Nonlinear Physics

PREFACE.

CONTENTS.

An integrable discretization of the vector nonlinear Schrödinger equation is considered within the framework of the inverse scattering method. The problem of multisoliton collisions is investigated. An analytic formula for the polarization shift of discrete vector solitons as a consequence of their interaction is obtained.

The coupled nonlinear Klein-Gordon equations are analyzed for their integrability properties in a systematic manner through Painlevé(P-) test. From the P-test, one-soliton solutions are identified. The connection between the system equations and sine-Gordon equation has been pointed out from the bilinear form. Then, the results are generalized to the two-coupled Klein-Gordon equations.

A unified general approach is presented for construction of solutions of the characteristic initial value problems for various integrable hyperbolic reductions of Einstein’s equations for space-times with two commuting isometries in General Relativity and in some string theory induced gravity models. In all cases the associated linear systems of similar structures are used and their fundamental solutions admit an alternative representation by two “scattering” matrices of a simple analytical structure on the spectral plane. The condition of equivalence of these representations leads to the linear “integral evolution equations” whose scalar kernels and right hand sides are determined completely by the initial data for the fields specified on the two initial characteristics. If the initial data for the fields are given, all field components of the corresponding solution can be expressed in quadratures in terms of a unique solution of these quasi - Fredholm integral evolution equations.

The inverse scattering theory for the sine-Gordon equation discretized in space and both in space and time is considered.

We consider a one-parameter family of non-evolutionary partial differential equations which includes the integrable Camassa-Holm equation and a new integrable equation first isolated by Degasperis and Procesi. A Lagrangian and Hamiltonian formulation is presented for the whole family of equations, and we discuss how this fits into a bi-Hamiltonian framework in the integrable cases. The Hamiltonian dynamics of peakons and some other special finite-dimensional reductions are also described.

A free boundary problem for a nonlinear diffusion-convection equation is considered. The problem is reduced to a nonlinear integral equation in time; by means of the contraction mapping technique we show that the nonlinear integral equation admits a unique solution for small times.

In a previous work⁷, we have obtained the Lax pair and Darboux transformations for an integrable equation in (2 + 1) dimensions. In the present paper we derive an iterative method of obtaining solutions in a rather easy way. Several solutions are presented for real and complex version of the equation.

The anticontinuum limit (i.e. the limit of weakly coupled oscillators) is used to obtain two surprising results. First we prove the continuation of discrete breathers of weakly interacting harmonic oscillators, provided a suitable coupling is chosen. Secondly we derive an analytical result for the wave transmission by a breather of the discrete nonlinear Schrödinger equation at weak coupling. We obtain a resonant full reflection due to a Fano resonance.

The complex Toda chain (CTC) is known to describe the N-soliton train interactions in adiabatic approximation for several nonlinear evolution equations: the nonlinear Schrödinger equation (NLS), the modified NLS, higher NLS and s-G equation. The integrability of the CTC is used to describe all possible dynamical regimes of the N-soliton trains and also to describe analytically the submanifolds of initial soliton parameters responsible for each of these regimes. The comparison of the numerical solutions for both NLS and MNLS with the results of the CTC model shows very good agreement. We compare the Hamiltonian properties of the N-soliton solutions of the NLS in the adiabatic approximation and show how it matches the Hamiltonian formulation for the complex Toda chain which describes the adiabatic N-soliton interactions.

The reductions for the first order linear system of the type (1) are studied. This system generalizes the Zakharov-Shabat system and the systems studied by Caudrey, Beals and Coifman (CBC systems)². Here J is a regular complex element of the Cartan subalgebra h ⊂ g of the simple Lie algebra g and the potential q(x, t) takes values in the image g_J of ad_J. Our investigation is based on the reduction group introduced by A. V. Mikhailov⁶. The properties of the fundamental analytical solutions (FAS) of (1) under the reductions are studied. Special attention is paid to the behavior of the corresponding scattering data of CBC systems under the Weyl group reductions.

The Hirota bilinear representation for two types of modified NLS equations (MNLS_1,2) and corresponding deformations by quantum potential are considered. For MNLS₁ it allows us to construct rather simple formulas for ultrashort solitons in a nonlinear optics. Addition of quantum potential, changing the dispersion qualitatively in a similar way as in dispersion-managed fibers, leads to a novel chiral properties of solitons.

The Moyal *-deformed noncommutative Burgers’ equation is considered. Using the *-analog of the Cole-Hopf transformation, the linearization of the model in terms of the linear heat equation is found. Deformations of one and two shock soliton solutions are described.

The recent formulation of the quasi-classial -dressing method shows that the weakly dispersive limit of the standard -problem provides the dispersionless limit and the corresponding dispersive corrections for some integrable hierarchy (KP and 2DTL). For a rather general choice of the kernel we provide the formula of quasi-classical -problem at each order on dispersive parameter and in the KP equation case we obtain the linear problems giving the corrective equation at second order.

We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals using the notion of tau function of matrix argument.

A new approach for derivation of Benney-like momentum chains and integrable hydrodynamic type systems is presented. A new integrable hydrodynamic chain is constructed, all its reductions are described and integrated.

Reconsidering the IST method for the KPII equation, we prove that condition of existence of the generator of integrals of motion leads to nonlinear equation on the spectral data of the associated linear problem (the heat equation). This equation is preserved under time evolution and leads to a specific behavior of the spectral data on the complex plane. In particular, these data cannot be local or rapidly decaying. We suggest and consider different generalizations of the IST beyond the class of decaying initial data.

Seeking Inverse Transforms for KdV is considered as a special chapter of Potential Scattering where the dynamic equation is a set of coupled “Lax” equations. With this view of the problem, one writes down transformation properties, either classical or new, consistency relations, explicit expansions of solutions - here in the multisolitonic case and in the linear approximation limit. The results are collected and gathered in view of checking tools or examples for studies of boundary value problems.

We propose a spatio-temporal discretization of the Schrödinger equation for field operators of noninteracting bosons. The scheme, being based on the associated space of the creation and annihilation operators and on discretization of the Schrödinger equation possesses all relevant properties of the underline continuum model. The discretization appears to be a linear part of an integrable discretization of classical nonlinear Schrödinger equation which allows one to obtain some exact solutions.

We solve the monodromy problem and prove the Painlevé property for self-similar ZS-AKNS flows with a quadratic spectral variable in this report. In particular, meromorphic solutions for the Cauchy problem of the self-similar derivative nonlinear Schrödinger equation (DNLS) are obtained.

Fourth-order nonlinear diffusion equations appear frequently in the description of physical processes, among these processes, the lubrication equation u_t = (u⁻ⁿu_xxx)_x plays an important role in the study of interface movements. For this equation new classes of symmetries are derived. These symmetries allow us to increase the number of exact explicit solutions of this equation. These solutions are neither solutions arising from nonclassical symmetries, nor solutions arising from classical potential symmetries.

We define quantum solitons for physical systems governed by the quantum attractive nonlinear Schrödinger model (such as optical fibres) and for physical systems governed by the quantum sine-Gordon model (such as large area Josephson junctions). We indicate how these quantum solitons can be viewed as “qubits” in the quantum information sense. The picture of the “quantum soliton” which emerges is rather different from that presented in a recent popular article¹ with this title.

Renormalization Group Equations in integro-differential form describing the evolution of cascades or resumming logarithmic scaling violations have been known in quantum field theory for a long time. These equations have been traditionally solved by turning to Mellin moments, since in this space they become algebraic. x-space solutions are less known, but special asymptotic expansions exist which allow a fast numerical implementation of these equations. We illustrate how the equations can be solved using recursion relations in the next-to-leading order approximation.

No abstract received.

We introduce the notion of a real Hamiltonian form (RHF) of a dynamical system in analogy with the notion of real forms for simple Lie algebras. We first complexify the initial dynamical system and then restrict it to a subspace isomorphic (but not symplectomorphic) to the initial phase space. Thus to each real Hamiltonian system we are able to associate a family of nonequivalent RHF’s. A crucial role in this construction is played by a compatible involution leaving invariant the Hamiltonian mimicking the properties of complex conjugation. If the initial system is integrable, its RHF’s will also be integrable. This provides a method of finding new integrable systems starting from known ones. We construct several RHF’s for a family of Calogero–Moser Hamiltonians.

Classical super-integrable systems are often studied in the framework of Hamilton-Jacobi approach to classical dynamics via separability procedures. On the other hand it is known that integrable systems (and, a fortiori, super-integrable ones) admit more than one Hamiltonian description, therefore it seems unfair to select one symplectic structure among the many invariant ones to study integrability properties. In this report, we argue that a more appropriate setting would be the framework of Lie–Scheffers systems. These systems may be defined also in the quantum case, providing a unified approach to classical and quantum super-integrable systems.

Vortex dynamics in a thin superfluid ⁴He film as well as in a type II superconductor is described by the classical counterpart of the model advocated by Peierls, and used for deriving the ground states of the Fractional Quantum Hall Effect. The model has non-commuting coordinates, and is obtained by reduction from a particle associated with the “exotic” extension of the planar Galilei group.

We consider the structure of multi-meron knot action in the Yang-Mills theory and in the CP¹ Ginzburg-Landau (GL) model. Self-dual equations have been obtained without identifying orientations in the space-time and in the color space. The dependence of the energy bounds on topological parameters of coherent states in planar systems is also discussed. In particular, it is shown that a characteristic size of a knot in the Faddeev-Niemi model is determined by the Hopf invariant.

The problem of description for compatible nonlocal Poisson brackets of hydrodynamic type is solved. The nonlinear equations describing all compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of the inverse scattering problem is proved. A Lax pair with a spectral parameter is found for these nonlinear equations. In the special case, when one of compatible Poisson brackets of hydrodynamic type is local, the derived equations give integrable reductions of the classical Lamé equations. Corresponding Lax pairs with a spectral parameter are found for all these special reductions of the Lamé equations.

We introduce the concept of pseudoanti-Hermiticity in quaternionic quantum mechanics (QQM) and discuss some physical properties related to time-reversal symmetry of the pseudoanti-Hermitian Hamiltonians. We detect a Kramers like degeneracy for a subclass of such Hamiltonians.

A method for constructing integrable equations involving bosonic and fermionic variables and sharing the same Wahkquist–Estabrook prolongation superalgebra is proposed. New integrable systems and the corresponding spectral problems are constructed using the prolongation structure associated with the Manin–Radul super KdV.

A system of partial differential equations describing bilayer amphiphilic membranes is studied by the Lie group analysis. This algorithmic approach allows us to show all the symmetries of the system, determine all the possible symmetry reductions, recover the axisymmetric solutions and, finally, address the question of new similarity solutions.

An extension of the classic Enneper-Weierstrass representation for conformally parametrised surfaces in 3-dimensional Euclidean space R³ is performed. This is carried out by using CP¹ sigma model, which allows us to study constant mean curvature (CMC) surfaces immersed into R³. In particular it is demonstrated that the generalised Weierstrass representation can admit different CMC-surfaces in R³ which have globally the same Gauss map.

We describe our recent work on deformations of hyperelliptic curves by means of integrable hierarchy of hydrodynamic type, and discuss a further extension to the cases of non-hyperelliptic curves.

The study of features of rigid surfaces systems is commonly performed by making use of the Monge representation. We briefly outline two alternative ways to proceed and show some new results they lead to.

In a geometrical framework, we prove the integrability of two (1+1)-dimensional spin systems. A unifying approach, based on the integrable spin systems, to the construction of integrable classes of surfaces is proposed. In particular, two classes of integrable surfaces are studied.

The geometry of an admissible Bäcklund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite–dimensional skeleton which is the universal prolongation of a |1|–graded semi-simple Lie algebra.

We consider a system of coupled nonlinear Schrödinger equations with even, periodic boundary conditions, which are damped and quasi-periodically forced. Under certain conditions, we establish criteria for the existence of homoclinic orbits to a spatially independent invariant torus. We compare the analysis with rigorous numerical simulation.

Evolution equations associated with the multidimensional Schrödinger equation are discussed. It is shown that the problem is nonisospectral for an arbitrary time-dependent potential. In particular, the eigenvalues evolve according to the Hellmann-Feynman theorem, while the eigenfunctions follow from a system of coupled nonlinear ODE’s. A novel approach to solving a class of eigenvalue problems is also discussed and illustrated by computing energy levels of an anharmonic oscillator.

We discuss motion of vortex filament where the filament obeys the localized induction equation with corrections one of which gives rise to stretching and/or shrinking of the filament. Some numerical results are presented.

The Wannier function basis is used to construct lattice approximations of the nonlinear Schrödinger equation with a periodic potential. The approach is based on the averaging procedure which leads to a vector lattice with long-range interactions.

We illustrate the behaviour of a system of nonlinear dispersive PDEs modelling the propagation of water waves in a finite space interval, by comparing it with the behaviour of the corresponding linearized PDE system. We review how in general the interaction with the boundary destroys the nonlinear effects, reducing the evolution to an essentially linear dispersive phenomenon. In one particular case of homogeneous boundary conditions, it has been observed numerically that the evolution and in particular the solitary waves generated from the initial waveform persist beyond the interaction with the boundary, which is in this sense transparent. We show how in this case the solution of the linearized problem has a specific structure and propose that this can be used as a characterizing property of those boundary conditions that lead to “transparent” boundaries.

Rogue or “freak” waves are high amplitude waves whose heights exceed two times the significant wave height of the background sea ¹. We investigate rogue wave dynamics in deep water using the nonlinear Schrödinger (NLS) equation, as well as higher order generalizations, and observe that a chaotic regime greatly increases the likelihood of rogue wave formation. These large amplitude waves are well modeled by higher order homoclinic solutions of the NLS equation.

We study, in the underdamped regime, the nonlinear dynamics of a Frenkel-Kontorova type model subject to an external DC driving force and a substrate potential defined by the sum of two sinusoidal functions with different spatial periodicity. Particularly focusing on mutually incommensurate choices for the three inherent length scales characterizing the system, we simulate the particle dynamics over a quasiperiodic substrate. We find that when the length scales are related by the cubic spiral mean there exists a value of the interparticle interaction strength above which the static friction vanishes, while when they are related by the quadratic golden mean the static friction is always nonzero. We also analyze the nature of the depinning mechanisms and the steady states achieved by the moving chain.

We report on the possibility of the ratchet-like (unidirectional) motion of a topological soliton of a dissipative Klein-Gordon equation in the presence of AC forces with zero mean. The role played by the temporal asymmetry of the system in establishing soliton DC motions is emphasized. In particular, we show that effective soliton transport is achieved when the asymmetric internal mode of the kink and the external force get phase locked. The dependence of the soliton velocity on different system parameters has been studied.

In this paper, an asymptotic equation is derived from first principles which governs the propagation of electromagnetic waves in a waveguide array in the presence of both normal and anomalous diffraction. This is termed diffraction management. The theory is then extended to the vector case of coupled polarization modes.

Quantum Finance represents the synthesis of the techniques of quantum theory (quantum mechanics and quantum field theory) to theoretical and applied finance. After a brief overview of the connection between these fields, we illustrate some of the methods of lattice simulations of path integrals for the pricing of options. The ideas are sketched out for simple models, such as the Black-Scholes model, where analytical and numerical results are compared. Application of the method to nonlinear systems is also briefly overviewed. More general models, for exotic or path-dependent options are discussed.

Using an asymptotic theory and a momentum method, we identify a family of dispersion management schemes with distributed Raman amplification, which are advantageous for massive multichannel soliton transmission. For the case of two-step dispersion maps, special schemes are found that have optimal (chirp-free) launch point locations that are independent of the fibre dispersion. Despite the variation of dispersion with wavelength due to the fibre dispersion slope, the transmission in several different channels can be optimized simultaneously using the same optimal launch point. The theoretical results are verified by direct numerical simulations.

The standard procedure for detection of gravitational wave coalescing binaries signals is based on Wiener filtering with an appropriate bank of template filters. This is the optimal procedure in the hypothesis of addictive Gaussian and stationary noise. We study the possibility of improving the detection efficiency with a class of adaptive spectral identification techniques, analyzing their effect in presence of non stationarities and undetected non linearities in the noise.

Using variational methods, we derive a reduced system of equations from the non-local equation that governs the average dynamics in dispersion-managed (DM) systems. The reduced equations apply for any type of return-to-zero pulses and describe the slow evolution of the pulse parameters bypassing the fast variations inside each dispersion map. We investigate the stationary solutions assuming a Gaussian pulse shape and obtain analytical estimates for the parameters of DM solitons as functions of dispersion map strength. Also, in the limit of large map strength we integrate the equations to obtain explicit formulas for the parameters of a general return-to-zero pulse. The analytical results are in excellent agreement with numerical simulations of pulse propagation in optical fibers with dispersion management.

An influence of external potentials on production of kinks is investigated. The meaning of the presence of the additional length scales in the number density formula of produced kinks is commented.

A method for determining envelope solitonlike solutions of a wide class of nonlinear Schrödinger equations, based on a recently–developed correspondence between the generalized Korteweg-de Vries equation and the generalized nonlinear Schrödinger equation (with arbitrary nonlinearity), is presented. A review of the applications of this novel approach for determining, in particular, bright and grey/dark envelope solitonlike solutions of the cubic-quintic nonlinear Schrödinger equation is also presented. Finally, the results of a recent analytical-numerical study of the stability of these solutions in the case of the nonlinear interaction of an intense circularly polarized light beam with an uniform unmagnetized plasma are discussed.

We introduced the tomographic representation of soliton solutions of nonlinear Schrödinger equations (NLSE). Some examples of known solitons are studied in the tomographic representation.

No abstract received.

We discuss the application of importance sampling techniques to the numerical simulation of transmission effects induced by amplifier noise in soliton-based optical transmission systems. The method allows one to concentrate numerical simulations on the noise realizations that are most likely to result in transmission errors, thus leading to speedups of several orders of magnitude over standard Monte Carlo methods. We demonstrate the technique by calculating the probability distribution function of amplitude and timing fluctuations.

An electronic receiver based on stochastic resonance is presented to rescue subthreshold modulated or not digital data. A complete data restoration is achieved for both uniform and gaussian white noise, confirming the interest of stochastic resonance in the data transmission field.

Kink propagation failure induced by coupling inhomogeneities in a Nagumo lattice is investigated numerically, experimentally and theoretically.

The nonlinear dynamics of laser beams carrying phase singularity in media with cubic-quintic nonlinearity is studied. In such media can be generated not only localized vortex solitons but also a novel kind of stable nonlocalized optical vortices. Stability in the defocusing regime is confirmed numerically. In the focusing regime such a vortex breaks into filaments. Dynamics of a singular Gaussian beam is investigated. Numericals simulations show a new behavior of the supercritical Gaussian beam which first breaks into filaments coalessing after.

The interaction of very short (tens of femtoseconds) and ultraintense (I > 10¹⁸W/cm²) laser pulse on a low density-plasma (n_e ~ 10⁻³ − 10^-2n_c, being n_c the critical density of the plasma at the laser wavelength) can generate longitudinal electron plasma waves with very large amplitude (wake-field generation process). The evolution of such a wake field is higly nonlinear and a lot of interesting phenomena occur, one of them being the trapping and acceleration of electrons of several MeV’s in few tens of micrometers. The process of acceleration of the relativistic electrons in the regime of “Laser Wake Field Acceleration” has been demonstrated experimentally by using both gas-jet and (very recently) explodingfoils techniques. Here we present the result of a full 3D Particle-In-Cell (PIC) simulation of the interaction of a 35fs at relativistic intensity I = 10¹⁹W/cm², focussed onto a plasma having density equal to the “resonant” density n_R, which depends on the pulse length. The results of the PIC simulations agree with experimental data and confirm the production of a highly collimated and energetic electron bunch.

Modulational instability (MI) in a discrete nonlinear LC transmission line is investigated. The higher order nonlinear Schrödinger (HONLS) equation modelling modulated waves propagation in the network allows to predict the MI conditions, with additional features, compared to the standard NLS model.

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