Optical solitons with differential group delay for Kudryashov’s model by the auxiliary equation mapping method

In this paper, the auxiliary equation mapping method is employed to extract optical solitons and other solutions for special cases of Kudryashov’s model in birefringent fibers that is studied without the effect of four wave mixing effects. Bright, dark and singular solitons and other solutions emerge from the auxiliary equation mapping method.     

Stationary optical solitons with nonlinear chromatic dispersion having quadratic–cubic law of refractive index

Stationary optical solitons with quadratic–cubic law of nonlinear refractive index and nonlinear chromatic dispersion are retrieved. The model is considered with linear temporal evolution as well as with generalized temporal evolution. The results are in terms of Appell's hypergeometric function whose convergence criteria are also presented.     

Solitons for perturbed Gerdjikov–Ivanov equation in optical fibers and PCF by extended Kudryashov’s method

This paper reveals bright, dark and singular soliton solutions to the perturbed Gerdjikov–Ivanov equation by the aid of extended Kudryashov’s method. The nonlinear terms appear with full nonlinearity in order to give a generalized flavor to the model. As a byproduct of this scheme, plane waves and singular periodic solutions fall out and these solutions are listed as well.     

Solitons and conservation laws in magneto–optic waveguides having parabolic–nonlocal law of refractive index

This paper reveals soliton solutions to magneto–optic waveguides that maintain parabolic–nonlocal law of refractive index. The unified Riccati equation expansion together with extended auxiliary equation approach together reveal bright, dark, singular as well as straddled optical solitons. These soliton solutions are obtained through a limiting process when the modulus of ellipticity approaches unity. Finally, the conservation laws are also listed.     

Exact solutions of nonlinear Schrodinger’s equation with dual power-law nonlinearity by extended trial equation method

In this paper, we acquire the soliton solutions of the nonlinear Schrodinger’s equation with dual power-law nonlinearity. Primarily, we use the extended trial equation method to find exact solutions of this equation. Then, we attain some exact solutions including soliton solutions, rational and elliptic function solutions of this equation using the extended trial equation method.     

The extended (G′/G)-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law non-linearity by the extended (G′/G)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.     

Solitons and conservation laws in magneto–optic waveguides with generalized Kudryashov’s equation

The modified sub–ODE approach secures optical soliton solutions in magneto–optic waveguides with generalized Kudryashov’s equation. The solutions are initially drafted in terms of Jacobi’s elliptic functions. The limiting process, when the modulus of ellipticity approaches zero or unity, the soliton solutions emerge. A few solutions in terms of Weierstrass’ elliptic functions are also revealed. Finally, the conservation laws are computed for the model using the multiplier approach.     

Optical solitons for non-Kerr law nonlinear Schrödinger equation with third and fourth order dispersions

In this paper, we obtain optical soliton solutions for non-Kerr law nonlinear Schrödinger equation (NLSE) with third order (3OD) and fourth order dispersions (4OD). We will use two integration schemes, namely sin-cosine method and Bernoulli’s equation approach with five laws of nonlinearities. Sine-cosine method is applicable to Kerr, power and anti-cubic laws, this method provides bright soliton solutions. The second method is applicable to parabolic and cubic quintic laws, this method generates dark soliton. The results may be used in discussing the propagation of optical solitons in highly dispersive media with Kerr, power, anti-cubic, parabolic and cubic quintic law nonlinearities.     

On optical solitons of the resonant Schrödinger’s equation in optical fibers with dual-power law nonlinearity and time-dependent coefficients

In this paper, the generalized tanh method is used to construct dispersive optical solitons for the resonant Schrödinger’s equation with dual-power law nonlinearity and time-dependent coefficients. Some optical solitons of this equation have been obtained. We have proved that the terms of equation, like velocity, are effected by examined dependent coefficients terms of the equation.     

Wavelength switching in a mixed structure of a long-period and a Bragg fiber gratings

We built a numerical model for evaluating the coupling processes of a mixed structure of a Bragg fiber grat-ing and a long-period grating. From the numerical results, we not only confirmed the wavelength switchingphenomena observed in previously reported experiments, but also discovered a new coupling mechanism,which generated the reflection of a signal with its wavelength longer than the Bragg wavelength. Thedependencies of the wavelength switching behaviors on various parameters of the mixed grating structure     

Measurement of temperature and refractive index based on surface long-period gratings deposited onto a D-shaped photonic crystal fiber

We propose a surface long-period grating (LPG) based on a D-shaped photonic crystal fiber (PCF). The D-shaped PCF is fabricated by a side-polishing technique. The surface LPG based on periodic patterns of photoresist (PR) is formed by using the spin-coating and the standard contact lithography methods. Resonant coupling is created by the surface PR-LPG in the D-shaped PCF. The resonant peak shifts to longer wavelength as the ambient index is increased and shifts to shorter wavelength as the temperature is increased. The total wavelength shift is measured to be 122 nm in the refractive index range from 1 to 1.45 and the temperature sensitivity is measured to be −0.3 nm/°C in the temperature range from 30 to 100°C.     

Optical solitons and other solutions to the conformable space–time fractional complex Ginzburg–Landau equation under Kerr law nonlinearity

This study reveals the dark, bright, combined dark–bright, singular, combined singular optical solitons and singular periodic solutions to the conformable space–time fractional complex Ginzburg–Landau equation. We reach such solutions via the powerful extended sinh-Gordon equation expansion method (ShGEEM). Constraint conditions that guarantee the existence of valid solitary wave solutions are given. Under suitable choice of the parameter values, interesting three-dimensional graphs of some of the obtained solutions are plotted.     

Generalized Ohm’s law and potential equation in computational weakly-ionized plasmadynamics

A variant of the generalized Ohm’s law that is suited for a weakly-ionized multicomponent plasma in a magnetic field is here derived. The latter takes into consideration the current due to the non-neutrality of the plasma, the current due to the Hall effect, and the currents due to the ion slip associated with each type of ion. An equation for the electric field potential applicable to a non-neutral multicomponent plasma in the presence of a magnetic field is then presented. Despite some similarities between the potential equation and the Poisson equation, it is argued that the discretization of the potential equation cannot be accomplished in the same manner by using only central differences. It is here proven (and subsequently verified through a test case) that when the plasma exhibits conjunctly a high Hall parameter and a high electrical conductivity gradient, the centered stencils introduce spurious oscillations which can lead to severe numerical error. A novel discretization of the potential equation consisting of a blend of central and upwind differences is then presented. The proposed scheme is consistently monotonic for any value of the Hall parameter and is second-order accurate except in the vicinity of discontinuities.     

New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations

In this paper, we construct many new types of Jacobi elliptic function solutions of nonlinear evolution equations using the so-called new extended auxiliary equation method. The effectiveness of this method is demonstrated by applications to three higher order nonlinear evolution equations, namely, the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, the higher order dispersive nonlinear Schrödinger equation and the generalized nonlinear Schrödinger equation. The solitary wave solutions and periodic solutions are obtained from the Jacobi elliptic function solutions. Comparing our new results and the well-known results are given.     

Optical solitons in (2+1)–Dimensions with Kundu–Mukherjee–Naskar equation by extended trial function scheme

This paper addresses Kundu–Mukherjee–Naskar equation by the aid of extended trial function method to recover optical soliton solutions in (2+1)–dimensions. The integration algorithm revealed doubly periodic functions. Upon taking the limiting values of the modulus of ellipticity, bright and singular solitons as well as singular periodic solutions emerge. Additional solutions such as plane waves also fall out of the scheme.     

Zipf’s law and distribution of population in Indian cities

The Zipf’s law is studied here in the context of size distribution of Indian cities and the power law exponent is estimated. We have used the data from the Indian censuses of 1981,1991 and 2001. The analysis shows that the population distribution in Indian cities do follow a power law similar to the ones found in other countries. The scaling exponent are found to be 2.15 ± 0.01 for 1981, 2.11 ± 0.01 for 1991 and 2.05 ± 0.02 for 2001 from the linear fit. We have also estimated the scaling exponent from the maximum likelihood estimator technique which is found to be 2.04 ±.07 for the year 2001.     

A method of measuring the refractive index of extraordinary ray in uniaxial crystal with optic axis at an arbitrary orientation

A new equation to measure the refractive index of extraordinary ray in uniaxial crystal with the optic axis at an arbitrary orientation has been given in this letter,and the term in this equation makes the measurements to be relatively easy.The theoretical study shows that the accuracy achieved in the experiments attains to the order of magnitude in 10~(-3).     

Exact spatial soliton solution for nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index

In this paper, we analyze (2 + 1)-dimensional nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index in the transverse direction, and obtain an exact solution in explicit form using an ansatz method. Finally, the stability of the solution is discussed numerically, and the result reveals that the solution is stable to the finite initial perturbations.     

Optical solitons in birefringent fibers with Radhakrishnan–Kundu–Lakshmanan equation by a couple of strategically sound integration architectures

This paper addresses optical solitons in birefringent fibers modeled by Radhakrishnan–Kundu–Lakshmanan equation in coupled vector form. Bright, dark and singular solitons are recovered by trial and modified simple function principles.     

Optical solitons and periodic solutions of the (2 + 1)-dimensional nonlinear Schrödinger's equation

In this Letter, we consider the $(2+1)$-dimensional nonlinear Schrödinger's equation. With the aid of the Jacobian elliptic equation, we derive the exact bright soliton, dark soliton, singular soliton and periodic solutions of this equation expressed in terms of trigonometric functions, hyperbolic functions and Jacobian elliptic functions, respectively. Finally, for certain parametric values, we plot three dimensional graphics of modulus, real and imaginary parts of some solutions, which can help one better understand their dynamical behavior via their graphics analysis.