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.     

The Riemann–Hilbert Problem for Analytic Description of the DM Solitons

A simple exact formula is derived for the profile of the optical pulse propagating over a DM fiber with zero mean dispersion. The dissipation is neglected, and dispersion is assumed to be constant along the adjacent legs of the waveguide, thus providing the applicability of the integrable NLS models within each leg. The formula describes a class of solutions called dispersion-managed solitons (DM solitons), which are periodic along the waveguide and exponentially localized in time. The DM solitons are parameterized by a certain class of spectral data, specified from numerical simulations. Using a related Riemann–Hilbert problem, we reconstruct a profile of the DM soliton from the given spectral data. For sufficiently long legs, the leading term of DM soliton is found in explicit form by asymptotic undressing of the Riemann–Hilbert problem. The analytic results are compared with numerical simulations.     

Frequency shifting for solitons based on transformations in the Fourier domain and applications

We develop the theoretical procedures for shifting the frequency of a single soliton and of a sequence of solitons of the nonlinear Schrödinger equation. The procedures are based on simple transformations of the soliton pattern in the Fourier domain and on the shape-preserving property of solitons. These theoretical frequency shifting procedures are verified by numerical simulations with the nonlinear Schrödinger equation using the split-step Fourier method. In order to demonstrate the use of the frequency shifting procedures, two important applications are presented: (1) stabilization of the propagation of solitons in waveguides with frequency dependent linear gain-loss; (2) induction of repeated soliton collisions in waveguides with weak cubic loss. The results of numerical simulations with the nonlinear Schrödinger model are in very good agreement with the theoretical predictions.     

Multiscale expansions avector solitons of a two-dimensional nonlocal nonlinear Schrödinger system

One- and two-dimensional solitons of a multicomponent nonlocal nonlinear Schrödinger (NLS) system are constructed. The model finds applications in nonlinear optics, where it may describe the interaction of optical beams of different frequencies. We asymptotically reduce the model, via multiscale analysis, to completely integrable ones in both Cartesian and cylindrical geometries; we thus derive a Kadomtsev-Petviashvili equation and its cylindrical counterpart, Johnson's equation. This way, we derive approximate soliton solutions of the nonlocal NLS system, which have the form of: (a) dark or antidark soliton stripes and (b) dark lumps in the Cartesian geometry, as well as (c) ring dark or antidark solitons in the cylindrical geometry. The type of the soliton, namely dark or antidark, is determined by the degree of nonlocality: dark (antidark) soliton states are formed for weaker (stronger) nonlocality. We perform numerical simulations and show that the derived soliton solutions do exist and propagate undistorted in the original nonlocal NLS system.     

Numerical computation of high dimensional solitons via darboux transformation

Darboux transformation gives explicit soliton solutions of nonlinear partial differential equations. Using numerical computation in each step of constructing Darboux transformation, one can get the graphs of the solitons practically. In n dimensions (n ≥ 3), this method greatly increases the speed and deduces the memory usage of computation comparing to the software for algebraic computation. A technical problem concerning floating overflow is discussed.     

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.     

非传播孤立波及其相互作用的数值模拟

通过数值求解由Miles导出的目前公认的的非传播孤立波的控制方程——一个带复共轭项的非线性立方SchrLdinger方程,对非传播孤立波进行研究。讨论了Miles方程中的线性阻尼系数α的值,计算表明,线性阻尼α对形成稳定的非传播孤立波影响很大,Laedke等人关于非传播孤立波的稳定性条件只是一个必要条件,而不是充分条件。模拟了两个非传播孤立波的相互作用,数值模拟表明,两个波的作用模式依赖于系统的参数,对不同的初始扰动及其演化的计算表明,只有适当的初始扰动才能形成单个稳定的非传播孤立波,否则扰动可能消失或发展成多个孤立波。     

Exponential Asymptotics for Line Solitons in Two‐Dimensional Periodic Potentials

As a first step toward a fully two‐dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two‐dimensional periodic potentials. For this two‐dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one‐dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence‐relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch‐band edge; and for each rational slope, two line‐soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline‐soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.     

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 .     

Instability of the finite‐difference split‐step method applied to the nonlinear Schrödinger equation. II. moving soliton

We analyze a mechanism and features of a numerical instability (NI) that can be observed in simulations of moving solitons of the nonlinear Schrödinger equation (NLS). This NI is completely different than the one for the standing soliton. We explain how this seeming violation of the Galilean invariance of the NLS is caused by the finite‐difference approximation of the spatial derivative. Our theory extends beyond the von Neumann analysis of numerical methods; in fact, it critically relies on the coefficients in the equation for the numerical error being spatially localized. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1024–1040, 2016     

APPLICATIONS OF THE P-R SCHEME FOR GENERALIZED NONLINEAR SCHRODINGER EQUATIONS IN SOLVING SOLITON SOLUTIONS

Spatial soliton solutions of a class of generalized nonlinear Schrodinger equations in N-space are discussed analytically and numerically. This achieved using a traveling wavemethod to formulate one-soliton solution and the P-R method is employed to the numerlcal solutions and the interactions between the solirons for the generalized nonlinear systems in Z-pace.The results presented show that the soliton phenomena are characteristics associated with the nonlinearhies of the dynamical systems.     

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.      

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.     

Strong interference of sound pressure generated by vortex soliton with axial flow

Sound pressure generated by a thin vortex filament with axial flow is investigated on the base of soliton theory. The pressure is estimated by using the experimental results of Maxworthy et al. [J. Fluid Mech. 151, 141] where one-soliton propagation and head-on collision of two solitons are observed. Collision of two solitons gives rise to very enhanced interference of the sound pressure.     

A theoretical description for solitons in polyacetylene

The bond-alternation domain walls or the solitons in the dimerized polyacetylene are analyzed theoretically. The width of the soliton is many times the period of the chain, so that the soliton can be reasonably well described by a continuum model. Because of the existence of the bond-alternation domain walls, the electron density is different definitely. Thus the electron density can be used to describe the formation of the domain walls, and a self-trapped potential is discussed and introduced in the Hamiltonian. It is shown that the envelope of the wave functions of the chain is governed by the nonlinear Schr?dinger equation which has soliton solutions. Then the shape of the soliton is determined analytically which is in accordance with the numerical calculations by Su, Schrieffer and Heeger. This implies that the bond-alternation domain wall or the soliton is observed as the envelope of the wave function.     

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.     

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.     

Long-time behaviour of soliton ensembles. Part I––Emergence of ensembles

This paper is focused on the emergence of the KdV solitons from an initial harmonic excitation. In the long run this process is characterized not so much by regular soliton trains but rather by soliton ensembles. It has been shown explicitly that under indicated initial conditions the width of emerging solitons are mostly larger than the distance between maxima of wave profiles. Consequently, visible are the ensembles formed by several simultaneously interacting solitons including also hidden (virtual) solitons. The conditions for emerging such ensembles are studied over the wide range of amplitude ratios for typical dispersion parameters. Based on that analysis, it is possible to cast more light to the recurrence and periodicity in the long run (see Part II).     

Long-range interaction of four-vector field solitons of Minkowski space

The interaction of solitons of a system of nonlinear equations at distances greatly exceeding their characteristic dimensions is studied by perturbation methods. The slow modulation of soliton parameters under the influence of a small perturbing field of distant solitons is considered. It is shown that the equation of the soliton trajectory in the first and second orders of the method has the form of the classical equation of motion of a particle in electromagnetic and gravitational (in the sense of the bimetric theory) fields.Electrotechnical Institute, Leningrad. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 380–387, March, 1992.