Optical soliton and elliptic functions solutions of Sasa-satsuma dynamical equation and its applications

The Sasa-satsuma(SS) dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers. This dynamical model has important physical significance.In this article, two mathematical techniques namely, improved F-expansion and improved auxiliary methods are utilized to construct the several types of solitons such as dark soliton, bright soliton, periodic soliton, Elliptic function and solitary waves solutions of Sasa-satsuma dynamical equation. These results have imperative applications in sciences and other fields, and constructive to recognize the physical structure of this complex dynamical model. The computing work and obtained results show the influence and effectiveness of current methods.     

Solitons and nonlinear waves in the spiral magnetic structures

This paper presents a procedure to integrate the sine-Gordon model against the background of the stripe domain structure. The nonlinear dynamics of solitons and dispersive waves in the helical (stripe domain) structure of a ferromagnet with the easy plane anisotropy in the magnetic field, which is perpendicular to the spiral axis, has been investigated in detail. It has been shown that the formation and motion of solitons are accompanied by the local translations of the stripe structure and by the oscillations of its domain walls, which manifest themselves as “precursors” and “tails” of the solitons. The large time behavior of the weak-nonlinear dispersive wave field generated by an initial localized perturbation of the structure has been investigated. The ways of observing and exciting the solitons in the spiral structure of magnets and multiferroics are discussed.     

Stability of non compact steady and expanding gradient Ricci solitons

We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the linear stability of such gradient Ricci solitons.     

Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg-de Vries equations

This paper considers the coupled complex modified Korteweg-de Vries (mKdV) equations and presents a binary Darboux transformation for the equations. As a direct application, we give a classification of general soliton solutions derived from vanishing and non-vanishing backgrounds, on the basis of the dynamical behavior of the solutions. Special types of solutions in the presented solutions include breathers, bright-bright solitons, bright-dark solitons, bright-W-shaped solitons, and rogue wave solutions. Furthermore, dynamics and interactions of vector bright solitons are exhibited.     

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.     

Dynamical Behavior of Two-Soliton Solution Exhibited by Perturbed Sine-Gordon Equation

We use perturbation methods based on inverse scattering transform to study the dynamical behavior of the kink-antikink solitons exhibited by a driven-damped sine-Gordon equation with infinite boundary condition. It is shown that such solitons may be forced by damping and external forces to remain in a finite region instead of approaching infinity.     

On steady states in a collisionless plasma

Motivated by the collisionless shock problem in plasma, we study the stationary Vlasov-Maxwell system in one space dimension via a dynamical system approach. We construct flat-tail and oscillatory-tail solutions, as well as solitons in the presence of a nontrivial magnetic field. Flat-tail solutions connect different densities, pressures, and strengths of constant fields at x = ±∞, while oscillatory-tail solutions oscillate as x → −∞. We also construct topological horseshoes, which imply very complicated structures of some steady states. © 1996 John Wiley & Sons, Inc.. Inc.     

On the formation of gap solitons in nonlinear electromagnetic metamaterials

A nonlinear (Kerr‐type) electromagnetic metamaterial, characterized by a double‐Lorentz model of its frequency‐dependent linear effective dielectric permittivity and magnetic permeability, is considered. The formation of gap solitons in the low‐ and high‐frequency band gaps of this metamaterial is investigated analytically. Evolution equations governing the gap solitons, of the form of a nonlinear Klein‐Gordon and a nonlinear Schrödinger equation, are obtained, and the structure of their solutions is discussed.     

Exact M/W-shape solitary wave solutions determined by a singular traveling wave equation

It had been found that some nonlinear wave equations have the so-called “W/M”-shape-peaks solitons. What is the dynamical behavior of these solutions? To answer this question, all traveling wave solutions in the parameter space are investigated for a integrable water wave equation from a dynamical systems theoretical point of view. Exact explicit parametric representations of all solitary wave solutions are given.     

A semidiscrete Gardner equation

We construct the Darboux transformations, exact solutions, and infinite number of conservation laws for a semidiscrete Gardner equation. A special class of solutions of the semidiscrete equation, called table-top solitons, are given. The dynamical properties of these solutions are also discussed.     

Singular Ricci Solitons and Their Stability under the Ricci Flow

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow “pushes away” from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. The main objective of this paper is to study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the singular initial metric. We prove a local well-posedness result for the Ricci flow in suitably weighted Sobolev spaces, which in particular implies that the “opening up” of the singularity persists for the perturbations as well.     

Solitons in the Domain Structure of the Ferromagnet

By the method of dressing on a torus, we obtain and study solutions of the Landau–Lifshitz equation, which describe solitons in the stripe domain structure of the easy-axis ferromagnet. A specific feature of these solitons is that they are directly related to the domain structure: they induce translations and local oscillations of the domains. We find integrals of motion stabilizing the solitons on the background of the structure.     

Bifurcation and nonsmooth dynamics of solitary waves in the generalized long-short wave equations

Generalized wave equations, which model the resonant interaction between the long wave and the short wave, are considered. To understand the underlying complex dynamics, the bifurcations and nonsmooth behaviors of solitary waves for this system are investigated by qualitative techniques in dynamical systems. These complex behaviors may serve as mechanisms for fascinating physical phenomena such as solitons, chaos and turbulence.     

Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrödinger equation, and a hydrodynamic model.     

Exact solutions and dynamics of the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity

This paper investigate the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity by using the method of dynamical systems. The functions $q(x,t)=\phi(\xi)\exp(i(-kx+\omega t))$ are solutions of the equation (1.1) that governs the propagation of Raman solitons through optical metamaterials, where $\xi=x-vt$ and $\phi(\xi)$ in the solutions satisfy a singular planar dynamical system (1.5) which has two singular straight lines. By using the bifurcation theory method of dynamical systems to the equation of $\phi(\xi)$, bifurcations of phase portraits for this dynamical system are obtained under 28 different parameter conditions. Based on those phase portraits, 62 exact solutions of system (1.5) including periodic solutions, heteroclinic and homoclinic solutions, periodic peakons and peakons as well as compacton solutions are derived.     

Evolution of solitons of nonlinear Schrödinger equation with variable parameters

The nonlinear Schrödinger equation with variable parameters is solved by means of variational technique. A set of evolution equations for the solitary-wave solution is derived. The propagation properties of the solitons in an adiabatic amplification system and in a dispersion-decreasing fiber are analyzed. An explicit analytical approximate soliton solution in the exponentially dispersion-decreasing fiber is obtained using the derived dynamical equations.     

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.     

Soliton model of elementary electric charge

Conclusions Thus, we have shown that in the electrodynamics of the Klein-Gordon field there exist two spectra of three-dimensional electrostatic soliton solutions. One of them is dynamically stable, the other topologically stable. For each of the spectra, the value of the electrostatic potential at the center of the soliton is quantized, while the value of the electric field vanishes. At the periphery of the solitons, the electrostatic potential corresponds to the Coulomb law.The topological charge of the soliton is related to quantization of the value of the electrostatic potential at infinity: ,p=0, 1, 2, .... The topological soliton can obviously be regarded as a model of an elementary electric charge, an attractive feature of which is the absence of divergences of the integrals of the motion.For the dynamical solitons, we have _p =0,p=0, 1, 2, ....A rotating electrostatic soliton can be regarded as a soliton model of an elementary electric charge possessing an intrinsic magnetic moment. The magnetic field at the center of the rotating soliton is quantized, while at its periphery the field has a magneticdipole nature.Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 3, pp. 349–359, March, 1990.