Dynamics of Embedded Solitons in the Extended Korteweg–de Vries Equations

Embedded solitons are solitary waves residing inside the continuous spectrum of a wave system. They have been discovered in a wide array of physical situations recently. In this article, we present the first comprehensive theory on the dynamics of embedded solitons and nonlocal solitary waves in the framework of the perturbed fifth-order Korteweg–de Vries (KdV) hierarchy equation. Our method is based on the development of a soliton perturbation theory. By obtaining the analytical formula for the tail amplitudes of nonlocal solitary waves, we demonstrate the existence of single-hump embedded solitons for both Hamiltonian and non-Hamiltonian perturbations. These embedded solitons can be isolated (existing at a unique wave speed) or continuous (existing at all wave speeds). Under small wave speed limit, our results show that the tail amplitudes of nonlocal waves are exponentially small, and the product of the amplitude and cosine of the phase is a constant to leading order. This qualitatively reproduces the previous results on the fifth-order KdV equation obtained by exponential asymptotics techniques. We further study the dynamics of embedded solitons and prove that, under Hamiltonian perturbations, a localized wave initially moving faster than the embedded soliton will asymptotically approach this embedded soliton, whereas a localized wave moving slower than the embedded soliton will decay into radiation. Thus, the embedded soliton is semistable. Under non-Hamiltonian perturbations, stable embedded solitons are found for the first time.     

Internal Oscillations and Radiation Damping of Vector Solitons

Internal modes of vector solitons and their radiation-induced damping are studied analytically and numerically in the framework of coupled nonlinear Schrödinger equations. Bifurcations of internal modes from the integrable systems are analyzed, and the region of their existence in the parameter space of vector solitons is determined. In addition, radiation-induced decay of internal oscillations is investigated. Both exponential and algebraic decay rates are identified.     

Soliton interactions and Yang–Baxter maps for the complex coupled short-pulse equation

The complex coupled short-pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so-called self-symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long-time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well-known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.     

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.     

How to excite the internal modes of sine-Gordon solitons

We investigate the dynamics of the sine-Gordon solitons perturbed by spatiotemporal external forces. We prove the existence of internal (shape) modes of sine-Gordon solitons when they are in the presence of inhomogeneous space-dependent external forces, provided some conditions (for these forces) hold. Additional periodic time-dependent forces can sustain oscillations of the soliton width. We show that, in some cases, the internal mode even can become unstable, causing the soliton to decay in an antisoliton and two solitons. In general, in the presence of spatiotemporal forces the soliton behaves as a deformable (non-rigid) object. A soliton moving in an array of inhomogeneities can also present sustained oscillations of its width. There are very important phenomena (like the soliton–antisoliton collisions) where the existence of internal modes plays a crucial role. We show that, under some conditions, the dynamics of the soliton shape modes can be chaotic. A short report of some of our results has been published in [Phys. Rev. E 65 (2002) 065601(R)].     

The superposition of algebraic solitons for the modified Korteweg–de Vries equation

Many real nonlinear evolution equations exhibiting soliton properties display a special superposition principle, where an infinite array of equally spaced, identical solitons constitutes an exact periodic solution. This arrangement is studied for the modified Korteweg–de Vries equation with positive cubic nonlinearity, which possesses algebraic solitons with nonvanishing far field conditions. An infinite sum of equally spaced, identical algebraic pulses is evaluated in closed form, and leads to a complex valued solution of the nonlinear evolution equation.     

Soliton Splitting by External Delta Potentials

We show that in the dynamics of the nonlinear Schrodinger equation a soliton scattered by an external delta potential splits into two solitons and a radiation term. Theoretical analysis gives the amplitudes and phases of the reflected and transmitted solitons with errors going to zero as the velocity of the incoming soliton tends to infinity. Numerical analysis shows that this asymptotic relation is valid for all but very slow solitons. We also show that the total transmitted mass, that is, the square of the L² norm of the solution restricted on the transmitted side of the delta potential, is in good agreement with the quantum transmission rate of the delta potential.     

非线性弹性杆波动方程的摄动分析

针对计入横向惯性效应后的非线性弹性杆纵向波动方程进行了分析．在小振幅、长波长的一般情况下,根据远方场简单波理论,采用约化摄动法,得到了NLS方程,并讨论了存在NLS孤子的条件．     

Complexity and stability of soliton management in periodically modulated and random systems

A brief introduction is given to the concept of the soliton management, i.e., stable motion of localized pulses in media with strong periodic (or, sometimes, random) inhomogeneity, or conditions for the survival of solitons in models with strong time‐periodic modulation of linear or nonlinear coefficients. It is demonstrated that a class of systems can be identified, in which solitons remain robust inherently coherent objects in seemingly “hostile” environments. Most physical models belonging to this class originate in nonlinear optics and Bose‐Einstein condensation, although other examples are known too (in particular, in hydrodynamics). In this paper, the complexity of the soliton‐management systems, and the robustness of solitons in them are illustrated using a recently explored fiber‐optic setting combining a periodic concatenation of nonlinear and dispersive segments (the split‐step model) for bimodal optical signals (i.e., ones with two polarizations of light), which includes the polarization mode dispersion, i.e., random linear mixing of the two polarization components at junctions between the fiber segment. © 2008 Wiley Periodicals, Inc. Complexity, 2008     

On the optimal focusing of solitons and breathers in long‐wave models

Conditions of optimal (synchronized) collisions of any number of solitons and breathers are studied within the framework of the Gardner equation (GE) with positive cubic nonlinearity, which in the limits of small and large amplitudes tends to other long‐wave models, the classic and the modified Korteweg–de Vries equations. The local solution for an isolated soliton or breather within the GE is obtained. The wave amplitude in the focal point is calculated exactly. It exhibits a linear superposition of partial amplitudes of the solitons and breathers. The crucial role of the choice of proper soliton polarities and breather phases on the cumulative wave amplitude in the focal point is demonstrated. Solitons are most synchronized when they have alternating polarities. The straightforward link to the problem of synchronization of envelope solitons and breathers in the focusing nonlinear Schrödinger equation is discussed (then breathers correspond to envelope solitons propagating above a condensate).     

Some properties of the spinor soliton

In this paper, the solitons of nonlinear Dirac equation are discussed in detail, and several functions which reflect their characteristics are computed. The numerical results show that, the nonlinear Dirac equation has only finite meaningful solitons, and these solitons have 1/2-spin and positive mass; the spinor soliton has two kinds of parity states, and each parity state has two kinds of energy states; the larger the self-coupling coefficientw, the more the excitation states, and ifw is less than a critical value, then the meaningful soliton does not exist. These properties may have relations with some fundamental particles.     

Interaction theory of mirror-symmetry soliton pairs in nonlocal nonlinear Schrödinger equation

In this work, we theoretically investigate the evolution of the soliton pairs in strongly nonlocal nonlinear media, which is modeled by the nonlocal nonlinear Schrödinger equation. Taking two pairs of solitons as an example, which initial incident directions have a mirror symmetry, a set of mathematical expressions are derived to describe the soliton pairs’ propagation, the soliton spacing, the area of the optical field. The results demonstrate that the motion state of the soliton pairs is mirror-symmetry. Numerical simulations are carried out to illustrate the quintessential propagation properties.     

Solitary waves in a quartic nonlinear elastic rod

The bright and dark solitons described by the nonlinear Schrödinger equation (NLSE) are given for a quartic nonlinear elastic rod. It has also been found that the KdV soliton does not exist in this system.     

Continuous Families of Embedded Solitons in the Third-Order Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation with a third-order dispersive term is considered. Infinite families of embedded solitons, parameterized by the propagation velocity, are found through a gauge transformation. By applying this transformation, an embedded soliton can acquire any velocity above a certain threshold value. It is also shown that these families of embedded solitons are linearly stable, but nonlinearly semi-stable.     

Analytic method for solving coupled nonlinear Schroedinger equations with oscillating terms describing polarized soliton oscillations

Using a variational method, a new model of the nonlinear propagation of optical solitons generated from semiconductor lasers through lossy elliptically low birefringent fiber, is presented within the framework of a system of coupled nonlinear Schroedinger (CNLS) equations with oscillating terms. This analytic model demonstrates polarized soliton oscillations in a lossy elliptically low birefringent optical fiber.     

Soliton Interactions for the Three‐Coupled Discrete Nonlinear Schrödinger Equations in the Alpha Helical Proteins

Three‐coupled discrete nonlinear Schrödinger equations, which describe the dynamics of the three hydrogen bonding spines in the alpha helical proteins with the interspine coupling at the discrete level, are investigated. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation of those equations. Propagation characteristics and interactions of the bound‐state solitons are discussed. Bound states of two and three bright solitons arise when all of them propagate in parallel. Elastic interaction between the bound‐state solitons and one bright soliton is given. Increase of the dipole‐dipole interaction energy can lead to the increase of the soliton velocity, that is, the one‐interaction period becomes shorter.     

Solitons in liquid crystals: Recent developments

Liquid crystal is a state of matter intermediate between isotropic liquid and anisotropic crystal. The mechanical and optical properties of liquid crystals are highly nonlinear. Consequently, they are naturally soliton-bearing media. After a brief general introduction, five topics in recent developments on solitons in liquid crystals are presented, namely (i) optical solitons, (ii) solitons in nematics under a rotating magnetic field, (iii) solitons in electroconvective nematics, (iv) incommensurate solitons in smectic A, and (v) the soliton model for the chevron structure in ferroelectric smectic C^* and in smectic A.     

Soliton experiments in transmission lines

Different structures of the nonlinear electrical transmission lines are presented, including coupled, inhomogeneous and two-dimensional lines, and the all basic soliton phenomena, such as the formation of solitons from the initial state, the recurrence of solitons, the envelope solitons, and a.c.-driven solitons, which are observed using transmission lines, are demonstrated.     

Applications of the weighted scheme for GNLS equations in solving soliton solutions

Soliton solutions of a class of generalized nonlinear evolution equations are discussed analytically and numerically, which is achieved using a travelling wave method to formulate one-soliton solution and the finite difference method to the numerical solutions and the interactions between the solitons for the generalized nonlinear Schrödinger equations. The characteristic behavior of the nonlinearity admitted in the system has been investigated and the soliton state of the system in the limit ofα → 0 andα → ∞ has been studied. The results presented show that soliton phenomena are characteristics associated with the nonlinearities of the dynamical systems.