Scalar Nonlinear Schrodinger Equation Research Articles

Nonlinear Waves and Coherent Structures in Laser Induced Plasmas and Polarized Vacuum

Starting from the Vlasov–Maxwell equations describing the dynamics of various species in a free quasi-neutral plasma, an exact relativistic hydrodynamic closure for a special type of water-bag distributions satisfying the Vlasov equation has been derived.It has been shown that the set of equations for the macroscopic hydrodynamic variables coupled to the wave equations for the self-consistent electromagnetic field is equivalent to the Vlasov–Maxwell system. In the case of magnetized quasi-neutral plasma, the hydrodynamic substitution has been used to derive the hydrodynamic equations for the plasma density and current velocity, coupled to the wave equations for the self-consistent electromagnetic fields. Based on the method of multiple scales, a system comprising a vector nonlinear Schrodinger equation for the transverse envelopes of the self-consistent plasma wakefield, coupled to a scalar nonlinear Schrodinger equation for the electron current velocity envelope for free plasma, has been derived. In the case of magnetized plasma, it has been shown that the whistler wave envelopes of the three basic modes satisfy a system of three coupled nonlinear Schrodinger equations. Numerical examples for typical plasma parameters have been presented, which demonstrate the relevance of the results thus obtained to the so-called shock laser-plasma acceleration. In addition, it has been shown that in the case of magnetized plasma, the whistler waves facilitate the transverse confinement considerably. The effect of the nonlinear corrections to the electric displacement vector and the intensity of the magnetic field in nonlinear electrodynamics of Heisenberg-Euler type has been studied. It has been shown that the slowly varying wave envelope in free polarized vacuum satisfies a nonlinear wave equation. In the case, where an external constant magnetic field is applied, the single nonlinear wave equation should be replaced by a system of two coupled nonlinear equations for the two independent wave polarizations. In both cases a novel effect of reduction of the speed of light can be observed, which is due to the vacuum polarization.

On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations

We extend to a specific class of systems of nonlinear Schrodinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative.

Nonlinear waves and coherent structures in quasi-neutral plasmas excited by external electromagnetic radiation

Starting from the Vlasov-Maxwell equations describing the dynamics of various species in a quasi-neutral plasma, an exact relativistic hydrodynamic closure for a special type of water-bag distribution satisfying the Vlasov equation has been derived. It has been shown that the set of equations for the macroscopic hydrodynamic variables coupled to the wave equations for the self-consistent electromagnetic field is fully equivalent to the Vlasov-Maxwell system. Based on the method of multiple scales, a system comprising a vector nonlinear Schrodinger equation for the transverse envelopes of the self-consistent plasma wakefield, coupled to a scalar nonlinear Schrodinger equation for the electron current velocity envelope, has been derived. Using the method of formal series of Dubois-Violette, a traveling wave solution of the derived set of coupled nonlinear Schrodinger equations in the case of circular wave polarization has been obtained. This solution is represented as a ratio of two formal Volterra series. The terms of these series can be calculated explicitly to every desired order.

Leech Lattice Extension of the Non-Linear Schrodinger Equation Theory of Einstein Spaces

Although the scalar nonlinear Schrodinger equation has provided valuable insights into how quantum mechanics might modify the classical general relativistic description of space-time, a detailed understanding of space-times with matter has remained elusive. In this paper, we propose generalizing the nonlinear Schrodinger equation theory of Einstein spaces to include matter by transplanting the 3 + 1 dimensional theory to the 24-dimensional Leech lattice plus 1 time dimension. The scalar wave function and Chern-Simons gauge potential which encode the classical Kahler potential become 11 × 11 complex matrices belonging to a 195,442 dimensional representation of the Mathieu group M11. This theory describes gravity coupled to internal degrees of freedom which include a supersymmetric E6 × E6 Yang-Mills theory of matter.

Dynamic of HNLS Solitons using Compact Split Step Padé Scheme

In this communication, we use a compact split step Pade scheme (CSSPS) to solve the scalar higher-order nonlinear Schrodinger equation (HNLS) with higher-order linear and nonlinear effects. The second part, consisting of two sections, the first section is dedicated to the study numerically the stabilization of high-order solitons dynamic in optical fibers by compensation or by the interplay of higher order nonlinearity – especially quintic nonlinearity- and the self-steepening. In the second section we study also numerically the propagation of conventional chirped or unchirped solitons in optical fibers with to the management of the nonlinearity, dispersion and loss (gain).

Branching behavior of field dynamics of resonators of left-handed materials

In this paper, we study the spatiotemporal dynamics of left-handed materials that contain nonlinear cavities. These materials, having a negative refractive index, exhibit a nonlinear electromagnetic behavior. The insertion of the left-handed material into an optical cavity results into a variety of branching behavior of optical oscillators that are degenerate. The optical Kerr cavity considered in this work contains a cubic nonlinearity. The objective of this work was to investigate how branching behavior can yield useful information about the relative amount of left-handed and right-handed material controlling the diffraction in cavity. We will apply the reductive perturbation method-based multi-equation bifurcation theory to a scalar nonlinear Schrodinger equation and show how period-doubling occurs during optical wave propagation.

Thermalization of incoherent nonlinear waves

The subject of incoherent nonlinear optics received a renewed interest since the first experimental demonstration of incoherent solitons in slowly responding photorefractive crystals. Several theories have been successfully developed to provide a detailed description of the dynamics of incoherent optical solitons. However, such theories leave unanswered the question regarding the long term evolution of a partially incoherent optical field propagating in a nonlinear medium. In analogy with kinetic gas theory, an incoherent optical field evolves, owing to nonlinearity, towards a thermodynamic equilibrium state. Wave turbulence theory describes the essential properties of this irreversible process of thermal wave relaxation to equilibrium. Such irreversible behavior is expressed by a H-theorem of entropy growth, whose origin is analogous to the celebrated Boltzmann’s H-theorem. In this review we analyze the thermalization of incoherent nonlinear optical fields in various circumstances. We shall begin with the simplest case where the optical field is ruled by the scalar NonLinear Schrodinger (NLS) equation. In this case wave thermalization is characterized by a condensation process of the optical field, whose thermodynamic properties are analogous to those of Bose-Einstein condensation, despite the fact that the considered optical wave is completely classical. We then study the thermalization of an ensemble of incoherent fields governed by the vector NLS equation. The influence of the relative intensity of the coupled fields reveals the existence of a process of coherence absorption: The condensation of the optical field is induced by the presence of another small-amplitude field, which absorbs almost all the noise of the system. Such a coherence absorption effect is shown to also occur in thermal quantum Bose gases. The influence of convection (group-velocity difference) on the thermalization process is also analyzed. A set of incoherent wave-packets is shown to irreversibly evolve towards an equilibrium state in which they propagate with an identical group-velocity. This velocity-locking effect has a thermodynamic origin and thus appears as a generic property of a system of incoherent nonlinear waves. The influence of a coherent coupling on wave thermalization leads to an unexpected process of spontaneous polarization of unpolarized incoherent light, without loss of energy. Finally, the influence of higher-order dispersion effects on optical thermalization is responsible for a dramatic spectral broadening phenomenon known as supercontinuum generation. This reveals that supercontinuum generation may be regarded as a consequence of the natural thermalization of an optical field to equilibrium. We finally show that the thermodynamic properties of incoherent nonlinear waves may be analyzed by means of the fundamental T dS equation of thermodynamics.

Instability of Standing Waves for a System of Nonlinear Schroedinger Equations with Three-Wave Interaction

We consider a three-component system of nonlinear Schrodinger equations related to the Raman amplification in a plasma. In dimension N ≤ 3, we study the orbital instability of standing wave solution of the form (0,0,e2iωtψ), where ψ is a ground state of scalar nonlinear Schrodinger equation. Using time derivative instead of space derivatives to estimate nonlinear terms, we improve an instability result in our previous paper [4], and also give a simpler proof.

Numerical modeling of (D+1)-dimensional solitons in a sign-alternating nonlinear medium with an adaptive fast Hankel split-step method

We present a novel technique to numerically solve transverse and pulsed optical beam or light bullet propagation in a layered alternating self-focusing and self-defocusing medium based on the scalar nonlinear Schrodinger equation in two and three dimensions with cylindrical and spherical symmetry, respectively. Using fast algorithms for Hankel transform along with adaptive longitudinal stepping and transverse grid management in a symmetrized split-step technique, it is possible to accurately study many nonlinear effects, including the possibility of spatiotemporal collapse, or the collapse-arresting mechanism due to a sign-alternating nonlinearity coefficient. Also, by using the variational approximation technique, we can prove that stable (D+1)-dimensional soliton beams and optical bullets exist in these media.

Interaction of nonlinearity and polarization mode dispersion

The derivation of the coupled nonlinear Schrodinger equation and the Manakov-PMD equation is reviewed. It is shown that the usual scalar nonlinear Schrodinger equation can be derived from the Manakov-PMD equation when polarization mode dispersion is negligible and the signal is initially in a single polarization state as a function of time. Applications of the Manakov-PMD equation to studies of the interaction of the Kerr nonlinearity with polarization mode dispersion are then discussed.

On the susceptibility of bright nonlinear Schrödinger solitons to long–wave transverse instability

A new theory for transverse instability of bright solitons of equations of nonlinear Schrodinger (NLS) type is presented, based on a natural deformation of the solitons into a four-parameter family. This deformation induces a set of four diagnostic functionals which encode information about transverse instability. These functionals include the deformed power, the deformed momentum and two new functionals. The main result is that a sufficient condition for long-wave transverse instability is completely determined by these functionals. Whereas longitudinal instability is determined by a single partial derivative (the Vakhitov-Kolokolov criterion), the condition for transverse instability requires 10 partial derivatives. The theory is illustrated by application to scalar NLS equations with general potential, and vector NLS equations for optical media with χ(2) nonlinearity.

Spatial modulation instability of coherent light in a weakly-relaxing Kerr medium

We report on a theoretical analysis of spatial modulation instability of coherent light propagating in a nonlinear medium with a noninstantaneous Kerr response. The latter is of a relaxing Debye type and originates from diffusive molecular reorientation. We consider the examples of harmonic and pulse-like perturbations superimposed either on a monochromatic temporal background or on a quasi-monochromatic Gaussian pulse. Our analysis reveals that the finite duration of the nonlinear response is responsible for spatiotemporal dynamic features that obviously do not exist within the framework of the usual scalar nonlinear Schrodinger equation, which models spatial modulation instability in an instantaneous Kerr medium.

Application of multiple-length-scale methods to the study of optical fiber transmission

It is natural to apply multiple-length-scale methods to the study of optical-fiber transmission because the key length scales span 13 orders of magnitude and cluster in three main groups. At the lowest scale, corresponding to micrometers, the full set of Maxwell's equations should be used. At the intermediate scale, corresponding to the range from one centimeter to tens of meters, the coupled nonlinear Schrodinger equation should be used. Finally, at the longest length scale, corresponding to the range from tens to thousands of kilometers, the Manakov-PMD equation should be used, and, when polarization mode dispersion can be neglected and the fiber gain and loss can be averaged out, one arrives at the scalar nonlinear Schrodinger equation. As an illustrative example of multiple-scale-length techniques, the nonlinear Schrodinger equation will be derived, carefully taking into account the actual length scales that are important in optical-fiber transmission.

Nonlinear propagation of electromagnetic pulses in two-level media under strong coupling

We consider the problem of the propagation of short-duration light pulses in a two-level medium in the regime of strong interaction, near the resonance, with no restriction on the strength of the coupling between field and medium. We prove that the local variations of the population difference created by the electric field actually induce a nonlinear effect by coupling the fundamental of the field Fourier component to its harmonics. As a result, the multiscale averaging limit of the Maxwell-Bloch system is a new system of coupled nonlinear Schrodinger equations for the slowly varying envelope of the electric field. This system has one integrable limit when the electric field is circularly polarized: it is the scalar nonlinear Schrodinger equation. Moreover, it possesses three constants of the motion and a Hamiltonian formulation. Then we prove that our system is nonintegrable as it does not pass the Painleve test. Finally, a stability analysis shows that it is modulationally unstable and thus it propagates steady pulses of coherent light.