Zakharov-Shabat Problem Research Articles

Introducing phase jump tracking - a fast method for eigenvalue evaluation of the direct Zakharov-Shabat problem

We propose a new method for finding discrete eigenvalues for the direct Zakharov-Shabat problem, based on moving in the complex plane along the argument jumps of the function a(ζ), the localization of which does not require great accuracy. It allows to find all discrete eigenvalues taking into account their multiplicity faster than matrix methods and contour integrals. The method shows significant advantage over other methods when calculating a large discrete spectrum, both in speed and accuracy.

Conservative multi-exponential scheme for solving the direct Zakharov-Shabat scattering problem.

The direct Zakharov-Shabat scattering problem has recently gained significant attention in various applications of fiber optics. The development of accurate and fast algorithms with low computational complexity to solve the Zakharov-Shabat problem (ZSP) remains an urgent problem in optics. In this Letter, a fourth-order multi-exponential scheme is proposed for the Zakharov-Shabat system. The construction of the scheme is based on a fourth-order three-exponential scheme and Suzuki factorization. This allows one to apply the fast algorithms with low complexity to calculate the ZSP for a large number of spectral parameters. The scheme conserves the quadratic invariant for real spectral parameters, which is important for various telecommunication problems related to information coding.

Преобразование собственных значений задачи Захарова–Шабата при столкновении солитонов

Background and Objectives: The Zakharov–Shabat spectral problem allows to find soliton solutions of the nonlinear Schrodinger equation. Solving the Zakharov–Shabat problem gives both a discrete set of eigenvalues λj and a continuous one. Each discrete eigenvalue corresponds to an individual soliton with the real part Re(λj) providing the soliton velocity and the imaginary part Im(λj) determining the soliton amplitude. Solitons can be used in optical communication lines to compensate both non-linearity and dispersion. However, a direct use of solitons in return-to-zero signal encoding is inhibited. The interaction between solitions leads to the loss of transmitted data. The problem of soliton interaction can be solved using eigenvalues. The latter do not change when the solitons obey the nonlinear Schrodinger equation. Eigenvalue communication was realized recently using electronic signal processing. To increase the transmission speed the all-optical method for controlling eigenvalues should be developed. The presented research is useful to develop optical methods for the transformation of the eigenvalues. The purpose of the current paper is twofold. First, we intend to clarify the issue of whether the dispersion perturbation can not only split a bound soliton state but join solitons into a short oscillating period breather. The second goal of the paper is to describe the complicated dynamics and mutual interaction of complex eigenvalues of the Zakharov–Shabat spectral problem. Materials and Methods: Pulse propagation in single-mode optical fibers with a variable core diameter can be described using the nonlinear Schrödinger equation (NLSE) which coefficients depends on the evolution coordinate. The NLSE with the variable dispersion coefficient was considered. The dispersion coefficient was described using a hyperbolic tangent function. The NLSE and the Zakharov– Shabat spectral problem were solved using the split-step method and the layer-peeling method, respectively. Results: The results of numerical analysis of the modification of soliton pulses under the effect of variable dispersion coefficient are presented. The main attention is paid to the process of transformation of eigenvalues of the Zakharov–Shabat problem. Collision of two in-phase solitons, which are characterized by two complex eigenvalues is considered. When the coefficients of the nonlinear Schrodinger equation change, the collision of the solitons becomes inelastic. The inelastic collision is characterized by the change of the eigenvalues. It is shown that the variation of the coefficients of the NLSE allows to control both real and imaginary parts of the eigenvalues. Two scenarios for the change of the eigenvalues were identified. The first scenario is characterized by preserving the zero real part of the eigenvalues. The second one is characterized by the equality of their imaginary parts. The transformation of eigenvalues is most effective at the distance where the field spectrum possesses a two-lobe shape. Variation of the NLSE coefficient can introduce splitting or joining of colliding soliton pulses. Conclusion: The presented results show that the eigenvalues can be changed only with a small variation of the NLSE coefficients. On the one hand, a change in the eigenvalues under the effect of inelastic soliton collision is an undesirable effect since the inelastic collision of solitons will lead to unaccounted modulation in soliton optical communication links. On the other hand, the dependence of the eigenvalues on the parameters of the colliding solitons allows to modulate the eigenvalues using all-fiber optical devices. Currently, the modulation of the eigenvalues is organized using electronic devices. Therefore, the transmission of information is limited to nanosecond pulses. For picosecond pulse communication, the development of all-optical modulation methods is required. The presented results will be useful in the development of methods for controlling optical solitons and soliton states of the Bose–Einstein condensate.

Exponential fourth order schemes for direct Zakharov-Shabat problem.

Nowadays, improving the accuracy of computational methods to solve the initial value problem of the Zakharov-Shabat system remains an urgent problem in optics. In particular, increasing the approximation order of the methods is important, especially in problems where it is necessary to analyze the structure of complex waveforms. In this work, we propose two finite-difference algorithms of fourth order of approximation in the time variable. Both schemes have the exponential form and conserve the quadratic invariant of Zakharov-Shabat system. The second scheme allows applying fast algorithms with low computational complexity (fast nonlinear Fourier transform).

Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem.

We propose a finite-difference algorithm for solving the initial problem for the Zakharov-Shabat system. This method has the fourth order of accuracy and represents a generalization of the second-order Boffetta-Osborne scheme. Our method permits the Zakharov-Shabat spectral problem to be solved more effectively for continuous and discrete spectra.

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.

Contour integrals for numerical computation of discrete eigenvalues in the Zakharov-Shabat problem.

We propose a novel algorithm for the numerical computation of discrete eigenvalues in the Zakharov-Shabat problem. Our approach is based on contour integrals of the nonlinear Fourier spectrum function in the complex plane of the spectral parameter. The reliability and performance of the new approach are examined in application to a single eigenvalue, multiple eigenvalues, and the degenerate breather's multiple eigenvalue. We also study the impact of additive white Gaussian noise on the stability of numerical eigenvalues computation.

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.

Inverse scattering transform for the Degasperis–Procesi equation

We develop the inverse scattering transform (IST) method for the Degasperis–Procesi equation. The spectral problem is an Zakharov–Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST, such as the construction of fundamental analytic solutions, the formulation of a Riemann–Hilbert problem, and the implementation of the dressing method, are presented.

On the stability of N-solitons in integrable systems

The dynamical stability of reflectionless N-solitons for a large class of integrable systems is considered. The underlying eigenvalue problem is the Zakharov–Shabat problem on for any r ≥ 1. Physical examples of interest include the vector nonlinear Schrödinger equation and the integrable (r + 1)-wave interaction problem. It is shown herein that under appropriate assumptions that these solitons are realized as a local minimum of an appropriate Lyapunov function, and are hence dynamically stable.

Elementary and binary Darboux transformations at rings

For a given idempotent p and some element σ from a differential associative ring, we introduce gauge transformation λp + σ with the spectral parameter λ that leaves some linear operators form invariant. The explicit form of the σ is derived for the generalized Zakharov-Shabat problem. The maps that factorize Darboux transformations are referred to as elementary ones. Binary transformations that correspond to the iteration of elementary maps with the special choice of solutions of the direct and conjugate problems are introduced and widen the potential reductions set. We show an infinitesimal version of iterated transforms. The spectral operator and soliton theories applications are outlined. The nonabelian N-wave interaction equation modification with additional linear terms, its zero curvature representation, soliton solutions, and their stability with respect to infinitesimal deformations are studied.

Intertwine operators and elementary darboux transforms in differential rings and modules

This paper presents an algebraic structure allowing to construct an elementary Darboux transformation. The set of necessary propositions is listed and the generic Darboux theorem is formulated for differential rings. The result is given for the generalized Zakharov-Shabat problem. The conditions for the appearance of the Schlesinger transformation and a construction of an analogue of supersymmetry algebra are shown starting from the intertwine relation. Some applications in the theory of nonlinear equations are also discussed.

Soliton solutions and gauge equivalence for the problem of Zakharov–Shabat and its generalizations

In a paper by Takhtadjan and Faddeev [Hamiltonian approach to the soliton theory (in Russian) (Nauka, Moskov, 1986)] the N-soliton solutions, related to the nonlinear Schrödinger equation (NSE), are given. A generalization of this approach allows us to apply it not only to the NSE, but to the whole hierarchy of the Zakharov–Shabat problem, to the quadratic bundle problem, and to the ones gauge equivalent to them [where one can find, for example, the Heisenberg ferromagnet equation, the relativistic Mikhailov model (which, in appropriate reduction, is equivalent to the massive Thirring model), the derivative nonlinear Schrödinger equation (which is equivalent to the derivative Landau–Lifshitz equation), etc.]. We have used an appropriate reduction, giving us interesting, from a physical point of view, results. Thus we manage to obtain the soliton solutions for the whole hierarchy of the quadratic bundle problem (free and under reduction) and for the ones gauge equivalent to it. This result was first announced in previous articles by Vaklev, but here we manage to solve the determinants given there and to present the searched result as simple fractions of products of numerical differences.

Using the complex WKB method for studying the evolution of initial pulses obeying the nonlinear Schr�dinger equation

Using the complex WKB method, we found an asymptotic solution of the associated Zakharov-Shabat problem in the limit of a small coefficient h → 0 multiplying the derivative of the potential that has a single hump. The obtained formulas can be used in describing the evolution of optical pulses of such shape obeying the nonlinear Schrodinger equation (NSE). Several examples are considered.

Effect of initial pulse shape modulation on spontaneous soliton formation in the NSE model

An initial pulse with fairly steep fronts whose evolution is described by the nonlinear Schrodinger equation, splits into soliton-like pulses (spontaneous soliton formation). The number of solitons formed in this process can be estimated by the number of spectral points of the associated linear Zakharov-Shabat problem for the initial pulse. Exact solutions of the Zakharov-Shabat problem are constructed for some classes of initial piecewise-continuous pulses by using the Darboux method. This allows us to estimate the effect of the shape of the initial pulse on the number of formed solitions and their parameters.

Darboux transforms for Davey-Stewartson equations and solitons in multidimensions

The algebraic structure of the set of solutions for the Davey-Stewartson equations is investigated. Simple and binary Darboux transforms are constructed. It is shown that the set of Darboux transforms for n auxiliary solutions of the corresponding linear Zakharov-Shabat problem comprise an Abelian group of order 2n. The Darboux technique allows nonlinear superposition formulas and dromion solutions to be obtained as in the 2+1 inverse scattering transform method. New solutions periodic in space and time for DS1 and new solutions exponentially decaying and oscillating in all directions except two (with discrete set of points on them) for DS2 are constructed. These solutions are analogues of dromions of DS1. The authors call these solutions, constructed for DS2, degenerate exultons.

Fast Exact Inversion of the Generalized Zakharov–Shabat Problem for Rational Scattering Data: Application to the Synthesis of Optical Couplers

An exact method is presented for the inversion of the generalized Zakharov–Shabat problem, for the case where the scattering data are (or are approximated by) rational functions. The presence of bound states is also accounted for. The problem is examined with the aim of obtaining an exact synthesis procedure for a class of two-mode optical couplers, including lossy and active structures. The present method yields a closed-form solution, thereby leading to very fast numerical implementation. The implications of the synthesis technique on the physical problem at hand are briefly discussed, and some numerical examples are presented.

On the two-dimensional Zakharov-Shabat problem

On the two-dimensional Zakharov-Shabat problem

The Moving Modified KdV Solitons In Polyacetylene

Abstract The fact is explored that the continuum model of polyacetylene is isomorphic to the well-known Zakharov-Shabat problem of 1-d scattering. It is shown that the gap parameter δ(x,=t) may be viewed as some solution to the modified Korteweg-de Vries equation.

Expansions over the ‘‘squared’’ solutions and difference evolution equations

The completeness relation for the system of ‘‘squared’’ solutions of the discrete analog of the Zakharov–Shabat problem is derived. It allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results. These include: (i) the expansion of the potential and its variations over the system of ‘‘squared’’ solutions, the expansion coefficients being the scattering data and their variations, respectively; thus the interpretation of the inverse scattering transform (IST) as a generalized Fourier transform becomes obvious; (ii) compact expressions for the trace identities through the operator Λ, for which the ‘‘squared’’ solutions are eigenfunctions; (iii) brief exposition of the spectral theory of the operator Λ; (iv) direct calculation of the action-angle variables based on the symplectic form of the completeness relation; (v) the generating functional of the M operators in the Lax representation; (vi) the quantum version of the IST.