Envelope Solitary Wave Solutions Research Articles

Envelope solitary wave and periodic wave solutions in a Madelung fluid description of generalized derivative resonant nonlinear Schrödinger equation

We have exploited the theory of Madelung fluid description in order to retrieve the correspondence between the envelope soliton like solutions of a generalized derivative resonant nonlinear Schrödinger equation (GDRNLSE) and the soliton like solutions of a stationary generalized Gardner equation. The motion under consideration was with stationary profile current velocity only and under suitable assumptions on current velocity in context of corresponding boundary conditions of fluid velocity and parametric constraints, the results were obtained. We mainly derived bright and dark (which involves grey and black) envelope soliton like solutions along with periodic wave envelope solutions having a phase depending on both space and time of the generalized derivative resonant nonlinear Schrödinger equation recovered from the corresponding solitary waves and periodic wave solutions of the stationary generalized Gardner equation.

Modulational Instability in the Interaction of Fast and Slow Magnetosonic Modes

The present paper studies the magnetic filaments produced by the interaction between the high-frequency magnetosonic wave and low-frequency magnetosonic wave. These phenomena can be explained by using an envelope solitary wave solution of the nonlinear Schrodinger equation (NLSE). The analytical results from NLSE are in agreement with the numerical result from Ref. Modi et al. (Phys. Plasmas 21: 102305, 2014). Both the linear and the nonlinear modulational instability are studied for the system. Two critical points are obtained. The first critical point gives the regions of both the instability and the stability, while the second one shows that if the perturbed wave number is close to the second critical point, the perturbed wave may be the most stable one. It is also found that all the higher order perturbed waves can be neglected in the stability region when the perturbed wave number is larger than that of the first critical point.

Nonlinear Wave Modulation in Nanorods Using Nonlocal Elasticity Theory

Abstract In this study, nonlinear wave modulation in nanorods is examined on the basis of nonlocal elasticity theory. Eringen's nonlocal elasticity theory is employed to derive nonlinear equations for the motion of nanorods. The analysis of the modulation of axial waves in nonlocal elastic media is performed, and the reductive perturbation method is used for the solution of the nonlinear equations. The propagation of weakly nonlinear and strongly dispersive waves is investigated, and the nonlinear Schrödinger (NLS) equation is acquired as an evolution equation. For the purpose of a numerical investigation of the nonlocal impacts on the NLS equation, it has been investigated whether envelope solitary wave solutions exist by utilizing the physical and geometric features of the carbon nanotubes. Amplitude dependent wave frequencies, phase and group velocities have been obtained and they have compared for the linear local, the linear nonlocal, the nonlinear local and the nonlinear nonlocal cases.

Solitons and other solutions of the nonlinear fractional Zoomeron equation

In order to investigate the nonlinear fractional Zoomeron equation, we propose three methods, namely the Jacobi elliptic function rational expansion method, the exponential rational function method and the new Jacobi elliptic function expansion method. Many kinds of solutions are obtained and the existence of these solutions is determined. For some parameters, these solutions can degenerate to the envelope shock wave solutions and the envelope solitary wave solutions. A comparison of our new results with the well-known results is made. The methods used here can also be applicable to other nonlinear partial differential equations. The fractional derivatives in this work are described in the modified Riemann–Liouville sense.

Modulation instability and rogue wave structures of positron-acoustic waves in q-nonextensive plasmas

Abstract A theoretical investigation is made to study envelope excitations and rogue wave structures of the newly predicted positron-acoustic waves (PAWs) in a plasma with nonextensive electrons and nonextensive hot positrons. The reductive perturbation technique (RPT) is used to derive a nonlinear Schrodinger equation-like (NLSE) which governs the modulational instability (MI) of the PAWs. The NLSE admits localized envelope solitary wave solutions of bright and dark type. These envelope solutions depend upon the intrinsic plasma parameters. It is found that the MI of the PAWs is significantly affected by nonextensivity and other plasma parameters. Further, the analysis is extended for the rogue wave structures of the PAWs. The findings of the present investigation should be useful in understanding the acceleration mechanism of stable electrostatic wave packets in four components nonextensive plasmas.

Modulational instability and exact solutions of nonlinear cubic complex Ginzburg-Landau equation of thermodynamically open and dissipative warm ion acoustic waves system

Using the standard reductive perturbation technique, a nonlinear cubic complex Ginzburg-Landau equation (CGL3) is derived to study the modulational instability of ion acoustic waves (IAWs) in an unmagnetized plasma consisting of warm adiabatic ions and non-thermal electrons, which form the background. The CGL3 is exactly solved by using two methods: the separation of variables and the complex tanh function which produces four solutions; the results are compared and good agreement exists in most predictions. The CGL3 admits localized envelope (solitary wave) solutions of bright and dark types. We study the effects of the thermal conductivity of ions and non-thermally distributed electrons $ \beta$ on the modulational stability. The effect of ion thermal conductivity makes the frequency $ \omega$ be complex in the nonlinear dispersion relation. In terms of the first mode $ \omega_{{1}}^{}$ , the amplitude of the wave moving in the (+ ve) x -direction decreases when any one of the parameters $ \beta$ and the temperature ratio $ \sigma$ (= T i/T eff) increases (where Teff and Ti are the effective and ion temperatures, respectively). It is found that the whole k - $ \sigma$ plane transforms to an unstable region as $ \beta$ $ \approx$ 0.85 . Referring to the internal energy formula, the order of magnitude ratios of different kinds and contributions of the internal energy changes are calculated for various sets { $ \sigma$ , $ \beta$ } . The obtained results encourage us to name the unstable system --which experiences loss and gain of ions escaping out or embedding themselves into the acoustic frequency domain, collision and recently experimentally due to ionization and deionization-- a non-equilibrium thermodynamic dissipative open IAW system.

Envelope ion-acoustic solitary waves in a plasma with positive-negative ions and nonthermal electrons

Modulation instability of ion-acoustic waves is investigated in a plasma composed of positive and negative ions as well as nonthermal electrons. For this purpose, a linear dispersion relation and a nonlinear Schrödinger equation are derived. The latter admits localized envelope solitary wave solutions of bright-(pulses) and dark-(holes, voids) type. The envelope soliton depends on the intrinsic plasma parameters. It is found that modulation instability of ion-acoustic waves is significantly affected by the presence of nonthermal electrons. The present model is used to investigate the solitary excitations in the (H+,O2−) and (H+,H−) plasmas, where they are presented in the D-region and F-region of the Earth’s ionosphere. The findings of this investigation should be useful in understanding the stable electrostatic wave packet acceleration mechanisms in positive-negative ion plasmas, and also enhance our knowledge on the occurrence of instability associated to the propagation of the envelope ion-acoustic solitary waves in space and in laboratory plasmas where two distinct groups of ions and non-Boltzmann distributed electrons are present.

Modulation instability of ion thermal waves in a pair-ion plasma containing charged dust impurities

Modulation instability of ion thermal waves (ITWs) is investigated in a plasma composed of positive and negative ions as well as a fraction of stationary charged (positive or negative) dust impurities. For this purpose, a linear dispersion relation and a nonlinear Schrödinger equation are derived. The latter admits localized envelope solitary wave solutions of bright (pulses) and dark (holes, voids) type. The envelope soliton depends on the intrinsic plasma parameters. It is found that modulation instability of ITWs is significantly affected by the presence of positively/negatively charged dust grains. The findings of this investigation should be useful in understanding the stable electrostatic wave packet acceleration mechanisms in pair-ion plasma, and also enhances our knowledge on the occurrence of instability associated to the existence of charged dust impurities in pair-ion plasmas. Our results should be of relevance for laboratory plasmas.

Envelope doubly periodic wave solutions of cubic nonlinear Schrödinger equation using the variational iteration method

One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The cubic nonlinear Schrödinger equation is used to illustrate effectiveness and convenience of the method, some new explicit exact travelling wave solutions are obtained which include envelope bell-type soliton solution, envelope solitary wave solution, and envelope doubly periodic wave solutions.

Exact solutions of Gerdjikov-Ivanov equation

The Gerdjikov-Ivanov equation which appears，in the fields of quanta field theory，weak nonlinear dispersive water wave，nonlinear optics， etc.， has been discussed. Nonlinear mathematical physics equation with higher order nonlinear terms is educed in the discussion of Gerdjikov-Ivanov equation. The Liénard equation is chosen as subsidiary ordinary differential equation, with the help of which and according to homogeneous balance principle，the Gerdjikov-Ivanov equation has been solved，and the envelope solitary wave solutions and envelope sinusoidal wave solutions have been obtained.

Linear and nonlinear propagation in negative index materials

We analyze linear propagation in negative index materials by starting from a dispersion relation and by deriving the underlying partial differential equation. Transfer functions for propagation are derived in temporal and spatial frequency domains for unidirectional baseband and modulated pulse propagation, as well as for beam propagation. Gaussian beam propagation is analyzed and reconciled with the ray transfer matrix approach as applied to propagation in negative index materials. Nonlinear extensions of the linear partial differential equation are made by incorporating quadratic and cubic terms, and baseband and envelope solitary wave solutions are determined. The conditions for envelope solitary wave solutions are compared with those for the standard nonlinear Schrodinger equation in a positive index material.

Thepth-order periodic solutions for a family ofN-coupled nonlinear Schrödinger equations

By using the solutions of an auxiliary Lamé equation, a direct algebraic method is proposed to construct the exact solutions of N-coupled nonlinear Schrödinger equations. The abundant higher-order exact periodic solutions of a family of N-coupled nonlinear Schrödinger equations are explicitly obtained with the aid of symbolic computation and they include corresponding envelope solitary and shock wave solutions.

Envelope exact solutions for the generalized nonlinear Schrödinger equation with a source

In this letter, the generalized nonlinear Schrodinger (GNLS) equation, phase-locked a source κ ei[χ(ξ)−ωt]: , is investigated. Firstly, we reduce this equation to a second-order non-homogeneous nonlinear ordinary differential equation via a plane wave transformation and some constraint conditions. Then we use some fractional transformations to study exact solutions in obtaining the GNLS equation. As a consequence, many types of exact solutions are deduced such as envelope rational solutions, envelope periodic wave solutions, envelope solitary wave solutions and envelope doubly periodic solutions. Similarly, the corresponding exact solutions can also be obtained for the Hirota-type GNLS equation with a source and their combined equation.

Construction of periodic and solitary wave solutions by the extended Jacobi elliptic function expansion method

In this paper, an extended Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the exact periodic solutions of some polynomials or nonlinear evolution equations. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in nonlinear mathematical physics. As a result, many exact travelling wave solutions are obtained which include new solitary or shock wave solution and envelope solitary and shock wave solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.

New solitons and periodic wave solutions for nonlinear evolution equations

The extended Jacobi elliptic function expansion methods with a computerized symbolic computation are used to construct the exact periodic solutions of some polynomials or nonlinear evolution equations. As a result, many exact travelling wave solutions are obtained which include new solitary or shock wave solution and envelope solitary and shock wave solutions.

New applications of developed Jacobi elliptic function expansion methods

The developed Jacobi elliptic function expansion methods are used to construct the exact periodic solutions of some non-polynomial or ( 2 + 1 ) -dimensional non-linear evolution equations. Some new results are presented, which include corresponding solitary or shock wave solutions and envelope solitary and shock wave solutions.

Exact theory for localized envelope modulated electrostatic wavepackets in space and dusty plasmas

Abstract. Abundant evidence for the occurrence of modulated envelope plasma wave packets is provided by recent satellite missions. These excitations are characterized by a slowly varying localized envelope structure, embedding the fast carrier wave, which appears to be the result of strong modulation of the wave amplitude. This modulation may be due to parametric interactions between different modes or, simply, to the nonlinear (self-)interaction of the carrier wave. A generic exact theory is presented in this study, for the nonlinear self-modulation of known electrostatic plasma modes, by employing a collisionless fluid model. Both cold (zero-temperature) and warm fluid descriptions are discussed and the results are compared. The (moderately) nonlinear oscillation regime is investigated by applying a multiple scale technique. The calculation leads to a Nonlinear Schrödinger-type Equation (NLSE), which describes the evolution of the slowly varying wave amplitude in time and space. The NLSE admits localized envelope (solitary wave) solutions of bright-(pulses) or dark- (holes, voids) type, whose characteristics (maximum amplitude, width) depend on intrinsic plasma parameters. Effects like amplitude perturbation obliqueness (with respect to the propagation direction), finite temperature and defect (dust) concentration are explicitly considered. Relevance with similar highly localized modulated wave structures observed during recent satellite missions is discussed.

Periodic solutions of nonlinear Schr dinger type equations

By using the solutions of an auxiliary elliptic equation, a direct algebraic method is proposed to construct the exact solutions of nonlinear Schrodinger type equations. It is shown that many exact periodic solutions of some nonlinear Schrodinger type equations are explicitly obtained with the aid of symbolic computation, including corresponding envelope solitary and shock wave solutions.

New Exact Solutions to NLS Equation and Coupled NLS Equations**The project supported by National Natural Science Foundation of China under Grant Nos. 40045016 and 40175016

A transformation is introduced on the basis of the projective Riccati equations, and it is applied as an intermediate in expansion method to solve nonlinear Schrödinger (NLS) equation and coupled NLS equations. Many kinds of envelope travelling wave solutions including envelope solitary wave solution are obtained, in which some are found for the first time.

Polariton effect in nonlinear pulse propagation

The joint influence of the polariton effect and Kerr-like nonlinearity on propagation of optical pulses is studied. The existence of different families of envelope solitary wave solutions in the vicinity of polariton gap is shown. The properties of solutions depend strongly on the carrier wave frequency. In particular, solitary waves inside and outside of the polariton gap exhibit different velocity and amplitude dependencies on their duration.