Zakharov Shabat Research Articles

Solitons: Conservation laws and dressing methods

We review some of the fundamental notions associated with the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux–Bäcklund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov–Shabat dressing scheme and the Gelfand–Levitan–Marchenko equation are also introduced. Via this method, generic solutions are produced and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete nonlinear Schrödinger model, the Korteweg–de Vries and nonlinear Schrödinger equations as well as the sine-Gordon and Liouville models.

On the All-Fiber Optical Methods of the Generation and Recognition of Soliton States

The state of a soliton is characterized by the eigenvalues of the Zakharov–Shabat problem. The data transfer rate in the fiber-optic communication lines that use eigenvalues for coding signals can be significantly increased only by developing optical eigenvalue control methods. We propose to use optical fibers with a sinusoidally changing dispersion to generate given sets of complex eigenvalues and to detect soliton states. Under the action of a periodic change in dispersion, multisoliton pulses divide into several solitons moving at different group velocities. This effect can be used to prepare fixed sets of eigenvalues. A soliton signal can be recognized by analyzing pulses and their spectra at the exit from a fiber with a periodically changing dispersion. Using the solitons specified by sets of four eigenvalues, we show that the field at the exit from a fiber corresponds to a unique combination of the spectrum and the number of pulses determined by initial eigenvalues.

Linear instability of the Peregrine breather: Numerical and analytical investigations

We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By constructing solutions of the linearization of the nonlinear Schrödinger equation in terms of quadratic products of components of the eigenfunctions of the Zakharov–Shabat system, we show that the Peregrine breather is linearly unstable. A numerical study employing a highly accurate Chebychev pseudo-spectral integrator confirms exponential growth of random initial perturbations of the Peregrine breather.

Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation

Starting from a comparison of some established numerical algorithms for the computation of the eigenvalues (discrete or solitonic spectrum) of the non-Hermitian version of the Zakharov–Shabat spectral problem, this article delivers new algorithms that combine the best features of the existing ones and thereby allays their relative weaknesses. Our algorithm is modelled within the remit of the so-called direct nonlinear Fourier transform (NFT) associated with the focusing nonlinear Schrödinger equation. First, we present the data for the calibration of existing methods comparing the relative errors associated with the computation of the continuous NF spectrum. Then each method is paired with different numerical algorithms for finding zeros of a complex-valued function to obtain the eigenvalues. Next we describe a new class of methods based on the contour integrals evaluation for the efficient search of eigenvalues. After that we introduce a new hybrid method, one of our main results: the method combines the advances of contour integral approach and makes use of the iterative algorithms at its second stage for the refined eigenvalues search. The veracity of our new hybrid algorithm is established by estimating the convergence speed and accuracy across three independent test profiles. Along with the development of a new approach for the computation of the eigenvalues, our study also addresses the problem of computation of the so-called norming constants associated with the eigenvalues. We show that our formalism effectively amounts to accurate and fast enough computation of residues of the reflection coefficient in the upper complex half-plane of the spectral parameter.

Fast inverse nonlinear Fourier transform.

This paper considers the non-Hermitian Zakharov-Shabat scattering problem which forms the basis for defining the SU(2)-nonlinear Fourier transform (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transform is quite well established in the Ablowitz-Kaup-Newell-Segur formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present two fast inverse NFT algorithms with O(KN+Nlog^{2}N) complexity and a convergence rate of O(N^{-2}), where N is the number of samples of the signal and K is the number of eigenvalues. These algorithms are realized using a new fast layer-peeling (LP) scheme [O(Nlog^{2}N)] together with a new fast Darboux transformation (FDT) algorithm [O(KN+Nlog^{2}N)] previously developed by V. Vaibhav [Phys. Rev. E 96, 063302 (2017)2470-004510.1103/PhysRevE.96.063302]. The proposed fast inverse NFT algorithm proceeds in two steps: The first step involves computing the radiative part of the potential using the fast LP scheme for which the input is synthesized under the assumption that the radiative potential is nonlinearly bandlimited, i.e., the continuous spectrum has a compact support. The second step involves addition of bound states using the FDT algorithm. Finally, the performance of these algorithms is demonstrated through exhaustive numerical tests.

Asymptotics of spectral quantities of Zakharov–Shabat operators

In this paper we provide asymptotic expansions of various spectral quantities of Zakharov–Shabat operators on the circle with new error estimates. These estimates hold uniformly on bounded subsets of potentials in Sobolev spaces.

Detecting Impedance and Shunt Conductance Faults in Lossy Transmission Lines

We describe a method that uses propagation measurements to reconstruct the spatial profiles of the distributed impedance and shunt conductance parameters of a lossy transmission line. From these reconstructed spatial profiles, we can deduce the presence and location of transmission line faults. The method makes use of return loss measurements from both ends of the transmission line as well as the transmission loss between the two ends. An inverse scattering approach based on the Born approximation is utilized to obtain the reconstructions. Its advantage compared to exact formulations, such as those based on the Zakharov–Shabat equations, is that it provides separate analytical expressions for the individual spatial profiles of the distributed impedance and shunt conductance transmission line parameters. It is shown by simulation and experiment that the method performs well in a variety of configurations and has the potential to be used in fault detection for cables and feed networks.

Imaginary eigenvalues of Zakharov–Shabat problems with non-zero background

The focusing Zakharov–Shabat scattering problem on the infinite line with non-zero boundary conditions for the potential is studied, and sufficient conditions on the potential are identified to ensure that the problem admits only purely imaginary discrete eigenvalues. The results, which generalize previous work by Klaus and Shaw, are applicable to the study of solutions of the focusing nonlinear Schrödinger equation with non-zero background.

Rogue periodic waves of the focusing nonlinear Schrödinger equation.

Rogue periodic waves stand for rogue waves on a periodic background. The nonlinear Schrödinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine's breather). The magnification factor of the rogue periodic waves is computed as a function of the elliptic modulus. Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.

Exact solution of the Zakharov–Shabat scattering problem for doubly-truncated multisoliton potentials

Recent studies have revealed that multisoliton solutions of the nonlinear Schrödinger equation, as carriers of information, offer a promising solution to the problem of nonlinear signal distortions in fiber optic channels. In any nonlinear Fourier transform based transmission methodology seeking to modulate the discrete spectrum of the multisolitons, choice of an appropriate windowing function is an important design issue on account of the unbounded support of such signals. Here, we consider the rectangle function as the windowing function for the multisolitonic signal and provide a recipe for computing the exact solution of the associated Zakharov–Shabat (ZS) scattering problem for the windowed/doubly-truncated multisoliton potential. The idea consists in expressing the Jost solution of the doubly-truncated multisoliton potential in terms of the Jost solution of the original potential. The proposed method allows us to avoid prohibitive numerical computations normally required in order to accurately quantify the effect of time-domain windowing on the nonlinear Fourier spectrum of the multisolitonic signals. Further, the method devised in this work also applies to general type of signals admissible as ZS scattering potential, and, may prove to be a useful tool in the theoretical analysis of such systems.

Fast inverse nonlinear Fourier transformation using exponential one-step methods: Darboux transformation.

This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU(2) nonlinear Fourier transformation (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transformation are quite well established in the Ablowitz-Kaup-Newell-Segur formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present a unified framework for the forward and inverse NFT using exponential one-step methods which are amenable to FFT-based fast polynomial arithmetic. Within this discrete framework, we propose a fast Darboux transformation (FDT) algorithm having an operational complexity of O(KN+Nlog^{2}N) such that the error in the computed N-samples of the K-soliton vanishes as O(N^{-p}) where p is the order of convergence of the underlying one-step method. For fixed N, this algorithm outperforms the classical DT (CDT) algorithm which has a complexity of O(K^{2}N). We further present an extension of these algorithms to the general version of DT which allows one to add solitons to arbitrary profiles that are admissible as scattering potentials in the ZS problem. The general CDT and FDT algorithms have the same operational complexity as that of the K-soliton case and the order of convergence matches that of the underlying one-step method. A comparative study of these algorithms is presented through exhaustive numerical tests.

Nonlinear Compensation in Optical Communications Systems With Normal Dispersion Fibers Using the Nonlinear Fourier Transform

We investigate the computational cost of the nonlinear Fourier transform (NFT) based on the Zakharov–Shabat scattering problem as a nonlinear compensation technique for quadrature-phase-shift keyed (QPSK) signals with raised cosine frequency characteristic in optical fiber transmission systems with normal dispersion fibers. We show that the primary sources of computational errors that arise from the use of the NFT is the finite eigenvalue resolution of the left and the right reflection spectra. We show that this effect and, consequently, the computational cost of the NFT as a nonlinear mitigation technique in the normal dispersion regime increases exponentially or faster with both the channel power and the number of symbols per data frame even using the most efficient NFT algorithms that are currently known. We find that the computational cost of this approach becomes unacceptably large at data frame lengths and powers that are too small for this approach to be competitive with standard transmission methods. We explain the physical reasons for these limits.

Scattering data computation for the Zakharov–Shabat system with non-smooth potentials

In this paper we present a variant of the method for the scattering data computation for the Zakharov–Shabat system, recently proposed by the authors. The algorithm that characterizes this variant allows us to compute the scattering data also in the presence of jump discontinuities of the initial potential.

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel–Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel–Darboux kernels are shown to be quasi-determinants—as well as Schur complements—of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov–Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss–Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted τ-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in RD, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference–differential equations that involve only one site in the integrable lattice behaving as a Kadomstev–Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

Almost structure-preserving analysis for weakly linear damping nonlinear Schrödinger equation with periodic perturbation

Exploring the dynamic behaviors of the damping nonlinear Schrödinger equation (NLSE) with periodic perturbation is a challenge in the field of nonlinear science, because the numerical approaches available for damping-driven dynamic systems may exhibit the artificial dissipation in different degree. In this paper, based on the generalized multi-symplectic idea, the local energy/momentum loss expressions as well as the approximate symmetric form of the linearly damping NLSE with periodic perturbation are deduced firstly. And then, the local energy/momentum losses are separated from the simulation results of the NLSE with small linear damping rate less than the threshold to insure structure-preserving properties of the scheme. Finally, the breakup process of the multisoliton state is simulated and the bifurcation of the discrete eigenvalues of the associated Zakharov–Shabat spectral problem is obtained to investigate the variation of the velocity as well as the amplitude of the solitons during the splitting process.

Efficient numerical method for solving the direct Zakharov–Shabat scattering problem

A new method to solve the direct Zakharov–Shabat scattering problem is proposed based on the solution for coupled Gel’fand–Levitan–Marchenko integral equations. Speedup of computations is achieved by using the Töplitz symmetry of the matrix and an “inner bordering” procedure. The new algorithm was tested on the exactly solvable potential. It is shown that the suggested algorithm significantly surpasses the traditional transfer matrix method in efficiency.

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szegő polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szegő polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.

The Wentzel-Kramers-Brillouin approximation of semiclassical eigenvalues of the Zakharov–Shabat problem

The WKB approximation of semiclassical eigenvalues of the non-self-adjoint Zakharov–Shabat problem is a standard element in the theory of the integrable focusing nonlinear Schrödinger equation; this approximation is based on a reformulation of the eigenvalue problem as the eigenvalue problem for a self-adjoint Schrödinger operator with a correction term that depends on the spectral parameter. The approximation results from neglecting the correction term. We perform a numerical experiment which gives new evidence to support the validity of this procedure in the context of the semiclassical limit problem for the focusing nonlinear Schrödinger equation. In particular, our results suggest that the rate of convergence of the approximate eigenvalues to the true ones is of the order of the square of the small parameter. This information is relevant to the task of rigorously incorporating this approximation into the asymptotic analysis of the singular limit for the focusing nonlinear Schrödinger equation.

Eigenvalue problems, spectral parameter power series, and modern applications

In the present review, we deal with the recently introduced method of spectral parameter power series (SPPS) and show how its application leads to an explicit form of the characteristic equation for different eigenvalue problems involving Sturm–Liouville equations with variable coefficients. We consider Sturm–Liouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and Floquet–Bloch solutions; quantum‐mechanical spectral and transmission problems as well as the eigenvalue problems for the Zakharov–Shabat system. In all these cases, we obtain a characteristic equation of the problem, which in fact reduces to finding zeros of an analytic function given by its Taylor series. We illustrate the application of the method with several numerical examples, which show that at present, the SPPS method is the easiest in the implementation, the most accurate, and efficient. We emphasize that the SPPS method is not a purely numerical technique. It gives an analytical representation both for the solution and for the characteristic equation of the problem. This representation can be approximated by different numerical techniques and leads to a powerful numerical method, but most important, it offers a different insight into the spectral and transmission problems. Copyright © 2014 John Wiley & Sons, Ltd.

AMBROSE–SINGER THEOREM ON DIFFEOLOGICAL BUNDLES AND COMPLETE INTEGRABILITY OF THE KP EQUATION

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose–Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math.128 (2004) 513–529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal q-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov–Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev–Petviashvili equation Adv. Math.54 (1984) 57–66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose–Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Difféologie du fibré d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc.28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513–529] is completed.