Instabilities of Multihump Vector Solitons in Coupled Nonlinear Schrödinger Equations

Spectral stability of multihump vector solitons in the Hamiltonian system of coupled nonlinear Schrödinger (NLS) equations is investigated both analytically and numerically. Using the closure theorem for the negative index of the linearized Hamiltonian, we classify all possible bifurcations of unstable eigenvalues in the systems of coupled NLS equations with cubic and saturable nonlinearities. We also determine the eigenvalue spectrum numerically by the shooting method. In case of cubic nonlinearities, all multihump vector solitons in the nonintegrable model are found to be linearly unstable. In case of saturable nonlinearities, stable multihump vector solitons are found in certain parameter regions, and some errors in the literature are corrected.     

Evans Function for Lax Operators with Algebraically Decaying Potentials

We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singularities in the essential spectrum, which may include embedded eigenvalues with algebraically decaying eigenfunctions. We construct a renormalized Evans function and study bifurcations of embedded eigenvalues, when an algebraically decaying potential is perturbed by a generic potential with a faster decay at infinity. We show that the bifurcation problem for embedded eigenvalues can be reduced to cubic or quadratic equations, depending on whether the algebraic potential decays to zero or approaches a nonzero constant. Roots of the bifurcation equations define eigenvalues which correspond to nonlinear waves that are formed from unstable algebraic solitons. Our results provide precise information on the transformation of unstable algebraic solitons in the time-evolution problem associated with the integrable nonlinear equation. Algebraic solitons of the modified KdV equation are shown to transform to either travelling solitons or time-periodic breathers, depending on the sign of the perturbation. Algebraic solitons of the derivative NLS and massive Thirring equations are shown to transform to travelling and rotating solitons for either sign of the perturbation. Finally, algebraic homoclinic orbits of the focusing NLS equation are destroyed by the perturbation and evolve into time-periodic space-decaying solutions.     

Vector Solitons and Their Internal Oscilliations in Birefringent Nonlinear Optical Fibers

In this article, the vector solitons in birefringent nonlinear optical fibers are studied first. Special attention is given to the single-hump vector solitons due to evidences that only they are stable. Questions such as the existence, uniqueness, and total number of these solitons are addressed. It is found that the total number of them is continuously infinite and their polarizations can be arbitrary. Next, the internal oscillations of these vector solitons are investigated by the linearization method. Discrete eigenmodes of the linearized equations are identified. Such modes cause to the vector solitons a kind of permanent internal oscillations, which visually appear to be a combination of translational and width oscillations in the A and B pulses. The numerically observed radiation shelf at the tails of interacting pulses is also explained. Finally, the asymptotic states of the perturbed vector solitons are studied within both the linear and nonlinear theory. It is found that the state of internal oscillations of a vector soliton is always unstable. It invariably emits energy radiation and eventually evolves into a single-hump vector soliton state.     

Multi-component vortex solutions in symmetric coupled nonlinear Schrödinger equations

A Hamiltonian system of incoherently coupled nonlinear Schrödinger (NLS) equations is considered in the context of physical experiments in photorefractive crystals and Bose-Einstein condensates. Due to the incoherent coupling, the Hamiltonian system has a group of various symmetries that include symmetries with respect to gauge transformations and polarization rotations. We show that the group of rotational symmetries generates a large family of vortex solutions that generalize scalar vortices, vortex pairs with either double or hidden charge, and coupled states between solitons and vortices. Novel families of vortices with different frequencies and vortices with different charges at the same component are constructed and their linearized stability problem is block-diagonalized for numerical analysis of unstable eigenvalues.     

Discrete Vector Solitons: Composite Solitons, Yang–Baxter Maps and Computation

Collisions of solitons for an integrable discretization of the coupled nonlinear Schrödinger equation are investigated. By a generalization of Manakov's well-known formulas for the polarization shift of interacting vector solitons, it is shown that the multisoliton interaction process is equivalent to a sequence pairwise interactions and, moreover, the net result of the interaction is independent of the order in which such collisions occur. Further, the order-invariance is shown to be related to the fact that the map that determines the interaction of two such solitons satisfies the Yang–Baxter relation. The associated matrix factorization problem is discussed in detail and the notion of fundamental and composite solitons is elucidated. Moreover, it is shown that, in analogy with the continuous case, collisions of fundamental solitons can be described by explicit fractional linear transformations of a complex-valued scalar polarization state. Because the parameters controlling the energy switching between the two components exhibit nontrivial information transformation, they can, in principle, be used to implement logic operations.     

On the formation of gap solitons in nonlinear electromagnetic metamaterials

A nonlinear (Kerr‐type) electromagnetic metamaterial, characterized by a double‐Lorentz model of its frequency‐dependent linear effective dielectric permittivity and magnetic permeability, is considered. The formation of gap solitons in the low‐ and high‐frequency band gaps of this metamaterial is investigated analytically. Evolution equations governing the gap solitons, of the form of a nonlinear Klein‐Gordon and a nonlinear Schrödinger equation, are obtained, and the structure of their solutions is discussed.     

Topological 1-soliton solution of the nonlinear Schrodinger’s equation with Kerr law nonlinearity in 1 + 2 dimensions

In this paper, the topological 1-soliton solution of the nonlinear Schrödinger’s equation in 1 + 2 dimensions is obtained by the solitary wave ansatze method. These topological solitons are studied in the context of dark optical solitons. The type of nonlinearity that is considered is Kerr type.     

Neumann Control of Unstable Parabolic Systems: Numerical Approach

The present article is concerned with the Neumann control of systems modeled by scalar or vector parabolic equations of reaction-advection-diffusion type with a particular emphasis on systems which are unstable if uncontrolled. To solve these problems, we use a combination of finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate gradient algorithms for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of test problems in two dimensions, including problems related to nonlinear models.     

Soliton solutions via auxiliary function method for a coherently-coupled model in the optical fiber communications

A coherently-coupled nonlinear Schrödinger system in the optical fiber communications, with the mixed self-phase modulation (SPM), cross-phase modulation (XPM) and positive coherent coupling terms, is studied through the bilinear method with an auxiliary function. Solutions for that system are found to be of two types: singular and non-singular ones, and the latter appear as the soliton-typed. Vector bright one- and two-solitons are derived with the corresponding phase-shift parameter constraints. In virtue of computerized symbolic computation and asymptotic behavior analysis, elastic collision mechanisms of such vector solitons are investigated. With the aid of graphical simulation, vector solitons are displayed to be of the single- or double-hump profiles. The formation and collision mechanisms of the vector bright solitons for that system are generated based on the combined effects of SPM, XPM and coherent coupling. Only elastic collisions of the vector solitons occur for that system, which is a distinctive feature amid those of other coherently-coupled nonlinear Schrödinger systems.     

Optical solitons with non-Kerr law nonlinearity and inter-modal dispersion with time-dependent coefficients

This paper studies optical solitons with non-Kerr law nonlinearity, in the presence of inter-modal dispersion. The coefficients of group velocity dispersion, nonlinearity and inter-modal dispersion terms have time-dependent coefficients. The types of nonlinearity that are considered are Kerr, power, parabolic and dual-power laws. The solitary wave ansatz is used to carry out the integration of the governing nonlinear Schrödinger’s equation with time-dependent coefficients. Both, bright and dark optical solitons, are considered, in this paper. Finally, numerical simulations are also given in each of these cases. The only necessary condition for these solitons to exist is that these time-dependent coefficients of group velocity dispersion and inter-modal dispersion are Riemann integrable.     

Dark solitons in external potentials

We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.      

Dark solitons in external potentials

We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.     

Nonlinearity management in optics

We study nonlinearity management in optics by investigating the propagation of localized pulses and plane waves in a layered, cubically nonlinear (Kerr) medium that consists of alternating layers of glass and air. We show that such nonlinearity management delays the blow-up/collapse of pulses and leads to a band structure of modulationally unstable regions for plane waves. We find excellent agreement between experiments, numerical simulations, and theory. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of topological optical solitons with time-dependent dispersion,nonlinearity and attenuation

In this paper, the dark or topological optical 1-soliton solution of the nonlinear Schrödinger’s equation is obtained. The time-dependent coefficients of the group velocity dispersion, Kerr nonlinearity and the attenuation terms are considered. This leads to the constraint relation between these coefficients for the topological solitons to exist. All what is necessary is that these time-dependent coefficients be simply Riemann integrable.     

Dynamical behavior of the bright incoherent spatial solitons in self-defocusing nonlinear media

We present a theory of bright incoherent photovoltaic (PV) solitons in self-defocusing nonlinear media by using a coherent density approach. It is shown that this theory model is effectively governed by an infinite set of coupled nonlinear Schrödinger equations, which are initially weighted with respect to the incoherent angular power spectrum of source. We then numerically study the particular case of spatially incoherent beam propagating in LiNbO₃:Fe crystal with split-step Fourier method. Numerical simulations indicate that the ratio of PV constant κ is a key parameter to spatial compression as well as the possible dark and bright PV solitons. Besides, the formation of bright incoherent PV solitons is affected by intensity ratios r_T and width of the source angular power spectrum θ₀. Better coherent property is found at margins of bright incoherent soliton through the associated coherence length calculation. These results are in good agreement with recent experimental observations.     

向量优化问题的一类非线性标量化定理

利用Gertewitz泛函研究向量优化问题的一类非线性标量化问题. 证明了向量优化问题的(C, \varepsilon)-弱有效解或(C, \varepsilon)-有效解与标量化问题的近似解或严格近似解间的等价关系, 并估计了标量化问题的近似解．     

Two-dimensional solitons in irregular lattice systems

We compute and study localized nonlinear modes (solitons) in the semi-infinite gap of the focusing two-dimensional nonlinear Schrödinger (NLS) equation with various irregular lattice-type potentials. The potentials are characterized by large variations from periodicity, such as vacancy defects, edge dislocations, and a quasicrystal structure. We use a spectral fixed-point computational scheme to obtain the solitons. The eigenvalue dependence of the soliton power indicates parameter regions of self-focusing instability; we compare these results with direct numerical simulations of the NLS equation. We show that in the general case, solitons on local lattice maximums collapse. Furthermore, we show that the Nth-order quasicrystal solitons approach Bessel solitons in the large-N limit.     

A numerical method for calculating solitons of the nonlinear Schödinger equation in the axially symmetric case

A numerical method is proposed for determination of the eigenfunctions and eigenvalues of the nonlinear Schrödinger equation in the axially symmetric case. Optical solitons interpreted in the physical sense are found for various values of the nonlinearity coefficient by means of the developed method. As has previously been shown by other authors, such solitons are unstable under small perturbations of their shape. Since the considered problem finds numerous applications, methods providing for soliton stabilization are widely discussed in the literature. One of these methods involves strong modulation of the medium nonlinearity or even the reversal of the nonlinearity sign, which necessitates taking into account the wave reflected from irregularities and analyzing additionally the applicability of the mathematical model. We show that, theoretically, it is possible to stabilize a soliton via weak modulation of the cubic-nonlinearity coefficient. Such modulation ensures alternation of the length of nonlinear layers and enables one to increase the path length by a factor of 70 without a beam collapse.     

Bistable Solitons in Single- and Multichannel Waveguides with the Cubic-Quintic Nonlinearity

We consider spatial solitons in a channel waveguide or in a periodic array of rectangular potential wells (the Kronig-Penney (KP) model) in the presence of the uniform cubic-quintic (CQ) nonlinearity. Using the variational approximation and numerical methods, we. nd two branches of fundamental (single-humped) soliton solutions. The soliton characteristics, in the form of the integral power Q (or width w) vs. the propagation constant k, reveal a strong bistability with two different solutions found for a given k. Violating the known Vakhitov-Kolokolov criterion, the solution branches with dQ/dk > 0 and dQ/dk < 0 are simultaneously stable. Various families of higher-order solitons are also found in the KP version of the model: symmetric and antisymmetric double-humped solitons, three-peak solitons with and without the phase shift π between the peaks, etc. In a relatively shallow KP lattice, all the solitons belong to the semi-infinite gap beneath the linear band structure of the KP potential, while finite gaps between the bands remain empty (solitons in the finite gaps can be found if the lattice is much deeper). But in contrast to the situation known for the model combining a periodic potential and the self-focusing Kerr nonlinearity, the fundamental solitons fill only a finite zone near the top of the semi-infinite gap, which is a manifestation of the saturable character of the CQ nonlinearity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 324–335, August, 2005. An erratum to this article is available at .     

Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Three new iteration methods, namely the squared-operator method, the modified squared-operator method, and the power-conserving squared-operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast-converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared-operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher-gap vortex solitons in the two-dimensional nonlinear Schrödinger equations with periodic potentials, and isolated solitons in Ginzburg–Landau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared-operator method delivers the best performance among the methods proposed in this article.