Analytical study of the nonlinear dust-acoustic waves in an unmagnetized dusty plasma

The nonlinear dust-acoustic waves in an unmagnetized dusty plasma, including consideration of the dust charge variation, is analytically investigated by using the formally variable separation approach. The exact analytical solutions in the general case are also obtained.     

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.     

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.     

Analytical study of nonlinear dust acoustic waves in two-dimensional dust plasma with vortex-like ion distribution

The nonlinear dust acoustic waves in two-dimensional dust plasma with vortex-like ion distribution are analytically investigated by using the formally variable separation approach. New analytical solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma for the first time. We derive exact analytical expressions for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with vortex-like ion distribution.     

The formally variable separation approach for the modified Zakharov–Kuznetsov equation

The formally variable separation approach is used for handling the modified Zakharov–Kuznetsov equation in plasmas. New analytical solutions of nonlinear waves are formally derived for the governing equation of the system. The work introduces entirely new solutions and emphasizes the power of the newly developed method that can be used in problems with identical nonlinearities.     

Theoretic study of dust acoustic waves in a dusty plasma with dust charge variation

The nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation is investigated by using the formally variable separation approach. New solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma firsthand. We derive exact mathematical expressions and numerical simulation studies for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation.     

The Riemann problem for an isentropic ideal dusty gas flow with a magnetic field

This paper deals with Riemann problem for one-dimensional inviscid, isentropic, and perfectly conducting ideal dusty gas flow with a transverse magnetic field. The explicit expressions of elementary waves are derived in terms of the density, velocity, and transverse magnetic induction of an ideal dusty gas flow. The analytical properties of elementary wave curves and the influence of parameter on the elementary waves are discussed. A new approach is proposed to resolve the Riemann problem. By applying this approach, we obtain 10 kinds of exact solutions and their corresponding criteria.     

Mathematical view with observational/experimental consideration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas

Plasmas are believed to be possibly the most abundant form of ordinary matter in the Universe, supporting a variety of the wave phenomena, while a dusty plasma is of interest as a non-Hamiltonian system of interacting particles. In this Letter, symbolic computation on an observationally/experimentally-supported (2+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili-Burgers-type equation is done, for certain dust-acoustic, electron-acoustic, positron-acoustic, magneto-acoustic, dust-magneto-acoustic, ion-acoustic, dust-ion-acoustic and/or quantum-dust-ion-acoustic waves in one of the cosmic/laboratory dusty plasmas. Auto-Bäcklund transformation and families of the solitonic solutions are obtained, for the electrostatic wave potential, perturbation of the magnitude of the magnetic field, fluctuation of electron or ion density, or radial-direction component of the velocity of ions or dust particles, relying on such plasma coefficient functions as the nonlinearity, dispersion, dusty-fluid-viscosity/Burgers-dissipation, geometric-effect and diffraction/transverse-perturbation coefficients. Shock structures presented in this Letter are very close to the experimental results previously reported. Future plasma observations/experiments might verify some other effects offered by our analytic results with respect to those plasma coefficient functions.     

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.     

3+1 dimensional envelop waves and its stability in magnetized dusty plasma

It is well known that there are envelope solitary waves in unmagnetized dusty plasmas which are described by a nonlinear Schrodinger equation (NLSE). A three dimension nonlinear Schrodinger equation for small but finite amplitude dust acoustic waves is first obtained for magnetized dusty plasma in this paper. It suggest that in magnetized dusty plasmas the envelope solitary waves exist. The modulational instability for three dimensional NLSE is studied as well. The regions of stability and instability are well determined in this paper.     

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.     

Construction of traveling wave solutions of the (2 + 1)-dimensional modified KdV-KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)-dimensional modified KdV-KP equation are derived by employing an ansatz method, named the enhanced (G′/G)-expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum-modified KdV-KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum-modified KdV-KP system supports solitary waves having different shapes and speeds for different values of the parameters.     

Electrostatic wave solutions for a nonlinear coupled system in an inhomogeneous collisional dusty magnetoplasma

In an inhomogeneous collisional dusty magnetoplasma, a new coupled (3 + 1)-dimensional nonlinear system is derived for the low-frequency electrostatic waves considering the collision between ions and neutrals. It is demonstrated that due to the collision, the scaling symmetry of the system is destroyed. By means of the classical Lie group approach, two types of exact similarity waves are obtained which show three important features. First, various waves can be constructed from these solutions, such as solitary waves, shock waves and periodic waves. Second, these waves may have shears that are time-dependent, linear and nonlinear. Third, the electrostatic potential and the parallel electron velocity possess more freedoms in the sense that they can have either same or different wave forms.     

Analytical existence of solutions to a system of nonlinear equations with application

We study the existence of analytical solutions to a system of nonlinear equations under constraints linked to the analysis of a road safety measure without computing second derivatives. We formally demonstrate this existence of solutions by applying a matrix inversion principle through Schur complement to a subsystem of equations derived from the main system. The analytical results thus obtained are used to construct a simple iterative procedure to look for optimal solutions as well as an initial solution adapted to data of each case study. We illustrate our results with simulated accident data obtained from the setting-up of a road safety measure. The numerical solutions thus obtained are then compared to those given through a classic Newton-Raphson type approach directly applied to the main system.     

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.     

Volterra series approximation for rational nonlinear system

Rational nonlinear systems are widely used to model the phenomena in mechanics, biology, physics and engineering. However, there are no exact analytical solutions for rational nonlinear system. Hence, the approximate analytical solutions are good choices as they can give the estimation of the states for system analysis, controller design and reduction. In this paper, an approximate analytical solution for rational nonlinear system is derived in terms of the solution of a polynomial system by Volterra series theory. The rational nonlinear system is transformed to a singular polynomial system with finite terms by adding some algebraic constraints related to the rational terms. The analytical solution of singular polynomial system is approximated by the summation of the solutions of Volterra singular subsystems. Their analytical solutions are derived by a novel regularization algorithm. The first fourth Volterra subsystems are enough to approximate the analytical solution to guarantee the accuracy. Results of numerical experiments are reported to show the effectiveness of the proposed method.     

On the formation of shock and soliton in a dense quantum dusty plasma with cylindrical geometry

Excitation of nonlinear waves in a quantum dusty plasma with various effects is analyzed when the geometry is cylindrical.This introduces the effect of finite boundary conditions on the solitary waves so generated. it is observed that the nonlinear equation deduced is cylindrical KP–Burger type leading to the generation of Shock Wave. Different situations which arises in various parameter regions are considered separately and the form of the nonlinear excitations are obtained explicitly.     

Generation mechanism of rogue waves for the discrete nonlinear Schrödinger equation

In this paper, we analyze the generation mechanism of rogue waves for the discrete nonlinear Schrödinger (DNLS) equation from the viewpoint of structural discontinuities. First of all, we derive the analytical breather solutions of the DNLS equation on a new nonvanishing background through the Darboux transformation (DT). Via the explicit expressions of group and phase velocities, we give the parameter conditions for existence of the velocity jumps, which are consistent with the derivation of rogue waves via the generalized DT. Furthermore, to verify such statement, we apply the Taylor expansion to the breather solutions and find that the first-order rogue wave can be obtained at the velocity-jumping point. Our analysis can help to enrich the understanding on the rogue waves for the discrete nonlinear systems.     

Novel rogue waves in an inhomogenous nonlinear medium with external potentials

Employing the similarity transformation connected with the standard constant coefficient nonlinear Schrödinger equation, we obtain the analytical rogue wave solutions to a generalized variable coefficient nonlinear Schrödinger equation with external potentials describing the pulse propagation in nonlinear media with transverse and longitudinal directions nonuniformly distributed. Based on the obtained solutions, abundant structures of rogue waves are constructed by selecting some special parameters. The main properties as well as the dynamic behaviors of these rogue waves are discussed by direct computer simulations.     

Remarks on a Strongly Nonlinear Model for Two‐Layer Flows with a Top Free Surface

We revisit in this paper the strongly nonlinear long wave model for large amplitude internal waves in two‐layer flows with a free surface proposed by Choi and Camassa [1] and Barros et al. [2]. Its solitary‐wave solutions were the object of the work by Barros and Gavrilyuk [3], who proved that such solutions are governed by a Hamiltonian system with two degrees of freedom. A detailed analysis of the critical points of the system is presented here, leading to some new results. It is shown that conjugate states for the long wave model are the same as those predicted by the fully nonlinear Euler equations. Some emphasis will be given to the baroclinic mode, where interfacial waves are known to change polarity according to different values of density and depth ratios. A critical depth ratio separates these two regimes and its analytical expression is derived directly from the model. In addition, we prove that such waves cannot exist throughout the whole range of speeds.