Physics and mathematics of dispersion-managed optical solitons

We review the main physical and mathematical properties of dispersion-managed (DM) optical solitons. Theory of DM solitons can be presented at two levels of accuracy: first, simple, but nevertheless, quantitative models based on ordinary differential equations governing evolution of the soliton width and phase parameter (the so-called chirp); and second, a comprehensive path-average theory that is capable of describing in detail both the fine structure of DM soliton form and its evolution along the fiber line. An analogy between DM soliton and a macroscopic nonlinear quantum oscillator model is also discussed. To cite this article: S.K. Turitsyn et al., C. R. Physique 4 (2003).     

Nonlinear waves in a fluid-filled thin viscoelastic tube

In the present paper the propagation property of nonlinear waves in a thin viscoelastic tube filled with incom-pressible inviscid fluid is studied.The tube is considered to be made of an incompressible isotropic viscoelastic material described by Kelvin-Voigt model.Using the mass conservation and the momentum theorem of the fluid and radial dynamic equilibrium of an element of the tube wall,a set of nonlinear partial differential equations governing the prop-agation of nonlinear pressure wave in the solid-liquid coupled system is obtained.In the long-wave approximation the nonlinear far-field equations can be derived employing the reductive perturbation technique (RPT).Selecting the expo-nent α of the perturbation parameter in Gardner-Morikawa transformation according to the order of viscous coefficient η,three kinds of evolution equations with soliton solution,i.e.Korteweg-de Vries (KdV)-Burgers,KdV and Burgers equations are deduced.By means of the method of traveling-wave solution and numerical calculation,the propagation properties of solitary waves corresponding with these evolution equations are analysed in detail.Finally,as a example of practical application,the propagation of pressure pulses in large blood vessels is discussed.     

Backlünd transformation and multiple soliton solutions for the (3+1)-dimensional Nizhnik－Novikov－Veselov equation

We develop an approach to construct multiple soliton solutions of the (3+1)-dimensional nonlinear evolution equation. We take the (3+1)-dimensional Nizhnik－Novikov－Veselov (NNV) equation as an example. Using the extended homogeneous balance method, one can find a Backlünd transformation to decompose the (3+1)-dimensional NNV into a set of partial differential equations. Starting from these partial differential equations, some multiple soliton solutions for the (3+1)-dimensional NNV equation are obtained by introducing a class of formal solutions.     

Flames in narrow circular tubes

We examine an axi-symmetric deflagration located in a tube with thermally active walls. It is noted that the flame-in-tube configuration defines a classical edge-flame, a flame in a shear flow for which there is a water-shed solution for a critical value of the Damköhler number (D), ignition front solutions for larger values of D, and failure wave solutions for smaller values. We examine semi-infinite tubes with a heat flux imposed at the tube wall ends, to simulate experiments reported in the 30th Symposium. We identify parameters for which stable solutions are obtained at certain flow rates, but unstable solutions are generated at higher flow rates, followed by stable solutions at still higher flow rates. These solutions are consistent with the experimental record. Instability leads either to regular oscillations or to a violent process characterized by quenching and re-ignition.     

Matrix KP: tropical limit and Yang–Baxter maps

We study soliton solutions of matrix Kadomtsev–Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines. There is a subclass of “pure line soliton solutions” for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang–Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a nonlinear map in the case of a more general matrix KP equation. We also consider the corresponding Korteweg–deVries reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain an apparently new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter dependence of the vector KP R-matrix.

     

Waveguides induced by steady-state gray solitons in biased photorefractive-photovoltaic crystals

We carry out a theoretical investigation of the properties of waveguides induced by photorefractive one-dimensional steady-state gray spatial solitons (i.e., screening solitons, photovoltaic solitons, and screening-photovoltaic solitons). We demonstrate that waveguides induced by photorefractive steady-state gray spatial solitons are only a single guided mode for both all soliton graynesses and all values of ρ, where ρ is the ratio between the soliton peak intensity and the dark irradiance, and moreover, waveguides induced by gray photovoltaic solitons for closed-circuit condition are also only a single guided mode for all electric current densities. We find that the confined energy near the center of a photorefractive steady-state gray spatial soliton increases with ρ and decreases with an increase in the soliton grayness. We also find that the confined energy near the center of a gray photovoltaic soliton for closed-circuit condition increases with the electric current density. On the other hand, waveguides induced by gray screening-photovoltaic solitons are gray screening soliton-induced waveguides when the bulk photovoltaic effect is neglectable and are gray photovoltaic soliton-induced waveguides when the external bias field is absent.     

Baeicklund Transformation and Multiple Soliton Solutions for （3＋1）-Dimensional Potential-YTSF Equation

We study an approach to constructing multiple soliton solutions of the (3 1)-dimensional nonlinear evolution equation. We take the (3 1)-dimensional potential- YTSF equation as an example. Using the extended homogeneous balance method, one can find a Backlund transformation to decompose the (3 1)-dimensional potential-YTSF equation into a set of partial differential equations. Starting from these partial differential equations, some multiple soliton solutions for the (3 1)-dimensional potential-YTSF equation are obtained by introducing a class of formal solutions.     

Heat transfer analysis for stretching flow over a curved surface with magnetic field

The analysis of a viscous fluid flow and heat transfer is carried out under the influence of a constant applied magnetic field over a curved stretching sheet. Heat transfer analysis is carried out for two heating processes, namely, prescribed surface temperature (PST) and prescribed heat flux (PHF). The equations governing the flow are modeled in a curvilinear coordinate system (r, s, z). The nonlinear partial differential equations are then transformed to nonlinear ordinary differential equations by using similarity transformations. The obtained system of equations is solved numerically by a shooting method using Runge-Kutta algorithm. The interest lies in determining the influence of dimensionless radius of curvature on the velocity, temperature, skin friction, and rate of heat transfer at the wall prescribed by the Nusselt number. The effects of Hartmann number are also presented for the fluid properties of interest.     

Heat transport by acoustic streaming within a cylindrical resonator

Several experiments on heat transport within a cylindrical resonance tube, mediated by acoustic streaming, are described. The amplitude dependence of the heat transfer coefficient, h, from a hot object located inside the tube depends on the size of the object. For an object short compared to the acoustic displacement amplitude, h is proportional to the square root of amplitude; for a long object, h is linear in amplitude. For an empty resonator with a heated wall segment, the radial heat flux varies with position in a manner consistent with the global streaming pattern within the tube. The magnitude of heat transport from the heated wall segment is increased by inserting an object into the tube because the localized streaming velocity induced by the object is larger than the global streaming velocity in the empty tube. These effects could find application in the cooling of hot objects like electronic components or in thermoacoustic engines.     

Cusp and loop soliton solutions of short-wave models for the Camassa–Holm and Degasperis–Procesi equations

A simple method is developed for constructing the solutions of the short-wave model equations associated with the Camassa–Holm (CH) and Degasperis–Procesi (DP) shallow-water wave equations. Taking an appropriate scaling limit of the N-soliton solution of the CH equation, we obtain the N-cusp soliton solution for the CH short-wave model. The similar procedure also leads to the N-loop soliton solution for the DP short-wave model. We describe the property of the solutions. In particular, we derive the large-time asymptotics of the solutions as well as the formulas for the phase shift.     

On exact N-loop soliton solution to nonlinear coupled dispersionless evolution equations

We investigate some nonlinear coupled dispersionless evolution equations (NLCDEE) modelling the dynamics of a current-fed string within an external magnetic field in 2D-space. Using a blend of transformations of independent variables, we derive from the previous equations a Schäfer-Wayne short pulse equation (SWSPE). By means of a transformation back to the original independent variables, we find the N-loop soliton solution to the coupled equations. We give some detail on the scattering behavior of two-loop solitons.     

The spectrum of multi-flavor QCD2 and the non-abelian Schwinger equation

Massless QCD₂ is dominated by classical configurations in the larre-N_f limit. We use this observation to study the theory by finding solutions to equations of motion, which are the non-abelian generalization of the Schwinger equation. We find that the spectrum consists of massive mesons with M² = e²N_f/2π, which correspond to abelian solutions. We generalize previously discovered non-abelian solutions and discuss their interpretation. We prove a no-go theorem ruling out the existence of soliton solutions. Thus the semi-classical approximation shows no baryons in the case of massless quarks, a result derived before in the strong-coupling limit only.     

Noncommutative M-branes from covariant open supermembranes

We discuss an open supermembrane in the presence of a constant three-form. The boundary conditions to ensure the κ-invariance of the action lead to possible Dirichlet branes. It is shown that a noncommutative (NC) M5-brane is possible as a boundary and the self-duality condition that the flux on the world-volume satisfies is derived from the requirement of the κ-symmetry. We also find that the open supermembrane can attach to each of infinitely many M2-branes on an M5-brane, namely a strong flux limit of the NC M5-brane.     

Novel domain wall and Minkowski vacua of D=9 maximal SO(2) gauged supergravity

We show that a generalised reduction of D=10 IIB supergravity leads, in a certain limit, to a maximally extended SO(2) gauged supergravity in D=9. We show the scalar potential of this model allows both Minkowski and a new type of domain wall solution to the Bogomol'nyi equations. We relate these vacua to type IIB D-branes.     

B(a)cklund Transformation and Multiple Soliton Solutions for (3+1)-Dimensional Potential-YTSF Equation

We study an approach to constructing multiple soliton solutions of the (3 1)-dimensional nonlinear evolu tion equation. We take the (3 1)-dimensional potential-YTSF equation as an example. Using the extended homogeneous balance method, one can find a Backlund transformation to decompose the (3 1)-dimensional potential-YTSF equa tion into a set of partial differential equations. Starting from these partial differential equations, some multiple soliton solutions for the (3 1)-dimensional potential-YTSF equation are obtained by introducing a class of formal solutions.     

Flat coordinates,topological Landau-Ginzburg models and the Seiberg-Witten period integrals

We study the Picard-Fuchs differential equations for the Seiberg-Witten period integrals in N = 2 supersymmetric Yang-Mills theory. For A-D-E gauge groups we derive the Picard-Fuchs equations by using the flat coordinates in the A-D-E singularity theory. We then find that these are equivalent to the Gauss-Manin system for two-dimensional A-D-E topological Landau-Ginzburg models and the scaling relation for the Seiberg-Witten differential. This suggests an interesting relationship between four-dimensional N = 2 gauge theories in the Coulomb branch and two-dimensional topological field theories.     

Thermal diffusion of envelope solitons on anharmonic atomic chains

We study the motion of envelope solitons on anharmonic atomic chains in the presence of dissipation and thermal fluctuations. We consider the continuum limit of the discrete system and apply an adiabatic perturbation theory which yields a system of stochastic integro-differential equations for the collective variables of the ansatz for the perturbed envelope soliton. We derive the Fokker-Planck equation of this system and search for a statistically equivalent system of Langevin equations, which shares the same Fokker-Planck equation. We undertake an analytical analysis of the Langevin system and derive an expression for the variance of the soliton position Var[x _s ] which predicts a stronger than linear time dependence of Var[x _s ] (superdiffusion). We compare these results with simulations for the discrete system and find they agree well. We refer to recent studies where the diffusion of pulse solitons were found to exhibit a superdiffusive behaviour on longer time scales.Received: 28 June 2004, Published online: 26 November 2004PACS: 05.10.Gg Stochastic analysis methods - 05.45.Yv Solitons - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.50. + q Lattice theory and statistics     

Phase jitter in periodically dispersion-managed optical transmission systems

We investigate the phase jitter in long-haul optical transmission systems with periodic dispersion management and amplification. We compare different dispersion-managed soliton systems and a conventional soliton system having the same pulse width and path-averaged dispersion. Using the variational method, we derive the ordinary differential equations for the pulse parameters perturbed by amplifier noise and hence calculate the phase jitter. We verify the analytical results by numerically solving the nonlinear Schrödinger equation using split-step Fourier algorithm. The results suggest that the reduction of nonlinear phase noise in dispersion-managed soliton systems is possible compared to a conventional soliton system. It is also revealed that the phase noise is enhanced in stronger dispersion-managed systems. We also explore the phase noise effect in dispersion-managed quasi-linear systems and find that phase jitter is mitigated in highly dispersive fibers.     

B?cklund transformation and multiple soliton solutions for the (3+1)-dimensional Jimbo-Miwa equation

We study an approach to constructing multiple soliton solutions of the (3+1)-dimensional nonlinear evolution equation. We take the (3+1)-dimensional Jimbo－Miwa (JM) equation as an example. Using the extended homogeneous balance method, one can find a B?cklund transformation to decompose the (3+1)-dimensional JM equation into a linear partial differential equation and two bilinear partial differential equations. Starting from these linear and bilinear partial differential equations, some multiple soliton solutions for the (3+1)-dimensional JM equation are obtained by introducing a class of formal solutions.     

Stochastic aspects of pattern formation during the catalytic oxidation of CO on Pd(111) surfaces

We present experimental results on rare transitions between two states due to intrinsic noise between two states in a bistable surface reaction, namely the catalytic oxidation of CO on Pd(111) surfaces. The mean time scales involved are typically of order 10⁴ s and the probability distribution shows two peaks over a large part of the bistable regime of this surface reaction. We use measurements of the resulting CO₂ rate as well as photoelectron emission microscopy (PEEM) to characterize these rare transitions. From our dynamic data we can extract probability distributions for the CO₂ rate. We use x-t plots from PEEM measurements to describe the transitions, which are-as we demonstrate-characterized by one wall moving through the field of view in PEEM measurements. The resulting probability distributions for the CO₂ rate are shown to depend strongly on the value, Y, of the CO fraction in the reactant flux inside the bistable regime. We find that the probability distribution is strongly asymmetric indicating that the two basins of attraction are rather different in depth and width. This is also concluded from the PEEM measurements, which show in one case a rather sharp and narrow domain wall going one way, while it is rather wide and diffuse for the motion in the opposite direction. To have two basins of attraction in the bistable regime, which are rather different in nature is reminiscent of other bistable systems such as, for example, optical bistability, although the time scales involved in the present system are entirely different.