Individual Solitons Research Articles

Scanning-Tunneling Microscope Imaging of Single-Electron Solitons in a Material with Incommensurate Charge-Density Waves

We report on scanning-tunneling microscopy experiments in a charge-density wave (CDW) system allowing visually capturing and studying in detail the individual solitons corresponding to the self-trapping of just one electron. This "Amplitude Soliton" is marked by vanishing of the CDW amplitude and by the π shift of its phase. It might be the realization of the spinon--the long-sought particle (along with the holon) in the study of science of strongly correlated electronic systems. As a distinct feature we also observe one-dimensional Friedel oscillations superimposed on the CDW which develop independently of solitons.

Non-linear transport by solitons in nanofibers of polymers in high magnetic field

Nonlinear local excitations like solitons, polarons, and bipolarons are known to be responsible for physical properties of conducting polymers. Recent experiments on nano-fibers in high electric and magnetic fields provide a further insight by demonstrating an effect of vanishing magnetoconductance (MC) in the polyacetylene (PA)—in contrast to other polymers. Here we present new experimental data and describe the theoretical model based on notion of solitons—dimerization kinks which can carry either the spin or the charge; they are allowed only in the PA with its degenerate ground state. The solitons experience a confinement force due to the interchange coupling which is erased by the electric field and disappears above critical field strength. The unbinding by tunneling allows for the transport of individual solitons, which sweeps off the spins residing at electronic intragap states associated with polarons, hence the vanishing MC.

Interaction properties of complex modified Korteweg–de Vries (mKdV) solitons

Interaction properties of complex solitons are studied for the two U ( 1 ) -invariant integrable generalizations of the modified Korteweg–de Vries (mKdV) equation, given by the Hirota equation and the Sasa–Satsuma equation, which share the same traveling wave (single-soliton) solution having a sech profile characterized by a constant speed and a constant phase angle. For both equations, nonlinear interactions in which a fast soliton collides with a slow soliton are shown to be described by 2 -soliton solutions that can have three different types of interaction profile, depending on the speed ratio and the relative phase angle of the individual solitons. In all cases, the shapes and speeds of the solitons are found to be preserved apart from a shift in position such that their center of momentum moves at a constant speed. Moreover, for the Hirota equation, the phase angles of the fast and slow solitons are found to remain unchanged, while, for the Sasa–Satsuma equation, the phase angles are shown to undergo a shift such that the relative phase between the fast and slow solitons changes sign.

Regular and stochastic motion of dissipative optical solitons

Investigations of the motion of dissipative optical solitons and their complexes in wide-aperture nonlinearly optical (with coherent pump radiation) and laser (with incoherent pump radiation) systems have been reviewed. An important characteristic of dissipative solitons is the topology of the energy fluxes, which determines the internal structure of individual solitons and makes it possible to certainly separate the cases of the weak and strong interactions between the solitons. It has been shown that the character of the regular motion of dissipative soliton structures in a homogeneous system is determined by the symmetry of the transverse distributions of the intensity and energy flux; the motion of asymmetric structures is curvilinear. This is also valid for complexes of three-dimensional dissipative optical solitons, “laser bullets.” The extreme possibilities of localization of solitons are determined by quantum noises. The corresponding Brownian motion of the center of the dissipative optical soliton is characterized by a much lower level of the statistic dispersion of the coordinates of its center and velocity than that in the case of conservative solitons.

Matter-Wave Dark Solitons: Stochastic versus Analytical Results

The dynamics of dark matter-wave solitons in elongated atomic condensates are discussed at finite temperatures. Simulations with the stochastic Gross-Pitaevskii equation reveal a noticeable, experimentally observable spread in individual soliton trajectories, attributed to inherent fluctuations in both phase and density of the underlying medium. Averaging over a number of such trajectories (as done in experiments) washes out such background fluctuations, revealing a well-defined temperature-dependent temporal growth in the oscillation amplitude. The average soliton dynamics is well captured by the simpler dissipative Gross-Pitaevskii equation, both numerically and via an analytically derived equation for the soliton center based on perturbation theory for dark solitons.

Channeling of high-power radio waves under conditions of strong anomalous absorption in the presence of an averaged electron heating source

Strong anomalous absorption of a high-power radio wave by small-scale plasma inhomogeneities in the Earth’s ionosphere can lead to the formation of self-consistent channels (solitons) in which the wave propagates along the magnetic field, but has a soliton-like intensity distribution across the field. The structure of a cylindrical soliton as a function of the wave intensity at the soliton axis is analyzed. Averaged density perturbations leading to wave focusing were calculated using the model proposed earlier by Vas’kov and Gurevich (Geomagn. Aeron. 16, 1112 (1976)), in which an averaged electron heating source was used. It is shown that, under conditions of strong electron recombination, the radii of individual solitons do not exceed 650 m.

Analytical proof of space-time chaos in Ginzburg-Landau equations

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDSs with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.

Soliton strings and interactions in mode-locked lasers

Soliton strings in mode-locked lasers are obtained using a variant of the nonlinear Schrödinger equation, appropriately modified to model power (intensity) and energy saturation. This equation goes beyond the well-known master equation often used to model these systems. It admits mode-locking and soliton strings in both the constant dispersion and dispersion-managed systems in the (net) anomalous and normal regimes; the master equation is contained as a limiting case. Analysis of soliton interactions show that soliton strings can form when pulses are a certain distance apart relative to their width. Anti-symmetric bi-soliton states are also obtained. Initial states mode-lock to these states under evolution. In the anomalous regime individual soliton pulses are well approximated by the solutions of the unperturbed nonlinear Schrödinger equation, while in the normal regime the pulses are much wider and strongly chirped.

Steady and oscillating multiple dissipative solitons in normal-dispersion mode-locked Yb-doped fiber laser

Multiple dissipative solitons are numerically studied in the normal-cavity-dispersion Yb-doped fiber laser. Soliton pairs and triplets of different types are found in parameter domain where single soliton mode-locking is unstable. Similar scenario is found for soliton pair and triplet: an additional weak soliton supplements already existing solitons, then with increasing gain the solitons start oscillating and at higher gain the soliton complex becomes stable. Spectral profiles of the solitons dynamically change taking either parabolic-like or Pi-like or M-like shape similar to those described in the literature for individual dissipative soliton at different pump level.

Soliton extraction from a bunch of solitons resulting from pulse breakup by using a nonlinear optical loop mirror

We numerically investigated the transmission of a bunch of solitons resulting from the breakup of pulses with duration of several tens of picoseconds (ps) through a nonlinear optical loop mirror (NOLM) and found that under some conditions an individual soliton can be extracted. The NOLM selectivity can be adjusted by the amplification of the bunch of solitons before it is launched into the NOLM. The results demonstrate that an appropriate choice of the amplification and of the NOLM loop length makes it possible to extract one fundamental soliton and to tune the soliton duration. For a particular case of 20 ps input pulses, the duration of the extracted soliton was tuned in the range between 0.23 and 0.61 ps. We believe that the suggested method can be useful for producing solitons with desirable duration.

Asymptotic Analysis of Pulse Dynamics in Mode‐Locked Lasers

Solitons of the power‐energy saturation (PES) equation are studied using adiabatic perturbation theory. In the anomalous regime individual soliton pulses are found to be well approximated by solutions of the classical nonlinear Schrödinger (NLS) equation with the key parameters of the soliton changing slowly as they evolve. Evolution equations are found for the pulse amplitude(s), velocity(ies), position(s), and phase(s) using integral relations derived from the PES equation. The results from the integral relations are shown to agree with multi‐scale perturbation theory. It is shown that the single soliton case exhibits mode‐locking behavior for a wide range of parameters, while the higher states form effective bound states. Using the fact that there is weak overlap between tails of interacting solitons, evolution equations are derived for the relative amplitudes, velocities, positions, and phase differences. Comparisons of interacting soliton behavior between the PES equation and the classical NLS equation are also exhibited.

Generation of high order multi-bound solitons and propagation in optical fibers

We demonstrate experimental generation of multi-bound solitons of up to sextuple in an active FM mode-locked fiber ring laser operating under power saturation in the locking state. The ring laser consists of two booster optical amplifiers operating in saturation regime, an electro-optic phase modulator driven by a sinusoidal electrical wave and a length of dispersive fiber. The periodic phase modulation generates phase chirp of the generated lightwaves in the ring laser. The chirped phase state plays an important role in the phase matching condition for mode-locking as well as the stabilization and the determination of the bound states of multi-solitons. The formation of such high order multi-bound solitons is explained based on the chirping of the phase and the behavior of the optical pulse sequence in the near field region of the dispersive fiber. The propagation of these multi-bound solitons through single mode optical fibers is observed. Experimental and simulation results of bound solitons have been shown to follow similar trends. The propagation of these multi-bound solitons through single mode optical fibers is described. Their mutual interaction through such quadratic phase media shows the influence of the quadratic phase property on the differential phase of individual solitons of the bound group. Simulated results confirm the evolution of the bound solitons over dispersive single mode optical fibers.

Diffusion stabilizes cavity solitons in bidirectional lasers

We study the influence of field diffusion on the spatial localized structures (cavity solitons) in bidirectional lasers. We find threefold positive role of the diffusion: 1) it allows for the existence of solitons in cavities with equal losses for the two fields; 2) it increases the stability range of the individual (isolated) solitons; and 3) it reduces the long-range interaction between cavity solitons, allowing for the independent manipulation (writing and erasing) of individual structures.

Spectral interference of frequency-shifted solitons in a photonic-crystal fiber

Frequency-shifted solitons in a highly nonlinear photonic-crystal fiber (PCF) are shown to give rise to high-visibility interference fringes in PCF output spectra, indicating flat spectral phase profiles of individual solitons in the PCF output. This experimental finding, supported by numerical simulations, suggests a promising method of fiber-format pulse shaping and an attractive technology for few-cycle pulse synthesis through a coherent addition of frequency-shifted solitons generated in a highly nonlinear fiber.

Anomalous pulse breakup in small-core photonic crystal fibers

Detailed numerical and experimental studies of propagation of 110 fs laser pulses at 800 nm in small-core photonic crystal fibers (γ≈100 W-1 km−1) reveal that pulse breakup occurs in two distinct regimes defined by the input power. At low peak power (soliton order N≤7) higher-order soliton fission occurs: individual solitons being ejected from the input pulse one after the other and are arranged in wavelength and in time by peak power. At higher levels of peak power (N>8), pulse breakup results in ejection of bound soliton pairs and the formation of single solitons that collide during propagation.

Solitons and Their Arrays: From Quasi 1D Conductors to Stripes

We suggest a short review of literature on various solitonic lattices and individual solitons in quasi one-dimensional conductors. This information seems to be quite relevant to topics of stripes and their melted phases correspondingly. We shall quote also the latest experiments, which access solitons as elementary excitations in organic conductors and in charge density waves. We shall outline a theory for ordered phases, where solitons should acquire forms of combined topological configurations (kink-roton complexes). The extension of this picture to cuprates allows interpretation of the latest STM observations on local rod-like structures.

Line Soliton Interactions of the Kadomtsev-Petviashvili Equation

We study soliton solutions of the Kadomtsev-Petviashvili II equation (-4u(t)+6uu(x)+3u(xxx))(x)+u(yy)=0 in terms of the amplitudes and directions of the interacting solitons. In particular, we classify elastic N-soliton solutions, namely, solutions for which the number, directions, and amplitudes of the N asymptotic line solitons as y-->infinity coincide with those of the N asymptotic line solitons as y-->-infinity. We also show that the (2N-1)!! types of solutions are uniquely characterized in terms of the individual soliton parameters, and we calculate the soliton position shifts arising from the interactions.

Incoherent weak coupling of laser solitons

A coupled state of a pair of spatial laser solitons with different topological charges in a wide-aperture class-A laser with saturable absorption was found by numerical simulation. In view of a difference in the frequencies of the individual solitons, periodic (with a period corresponding to the frequency difference) weak pulsations of the shape of the solitons and of the spacing between them take place, but the rotation of the pair and the motion of its center of inertia were not revealed.

Elastic and inelastic line-soliton solutions of the Kadomtsev–Petviashvili II equation

The Kadomtsev–Petviashvili II (KPII) equation admits a large variety of multi-soliton solutions which exhibit both elastic as well as inelastic types of interactions. This work investigates a general class of multi-solitons which were not previously studied, and which do not in general conserve the number of line solitons after interaction. The incoming and outgoing line solitons for these solutions are explicitly characterized by analyzing the τ -function generating such solutions. A special family of N-soliton solutions is also considered in this article. These solutions are characterized by elastic soliton interactions, in the sense that amplitude and directions of the individual line solitons as y → ∞ are the same as those of the individual line solitons as y → − ∞ . It is shown that the solution space of these elastic N-soliton solutions can be classified into ( 2 N − 1 ) ! ! disjoint sectors which are characterized in terms of the amplitudes and directions of the N line solitons.

津波の変形および波力特性に及ぼす海岸断面地形の効果に関する実験的研究

A big difference between the Andaman Sea coast and the coasts in the western countries of the Indian Ocean was found in damage state caused by the Indian Ocean Tsunami. It is thought that geographical features of the coastal cross-shore section affect the characteristics of the tsunami deformation and the damage situation of the coastal structures. In this paper, the model experiments were conducted to clarify the difference of deformation, breaking, runup or pressure of tsunami by the characteristic of the coastal cross section. The wave steepness and water depth on the reef are closely related to generation and development of dispersive solitons, and it is clarified that the impulsive breaking wave force takes place at the breaking of individual soliton. Tanimoto's empirical formula can be applied to estimate the wave force of tsunami on coastal structures when the dispersive solitons almost disappear due to these breaking or no dispersive solitons occur.