacton anti-compacton pairs for the hydrogen bond displacements.

We show that the law of the soliton adaptation to varying-in-time external potentials indicates conclusively the way for solitonlike bullets generation in three-dimensional nonautonomous nonlinear and dispersive systems. It turns out that the generation of matter-wave soliton bullets can be realized if periodic variations of non-linearity and confining sign-reversal varying-in-time harmonic oscillator potential are opposite in phases so that peaks of nonlinearity inside the atomic cloud coincide in time with repulsive character of trapping potential during reversal periodic transformations from cigar-shaped to ball-shaped trapping structures. In nonlinear optical applications, periodic graded-index nonlinear structures with alternating waveguiding and antiwaveguiding segments can be used to simulate complicated processes of matter-wave soliton bullets generation.

The method of mathematical analogies opens the possibility to study optical and matter-wave solitons in parallel and, due to the evident complexity of experiments with matter wave solitons, offers remarkable possibilities in studies of the BEC systems by performing experiments in the nonlinear optical systems and vice versa. We discuss previous misinterpretations of the binding soliton energy and investigate all possible scenarios of nonlinear soliton tunneling of Schrodinger solitons including soliton splitting on a potential barrier, leading to partial reflection and partial transmission, sub-barrier soliton tunneling and over-barrier soliton reflection. We reveal the most significant modifications of soliton tunneling scenarios due to increasing the binding energy, and show that the self-compressing soliton resembles more the classical particle case than a quantum mechanical behavior. The enhancement of the soliton binding energy is of decisive importance in the testing of a long-standing theoretical r...

The fundamental concept of colored nonautonomous solitons in nonlinear and dispersive nonautonomous physical systems is introduced. Novel soliton solutions for the nonautonomous nonlinear Schrodinger equation 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. The parallels between nonlinear guided wave phenomena in optics and nonlinear guided wave phenomena in Bose condensates are clearly demonstrated by considering optical and matter wave soliton dynamics in the framework of nonautonomous evolution equations. The exact analytical solutions and numerical experiments reveal many specific features of nonautonomous solitons. Fundamental laws of the soliton adaptation to the external potentials are derived. Bound sta...

The concept of nonautonomous soliton dispersion management pertaining to the propagation of chirped nonautonomous Raman colored solitons through a dispersion oscillating fiber in the presence of an electrooptic phase modulator and also by taking into consideration the effect due to self-induced Raman scattering on the propagation of optical solitons when the soliton amplitude does not vary significantly during self-scattering, has been studied. The amplitudes and speeds of these nonautonomous solitons, during the propagation and after the interaction are no longer found to be the same as those prior to the interaction. Moreover, intense soliton pulse amplification has been obtained, which forms the main results of this paper. In the context of Bose-Einstein condensates, the aforementioned scenario has also been interpreted as the dynamics of the condensates with linear and quadratic external potentials in an optical trap.

The concept of soliton dispersion management pertaining to the effect of varying dispersion with external harmonic oscillator potential for chirped solitons have been studied in detail with emphasis on the various aspects of dispersion management of solitons. The interaction scenarios pertaining to in-phase- and off-phase injection with equal and unequal amplitudes are dealt elaborately. Suppression of interaction forces between the dispersion managed solitons, optical soliton pulse amplification and robustness of bound soliton states to collisions, are some of the main results of this article.

The dynamics of nonlinear solitary waves is studied in the framework of the nonlinear Schrodinger equation model with time-dependent harmonic oscillator potential. The model allows one to analyse on general basis a variety of nonlinear phenomena appearing both in Bose–Einstein condensate, condensed matter physics, nonlinear optics, and biophysics. The soliton parametric resonance is investigated by using two complementary methods: the adiabatic perturbation theory and direct numerical experiments. Conditions for reversible and irreversible denaturation of soliton bound states are also considered.

The dynamics of dark solitons is studied within the framework of the mathematical model of nonlinear Schrodinger equation (NSE) with an external harmonic potential. A comparative analysis of the solutions of nonstationary problems is performed for a linear harmonic oscillator and the NSE model with a harmonic potential for different signs of the self-action potential. It is shown that the main specific feature of the dynamics of dark NSE solitons in a parabolic trap is the formation of solitons with dynamically changing form factors producing the periodic variation in the modulation depth (the degree of 'blackness') of dark solitons. The oscillation period of the dark soliton does not coincide with the oscillation period of a linear quantum-mechanical oscillator, which is caused by the periodic transformation of the black soliton to the grey one and vice versa. The conditions of applicability of the method of inverse scattering problem are presented, the generalised Lax pair is found, and exact soliton solutions are given for the mathematical NSE model with an external harmonic potential.

Stimulated Raman self-scattering (SRSS) is shown not to be a unique effect observed exclusively in the propagation of femtosecond optical solitons through optical fibres. SRSS always accompanies the self-action of high-power femtosecond pulses in a variety of media. Unlike classical SRS, SRSS has no lasing threshold and can appear when the pulse spectrum overlaps the band of vibrational resonances of the medium. A classification of the main SRSS regimes is given. The basic features of different regimes of ultrashort-pulse self-scattering are investigated by computer simulation techniques.

We present the algorithm based on the Lax pair generalization to reveal some properties of nonautonomous KdV solitons. Starting from the general method of solution for the nonisospectral IST problem, we demonstrate how the varying-coefficient KdV equation can arise in nonuniformed and inhomogeneous media. We write down the one- and two-soliton solution of the nonautonomous varying-coefficient KdV with the time-dependent spectral parameter and consider some special cases of the isospectral solutions of the KdV equation with varying dispersion, nonlinearity, and gain. Finally, we compare these solutions with the nonautonomous solitons of the NLSE emphasizing their common features.

The interpenetration of the main ideas and methods being used in different fields of science and technology has become one of the decisive factors in the progress of science as a whole. Mathematical analogies between different physical systems can be extremely fruitful in understanding the novel physical concepts. Accordingly to the new theory of bio-energy transport along protein molecules in living systems and modified Davydov molecular soliton theory, we propose a nonautonomous model that can be considered as a candidate of the bio-energy transport mechanism in protein molecules. Based on the generalized nonlinear Schrodinger equation model with varying-intime harmonic oscillator potential, we show that conditions of its exact integrability in one-dimensional case indicate conclusively the way for solitonlike pulse generation in polypeptide molecular systems. The most important properties of this soliton transport of bio-energy are related with periodically changed energy-release conditions and the influences of structure nonuniformity in protein molecules. By analogy with the corresponding optical phenomena in inhomogeneous optical fibers with varying properties along the length, we study the main features of modified Davydov soliton on the basis of the unified nonautonomous nonlinear Schrodinger model in the parameters region of the exact integrability of the model under consideration.

The nonlinear Schrodinger equation (NLSE) model with an external harmonic potential is one of the most important in modern science. This model makes it possible to analyze a variety of nonlinear phenomena, in nonlinear optics and laser physics, biophysics and in the theory of Bose-Einstein condensation of atoms. It is shown that the main specific feature of the dynamics of dark GP matter wave solitons in a parabolic trap is the formation of solitons with dynamically changing form-factors producing the periodic variation in the modulation depth (the degree of “blackness”) of dark solitons. In general, the period of dark soliton oscillations in trapping potential depends on the specific conditions of the experiment and does not coincide with the oscillation period of a linear quantum-mechanical oscillator. In the case of an immobile pedestal in the trap, the oscillation period of the black soliton considerably increases because of the periodic transformation of the black soliton to the gray one and vice versa. Surprisingly, that if the dark soliton is superimposed on the base pedestal oscillating in the trap and displaced from the trap center, the oscillation period of the dark soliton coincides with the period of oscillations of the linear harmonic oscillator, while the dynamics of the dark soliton is similar to that of a classical particle obeying the Newton mechanics laws.

The famous principle of wave-particle duality in quantum physics holds that matter and light exhibit the behaviors of both waves and particles, depending on the experiment being performed. Is there a similar analogy for the soliton? What other analogies have not been established so far? The key conceptual result of our work consists in the demonstration of a deep analogy between the Schrodinger soliton tunneling through the classically forbidden potential barrier and the Gamow scenario of quantum mechanical tunneling effect and alpha-particle decay. Guided by this constructive (but obviously only formal) analogy, we reveal a hidden role of the soliton self-interaction ("binding") energy and its dramatic impact on the wave-particle duality of scattered solitons. The solitonic analog of the de Broglie wavelength, "shooting out" of high-energy de Broglie video-solitons from arbitrary N-soliton superposition and phenomena similar to the Ramsauer-Taunsend effect and the Geiger--Nuttall law ought to be expected for solitons.

Nonlinear optical organic materials will be the key elements for future high-bit-rate telecommunications and ultrafast photonic technologies. We show that the process of nonlinear tunneling of optical solitons through a strong nonlinear thin film barrier, for example, a nonlinear organic thin film barrier, exhibits jump-like nonadiabatic evolution, which eventually leads to the soliton “fission reactions”. Nonlinear “fission reaction” is attended by generation of “colored” solitons with different frequencies and group velocities. It seems very attractive to use the nonlinear soliton tunneling effect through thin film of polydiacetylene para -toluene sulfonate in developing a whole class of basically novel all-optical soliton logic devices.

Methods for direct numerical integration of a system of nonlinear Maxwell's equations are used to establish a quantitative criterion of the validity of the method of slowly varying amplitudes and of a generalised model of the nonlinear Schroedinger equation in a description of the dynamics of femtosecond optical solitons. It is shown that Schroedinger solitons may be converted nonlinearly into Maxwellian wave solitons, whose special property is motion not only in the usual space and time, but also in the spectral space. Moreover, it should be possible to generate a pulse of duration amounting to one period of oscillations of the electromagnetic field in the course of amplification of a Maxwellian soliton.

The method of direct numerical integration of a system of nonlinear Maxwell's equations was used to investigate the dynamics of formation and interaction of Maxwellian wave solitons of a few femtosecond duration.

In this paper we discuss the propagation of two or three optical pulses in an isotropic media when they satisfy the phase-matching conditions of the kind 2ωx − 2ωy = 0 or 2ω3 = ω1 + ω2. In both cases in the corresponding nonlinear evolution equations appear terms connected with four photon parametric processes. These terms generate periodic exchange of energy between the optical pulses. In the present work we obtain new kind of stable soliton-like parametrically connected optical waves.