Soliton Solutions for a Generalized Inhomogeneous Variable-Coefficient Hirota Equation with Symbolic Computation

Under investigation in this paper is a generalized inhomogeneous variable- coefficient Hirota equation. Through the Hirota bilinear method and symbolic computation, the bilinear form and analytic one-, two- and N-soliton solutions for such an equation are obtained, respectively. Properties of those solitons in the inhomogeneous media are discussed analytically. We get the soliton with the property that the larger the amplitude is, the narrower and slower the pulse is. Dynamics of that soliton can be regarded as a repulsion of the soliton by the external potential barrier. During the interaction of two solitons, we observe that the larger the value of the coefficient β in the equation is, the larger the distance of the two solitons is.     

Perturbation of topological solitons due to sine-Gordon equation and its type

This paper studies the adiabatic dynamics of topological solitons in presence of perturbation terms. The solitons due to sine-Gordon equation, double sine-Gordon equation, sine–cosine Gordon equation and double sine–cosine Gordon equations are studied, in this paper. The adiabatic variation of soliton velocity is obtained in this paper by soliton perturbation theory.     

Dark solitonic excitations and collisions from a fourth-order dispersive nonlinear Schrödinger model for the alpha helical protein

Recent protein observations motivate the dark-soliton study to explain the energy transfer in the proteins. In this paper we will investigate a fourth-order dispersive nonlinear Schrödinger equation, which governs the Davydov solitons in the alpha helical protein with higher-order effects. Painlevé analysis is performed to prove the equation is integrable. Through the introduction of an auxiliary function, bilinear forms and dark N-soliton solutions are constructed with the Hirota method and symbolic computation. Asymptotic analysis on the two-soliton solutions indicates that the soliton collisions are elastic. Decrease of the coefficient of higher-order effects can increase the soliton velocities. Graphical analysis on the two-soliton solutions indicates that the head-on collision between the two solitons, overtaking collision between the two solitons and collision between a moving soliton and a stationary one are all elastic. Collisions among the three solitons are all pairwise elastic.     

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.     

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.     

Optical soliton solutions for the generalized Kudryashov equation of propagation pulse in optical fiber with power nonlinearities by three integration algorithms

In this paper, we employ three integration algorithms, namely, the well‐known Kudryashov method, the new Kudryashov method, and the unified Riccati equation expansion method to extract optical soliton solutions for the generalized Kudryashov equation with power nonlinearities. Straddled soliton, bright solitons, dark solitons, and singular solitons have been found.     

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.     

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。     

Dynamics of solitons in plasmas for the complex KdV equation with power law nonlinearity

This paper obtains the 1-soliton solution of the complex KdV equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration. The soliton perturbation theory for this equation is developed and the soliton cooling is observed for bright solitons. Finally, the dark soliton solution is also obtained for this equation.     

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.     

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).     

Unstable gap solitons in inhomogeneous nonlinear Schrödinger equations

A periodically inhomogeneous Schrödinger equation is considered. The inhomogeneity is reflected through a non-uniform coefficient of the linear and nonlinear term in the equation. Due to the periodic inhomogeneity of the linear term, the system may admit spectral bands. When the oscillation frequency of a localized solution resides in one of the finite band gaps, the solution is a gap soliton, characterized by the presence of infinitely many zeros in the spatial profile of the soliton. Recently, how to construct such gap solitons through a composite phase portrait is shown. By exploiting the phase-space method and combining it with the application of a topological argument, it is shown that the instability of a gap soliton can be described by the phase portrait of the solution. Surface gap solitons at the interface between a periodic inhomogeneous and a homogeneous medium are also discussed. Numerical calculations are presented accompanying the analytical results.     

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.     

Soliton perturbation theory for the modified nonlinear Schrödinger’s equation

The soliton perturbation theory is used to study the solitons that are governed by the modified nonlinear Schrödinger’s equation. The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained. In particular, the nonlinear gain (damping) and filters or the coefficient of finite conductivity are treated as perturbation terms for the solitons.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Dark–dark and dark–bright soliton interactions in the two-component defocusing nonlinear Schrödinger equation

We use the Inverse Scattering Transform machinery to construct multisoliton solutions to the 2-component defocusing nonlinear Schrödinger equation. Such solutions include dark–dark solitons, which have dark solitonic behaviour in both components, as well as dark–bright soliton solutions, with one dark and one bright component. We then derive the explicit expressions of two soliton solutions for all possible cases: two dark–dark solitons, two dark–bright solitons, and one dark–dark and one dark–bright soliton. Finally, we determine the long-time asymptotic behaviours of these solutions, which allows us to obtain explicit expressions for the shifts in the phases and in the soliton centers due to the interactions.     

Soliton chaos and lengthscale competition in nonlinear dynamics

Perturbing soliton-bearing completely integrable dynamics can give rise to rich and fascinating behaviour. If the perturbation introduces a lengthscale which is large compared to the spatial extent of the solitons present in the system, the solitons move like particles in an effective potential. Taking into account two-soliton interaction can result in chaotic behaviour called ‘soliton chaos’. In the opposite limit of a small-lengthscale perturbation the solitons acquire a dressing which effectively shields them from the perturbation. If the resulting ‘dressed solitons’ are subject to an additional long-wavelength perturbation they move like renormalised particles. Furthermore they can scatter nearly elastically. If the perturbation contains lengthscales which are comparable to one of the soliton's typical lengthscales then lengthscale competition can occur. Neither the particle approximation nor the dressed-particle approximation for the soliton is valid and complicated spatio-temporal behaviour is observed. We illustrate this scenario by means of the perturbed nonlinear Schrödinger equation. The perturbed sine-Gordon equation and the Ablowitz-Ladik equation are also discussed.     

Collision of delta-waves in a turbulent model studied via a distribution product

By using a product of distributions, the existence and collision of soliton delta-waves for a singular perturbation of the Burgers conservative equation are established. We also prove that singular solitons under collision behave as in classical soliton collision (for example, as described by the Korteweg-de Vries equation). The impossibility of two delta-wave collisions for the inviscid Burgers conservative equation is also verified. The introduction is dedicated to a motivation for our study.     

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.     

Adiabatic parameter dynamics of perturbed solitary waves

The soliton perturbation theory is used to study the solitons that are governed by the generalized Korteweg–de Vries equation in the presence of perturbation terms. The adiabatic parameter dynamics of the solitons in the presence of the perturbation terms are obtained.