Theoretical results of the dust acoustic waves in a magnetized dusty plasma with different dust particles

In this paper, the nonlinear dust acoustic waves (DAW) in a magnetized dusty plasmas with different dust grains are analytically investigated. New analytical solutions of the governing equation for this system have been obtained for the first time. The exact mathematical expressions of the nonlinear dust waves have been canvassed for the general case in magnetized dusty plasma containing different dust particles.     

The Kadomtsev–Petviashvili (KP), MKP,and coupled KP equations for two-ion-temperature dusty plasmas

It is well known that the Korteweg–de Vires (KdV) equation can describe small but finite amplitude dust acoustic waves in a dusty plasmas. In this paper, we use the reductive perturbation method and derive a Kadomtsev–Petviashvili (KP) equation, a modified KP (MKP) equation and a coupled KP equation for unmagnetized, collisionless, cold, and two-ion-temperature dusty plasmas with N different species of dust grains. We find that if a solitary wave exist in this system, the smaller grains have larger velocities and propagate longer distances than that of larger particles. The comparisons are given between the dusty plasma composed by different dust particles and the mono-sized dusty plasma.     

Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms

We report exact bright and dark solitary wave solution of the nonlinear Schrodinger equation (NLSE) in cubic–quintic non-Kerr medium adopting phase–amplitude ansatz method. We have found the solitary wave parameters along with the constraints under which bright or dark solitons may exist in such a media. Furthermore, we have also studied the modulation instability analysis both in anomalous and normal dispersion regime. The role of fourth order dispersion, cubic–quintic nonlinear parameter and self-steeping parameter on modulation instability gain has been investigated.     

Envelop solitons in dusty plasmas for warm dust

A nonlinear Schrödinger equation is obtained for the warm dusty plasmas. The modulational instability of envelop soliton is investigated in this paper. Both the temperature of the dust grains and the charge variations of dust grains affect the instability regions of the dusty plasmas. It also affect the amplitude and the width of the envelop soliton.     

Effects of the dust charge variation and non-thermal ions on multi-dimensional dust acoustic solitary structures in magnetized dusty plasmas

The combined effects of both adiabatic dust charge variation and non-thermally (fast) distributed ions on dust acoustic solitary structures are studied in a magnetized dusty plasmas consisting of the negatively and variably charged hot dust fluid, Boltzmann distributed electrons and non-thermally distributed ions. By using the reductive perturbation method, we derive the Korteweg-de Vries (KdV) equation governing the dust acoustic solitary waves. It is shown that the dust charge variation and the presence of non-thermally distributed ions would modify the nature of dust acoustic solitary structures significantly and may excite both dust acoustic solitary holes (soliton with a density dip) and positive solitons (soliton with a density hump).     

Obliquely propagating dust-acoustic solitary waves in cosmic dust-laden plasmas

Obliquely dust-acoustic solitary waves in a collisional, magnetized dusty plasmas having cold dust grains, isothermal electrons, two temperature isothermal ions and stationary neutrals are studied via a reductive perturbation method. It is found that the effects of two temperature ions, collisions, magnetic field and directional cosine of the waves vector k along the x-axis have vital roles in the behavior of the dust acoustic solitary waves. The present investigation can be relevance to the electrostatic solitary structures observed in various cosmic dust-laden plasmas, such as Saturn’s E-ring, noctilucent clouds, Halley’s comet and interstellar molecular clouds.     

Study of the dust acoustic waves in a magnetized dusty plasma with many different dust grains

The nonlinear dust waves in a magnetized dusty plasma with many different dust grains are analytically investigated. New analytical solutions for the governing equation of this system have been obtained for the dust acoustic waves in a dusty plasma for the first time. We derive exact mathematical expressions for the general case of the nonlinear dust waves in magnetized dusty plasma which contains different dust grains.     

Dust-acoustic solitary waves in a two-temperature electrons with charge fluctuations and nonisothermal ions

In this paper, a modified Korteweg–de Vries (mKdV) equation and Korteweg–de Vries (KdV) equation at critical ion density are derived for dusty plasmas consisting of hot dust fluid, nonisothermal ions and two-temperature electrons. The charge fluctuation dynamics of the dust grains has also been considered. It has been shown that the presence of a second component of electrons modifies the nature of dust acoustic (DA) solitary structures. The effects of two-temperature electrons, obliqueness and external magnetic field on the properties of DA solitary waves are discussed. Numerical investigations show that there exists only rarefactive solitary waves.     

Multidimensional dust acoustic solitary waves in inhomogeneous magnetized dusty plasmas with nonthermal ions

The linear dispersion relation and a modified variable coefficients Korteweg–de Vries (MKdV) equation governing the three-dimensional dust acoustic solitary waves are obtained in inhomogeneous dusty plasmas comprised of negatively charged dust grains of equal radii, Boltzmann distributed electrons and nonthermally distributed ions. The numerical results show that the inhomogeneity, the nonthermal ions, the external magnetic field and the collision have strong influence on the frequency and the nonlinear properties of dust acoustic solitary waves and both dust acoustic solitary holes (soliton with a density dip) and positive solitons (soliton with a density hump) are excited.     

EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHRDINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION

A system comprised of the nonlinear Schrodinger equation coupled to the Boussinesq equation (S-B equations) which dealing with the stationary propagation of coupled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed. To examine its solitary wave solutions, a reduced set of ordinary differential equations are considered by a simple traveling wave transformation. It is then shown that several new     

EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHR(O)DINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION

A system comprised of the nonlinear Schrodinger equation coupled to theBoussinesq equation (S-B equations) which dealing with the stationary propagation of cou-pled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed.To examine its solitary wave solutions, a reduced set of ordinary differential equations areconsidered by a simple traveling wave transformation. It is then shown that several newsolutions (either functional or parametrical) can be obtained systematically, in addition torederiving all known ones by means of our simple and direct algebra method with the helpof the computer algebra system Maple.     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.     

Propagation of the nonlinear damped Korteweg‐de Vries equation in an unmagnetized collisional dusty plasma via analytical mathematical methods

In this article, we consider the problem formulation of dust plasmas with positively charge, cold dust fluid with negatively charge, thermal electrons, ionized electrons, and immovable background neutral particles. We obtain the dust‐ion‐acoustic solitary waves (DIASWs) under nonmagnetized collision dusty plasma. By using the reductive perturbation technique, the nonlinear damped Korteweg‐de Vries (D‐KdV) equation is formulated. We found the solutions for nonlinear D‐KdV equation. The constructed solutions represent as bright solitons, dark solitons, kink wave and antikinks wave solitons, and periodic traveling waves. The physical interpretation of constructed solutions is represented by two‐ and three‐dimensional graphically models to understand the physical aspects of various behavior for DIASWs. These investigation prove that proposed techniques are more helpful, fruitful, powerful, and efficient to study analytically the other nonlinear nonlinear partial differential equations (PDEs) that arise in engineering, plasma physics, mathematical physics, and many other branches of applied sciences.     

Wave solitons in a coupled left-handed nonlinear transmission line: Effect of the coupling parameter

In this work, we investigate the dynamics of modulated waves in a discrete coupled Left-Handed nonlinear transmission line, assuming a one-dimensional (1-D) propagation variation. A nonlinear Schrödinger equation (NLSE) is derived, analytical solitons are found and the instability region is presented for this model.     

The formally variable separation approach for the dust-acoustic solitary waves with dust charge variation

The formally variable separation approach is used for handling the dust-acoustic solitary waves in a dusty plasma, including consideration of dust charge variation. New analytical solutions of nonlinear waves are formally derived for the governing equation of the system. We have triumphantly derived the exact analytical expressions and some approximate expressions of the nonlinear dust-acoustic waves in a dusty plasma under some special cases. The work introduces entirely new solutions and emphasizes the power of the newly developed method that can be used in problems with identical nonlinearities.     

Solitary wave solution of higher-order Korteweg–de Vries equation

An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Nonlinear modulation of SH waves in an incompressible hyperelastic plate

Propagation of nonlinear shear horizontal (SH) waves in a homogeneous, isotropic and incompressible elastic plate of uniform thickness is considered. The constituent material of the plate is assumed to be generalized neo-Hookean. By employing a perturbation method and balancing the weak nonlinearity and dispersion in the analysis, it is shown that the nonlinear modulation of waves is governed asymptotically by a nonlinear Schr?dinger (NLS) equation. Then the effect of nonlinearity on the propagation characteristics of asymptotic waves is discussed on the basis of this equation. It is found that, irrespective of the plate thickness, the wave number and the mode number, when the plate material is softening in shear then the nonlinear plane periodic waves are unstable under infinitesimal perturbations and therefore the bright (envelope) solitary SH waves will exist and propagate in such a plate. But if the plate material is hardening in shear in this case nonlinear plane periodic waves are stable and only the dark solitary SH waves may exist.     

Transverse nonlinear instability for two-dimensional dispersive models

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.     

Solitary Waves and Their Linear Stability in Nonlinear Lattices

Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schrödinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, which span a large number of lattice periods. In this limit, the allowed positions of solitary waves relative to the lattice, as well as their linear stability properties, hinge upon a certain recurrence relation which contains information beyond all orders of the usual two‐scale perturbation expansion. It follows that only two such positions are permissible, and of those two solitary waves, one is linearly stable and the other unstable. For a cosine lattice, in particular, the two possible solitary waves are centered at a maximum or minimum of the lattice, with the former being stable, and the analytical predictions for the associated linear stability eigenvalues are in excellent agreement with numerical results. Furthermore, a countable set of multi‐solitary‐wave bound states are constructed analytically. In spite of rather different physical settings, the exponential asymptotics approach followed here is strikingly similar to that taken in earlier studies of solitary wavepackets involving a periodic carrier and a slowly varying envelope, which underscores the general value of this procedure for treating multiscale solitary‐wave problems.