Bright and dark optical solitons with time-dependent coefficients in a non-Kerr law media

This paper obtains the 1-soliton solution of the nonlinear Schrödinger’s equations that governs the propagation of solitons through optical fibers. The study is conducted in presence of perturbation terms with non-Kerr law nonlinearity. The perturbation terms that are considered are third order dispersion, self-steepening and nonlinear dispersion. Both bright and dark soliton solutions are obtained.     

Quasi-particle theory of optical soliton interaction

The intra-channel collision of optical solitons, with non-Kerr law nonlinearities, is studied in this paper by the aid of quasi-particle theory. The perturbations terms that are considered in this paper are both of Hamiltonian as well as non-Hamiltonian type. The suppression of soliton–soliton interaction, in presence of these perturbation terms, is achieved. The nonlinearities that are studied in this paper are Kerr, power, parabolic and dual-power laws. The numerical simulations support the quasi-particle theory.     

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.     

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.     

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.     

Stochastic perturbation of dual-power law optimal solitons

The propagation of solitons through an optical fiber with dual-power law nonlinearity, in presence of stochastic perturbation term, is studied in this paper. The Langevin equations are derived. It is thus proved that solitons travel down an optical fiber with a fixed mean free velocity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic perturbation of solitons for Alfven waves in plasmas

The stochastic perturbation of solitons due to Alfven waves in plasmas, is studied in this paper, in addition to the deterministic perturbation terms. The Langevin equations are derived and it is proved that the soliton travels through the plasma with a fixed mean velocity.     

Soliton perturbation theory for the generalized Benjamin–Bona–Mahoney equation

The soliton perturbation theory is used to study the adiabatic parameter dynamics of solitons due to the Benjamin–Bona–Mahoney equations in presence of perturbation terms. The change in the velocity is also obtained in this paper.     

Soliton perturbation theory for the generalized fifth-order KdV equation

The soliton perturbation theory is used to study the adiabatic parameter dynamics of solitons due to the generalized fifth-order KdV equation in presence of perturbation terms. The adiabatic change of soliton velocity is also obtained in this paper.     

Optical soliton perturbation with time-dependent coefficients in a log law media

This paper obtains the exact 1-soliton solution to the nonlinear Schrödinger’s equation with log law nonlinearity in presence of time-dependent perturbations. The dispersion and nonlinearity are also taken to be time-dependent. The perturbation terms that are considered are linear attenuation and inter-modal dispersion. The constraint condition between the time-dependent coefficients also fall out as a necessary condition for the solitons to exist.     

Bäcklund transformation,superposition formulae and N-soliton solutions for the perturbed Korteweg–de Vries equation

With symbolic computation, under investigation in this paper is the perturbed Korteweg–de Vries equation for the nonlocal solitary waves and arrays of wave crests. Via the Hirota method, the bilinear form, Bäcklund transformation and superposition formulae are obtained. N-soliton solutions in terms of the Wronskian are constructed. Asymptotic analysis is used to analyze the collision dynamics, and figures are plotted to illustrate the influence of the perturbation. We find that the perturbation affects the propagation velocities of the solitons, but does not affect the amplitudes and widths of the solitons. Besides, the solitonic collisions turn out to be elastic.     

Optical solitons with non-Kerr law nonlinearity and inter-modal dispersion with time-dependent coefficients

This paper studies optical solitons with non-Kerr law nonlinearity, in the presence of inter-modal dispersion. The coefficients of group velocity dispersion, nonlinearity and inter-modal dispersion terms have time-dependent coefficients. The types of nonlinearity that are considered are Kerr, power, parabolic and dual-power laws. The solitary wave ansatz is used to carry out the integration of the governing nonlinear Schrödinger’s equation with time-dependent coefficients. Both, bright and dark optical solitons, are considered, in this paper. Finally, numerical simulations are also given in each of these cases. The only necessary condition for these solitons to exist is that these time-dependent coefficients of group velocity dispersion and inter-modal dispersion are Riemann integrable.     

1-Soliton solution of the complex KdV equation in plasmas with power law nonlinearity and time-dependent coefficients

In this paper, the complex Korteweg-de Vries equation with power law nonlinearity is studied in presence of perturbation terms. The exact 1-soliton solution is obtained. It will be seen that the time-dependent coefficients must be simply Riemann integrable for the solitons to exist. The solitary wave ansatz is used to carry out the integration.     

Coherent Structures in Weakly Birefringent Nonlinear Optical Fibers

This article studies two coupled nonlinear Schrodinger equations that govern the pulse propagation in weakly birefringent nonlinear optical fibers. The coherent structures for these equations, such as vector solitons and localized oscillating solutions, are studied analytically and numerically. Three types of localized oscillating structures are identified and their functional forms determined by perturbation methods. In some of these structures, infinite oscillating tails are present. The implications of these tails are also discussed.     

Soliton perturbation theory for the fifth order KdV-type equations with power law nonlinearity

The adiabatic parameter dynamics of solitons, due to fifth order KdV-type equations with power law nonlinearity, is obtained with the aid of soliton perturbation theory. In addition, the small change in the velocity of the soliton, in the presence of perturbation terms, is also determined in this work.     

Bright and dark solitons of the generalized nonlinear Schrödinger’s equation

This paper studies the generalized form of the nonlinear Schrödinger’s equation. The special cases of Kerr law, power law, parabolic law and the dual-power laws are considered. The 1-soliton solution is obtained in all of these four cases. The adiabatic parameter dynamics of the solitons due to perturbation terms are laid down. In addition, the analysis of dark soliton is also carried out. Finally, a few numerical simulations of these equations are given.     

Optical solitons with time-dependent dispersion,nonlinearity and attenuation in a power-law media

This paper studies optical solitons in a power-law media with time-dependent coefficients of dispersion, nonlinearity and attenuation. The 1-soliton solution is obtained for the nonlinear Schrödinger’s equation with power-law nonlinearity. In addition, a relation between these coefficients is obtained for the solitons to exist. Finally, the velocity of the soliton is also obtained in terms of these coefficients.     

Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation

In this paper, the higher-order generalized nonlinear Schrödinger equation, which describes the propagation of ultrashort optical pulse in optical fibers, is analytically investigated. By virtue of the Darboux transformation constructed in this paper, some exact soliton solutions on the continuous wave (cw) background are generated. The following propagation characteristics of those solitons are mainly discussed: (1) Propagation of two types of breathers which delineate modulation instability and bright pulse propagation on a cw background respectively; (2) Two types propagation characteristics of two-solitons: elastic interactions and mutual attractions and repulsions bound solitons. Those results might be useful in the study of ultrashort optical solitons in optical fibers.     

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.     

Three‐Dimensional Surface Water Waves Governed by the Forced Benney–Luke Equation

This paper studies the propagation of three‐dimensional surface waves in water with an ambient current over a varying bathymetry. When the ambient flow is near the critical speed, under the shallow water assumptions, a forced Benney–Luke (fBL) equation is derived from the Euler equations. An asymptotic approximation of the water's reaction force over the varying bathymetry is derived in terms of topographic stress. Numerical simulations of the fBL equation over a trough are compared to those using a forced Kadomtsev–Petviashvilli equation. For larger variations in the bathymetry that upstream‐radiating three‐dimensional solitons are observed, which are different from the upstream‐radiating solitons simulated by the forced Kadomtsev–Petviashvilli equation. In this case, we show the fBL equation is a singular perturbation of the forced Kadomtsev–Petviashvilli equation which explains the significant differences between the two flows.