Solitary Solutions Research Articles

Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives

We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider first the general case of presence of monomials of the both (odd and even) grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade. The methodology is illustrated by numerous examples.

Numerical Solutions of Two-Way Propagation of Nonlinear Dispersive Waves Using Radial Basis Functions

We obtain the numerical solution of a Boussinesq system for two-way propagation of nonlinear dispersive waves by using the meshless method, based on collocation with radial basis functions. The system of nonlinear partial differential equation is discretized in space by approximating the solution using radial basis functions. The discretization leads to a system of coupled nonlinear ordinary differential equations. The equations are then solved by using the fourth-order Runge-Kutta method. A stability analysis is provided and then the accuracy of method is tested by comparing it with the exact solitary solutions of the Boussinesq system. In addition, the conserved quantities are calculated numerically and compared to an exact solution. The numerical results show excellent agreement with the analytical solution and the calculated conserved quantities.

Homotopic mapping solitary traveling wave solutions for the disturbed BKK mechanism physical model

Using the trial equation method, a Broer—Kau—Kupershmidt (BKK) mechanism physical model is obtained, and the exact and approximate solitary traveling wave solutions are found. As an example, it is pointed out that the solitary traveling wave approximate solutions have better accurate degree by using the homotopic mapping theory.

Comments on “A new algorithm for automatic computation of solitary wave solutions to nonlinear partial differential equations based on the Exp-function method”

In their recent manuscript Lei Zhao et al. claim that they have constructed an algorithm for automatic construction of solitary wave solutions to nonlinear partial differential equations based on the Exp-function method. We argue that this algorithm does inherit all problems associated to the Exp-function method. We show that this algorithm produces solutions which do not satisfy the original differential equation. Even if the solutions would be correct, the Exp-function method would not be able to determine necessary and sufficient conditions for the existence of solitary solutions in the space of initial conditions and parameters.

Nonlinear vector waves of a flexural mode in a chain model of atomic particles

Flexural transverse waves in an anharmonic chain of atoms is considered and the nonlinear vector equation for the phonon modes in the long-wave approximation is derived taking into account the weak dispersion effects. Particular cases of the equation derived are discussed; among them the vector mKdV equation (Gorbacheva and Ostrovsky, 1983) [12], as well as the new model vector equations dubbed here the ‘second order cubic Benjamin–Ono (socBO) equation’ and ‘nonlinear pseudo-diffusion equation’. Stationary solutions to the equation derived are studied and it is found in which cases physically reasonable periodic and solitary type solutions may exist. The simplest non-stationary interactions of solitary waves of different polarisation are studied by means of numerical simulation. A new interesting phenomenon is revealed when two solitons of the same or opposite polarities interact elastically, whereas the interaction of two solitons lying initially in the perpendicular planes is essentially inelastic resulting in the survival of only one soliton and destruction of another one.

Applications of He’s variational principle method and modification of truncated expansion method to the coupled Klein–Gordon Zakharov equations

In this paper, we establish exact solutions for the coupled nonlinear Klein–Gordon Zakharov equations. The He’s semi-inverse and the modification of truncated expansion methods are used to construct exact solutions of this equation. We apply He’s semi-inverse method to establish a variational theory for the coupled nonlinear Klein–Gordon Zakharov equations. Based on this formulation, a solitary solution can be easily obtained using the Ritz method. The modification of truncated expansion method is used to construct exact solutions of the coupled nonlinear Klein–Gordon Zhakhrov equations. Moreover, it is observed that the suggested techniques are compatible with the physical nature of such problems.

On some solitary and cnoidal wave diffraction solutions of the Green–Naghdi equations

Nonlinear waves of the solitary and cnoidal types are studied in constant and variable water depths by use of the Irrotational Green–Naghdi (IGN) equations of different levels and the original Green–Naghdi (GN) equations (Level I). These equations, especially the IGN equations, have been established more recently than the classical water wave equations, therefore, only a handful applications of the equations are available. Moreover, their accuracies and the conditions under which they are applicable need to be studied. As a result, we consider a number of surface wave propagation and scattering problems that include soliton propagation and fission over a bump and onto a shelf, colliding solitons, soliton generation by an initial mound of water and diffraction of cnoidal waves due to a submerged bottom shelf, and compare the predictions with experimental data when available.

Exact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized $(\frac{G'}{G})$-Expansion Method

In this article, we construct the exact traveling wave solutions of the nonlinear (2+1)-dimensional Davey-Stewartson equation (D-S) using the generalized $(\frac{G'}{G})$-expansion method which play an important role in mathematical physics. As a result, hyperbolic, trigonometric and rational function solutions with parameters are obtained. When these parameters are taken special values, the solitary and periodic solutions are derived from the hyperbolic and trigonometric function solutions respectively. New complex type traveling wave solutions to the nonlinear (2+1)-dimensional Davey-Stewartson equation were obtained with Liu's theorem.

Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation

In the present study, the nonlinear Boussinesq type equation describe the bi-directional propagation of small amplitude long capillary–gravity waves on the surface of shallow water. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional fourth-order nonlinear Boussinesq equation with constant coefficient. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions are discussed.

Modulational instability of electrostatic acoustic waves in an electron-hole semiconductor quantum plasma

The modulational instability of quantum electrostatic acoustic waves in electron-hole quantum semiconductor plasmas is investigated using the quantum hydrodynamic model, from which a modified nonlinear Schrodinger equation with damping effects is derived using the reductive perturbation method. Here, we consider the combined effects of quantum recoil, quantum degenerate pressures, as well as the exchange-correlation effect standing for the electrons (holes) spin. The modulational instability for different semiconductors (GaAs, GaSb, and InP) is discussed. The collision between electron (hole) and phonon is also investigated. The permitted maximum time for modulational instability and the damping features of quantum envelope solitary wave are all determined by the collision. The approximate solitary solution with damping effects is presented in weak collision limit. The damping properties were discussed by numerical method.

Nonlinear Schrödinger equation on graphs: recent results and open problems

In this paper, an introduction to the new subject of nonlinear dispersive Hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of the nonlinear Schrödinger (NLS) equation. Special consideration is given to the existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning the NLS equation on a star graph: the standing waves of the NLS equation on a graph with a δ interaction at the vertex, and the scattering of fast solitons through a Y-junction in the cubic case. The emphasis is on a description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.

Approximate solution of solitary wave for nonlinear-disturbed time delay long-wave system

The solitary wave approximate solutions for a class of nonlinear-disturbed time delay long-wave system are considered. First, we introduce into exact solution of a non-disturbed typical long-wave system. Then, by using the homotopic mapping and an improved technique, the approximate expansions of the traveling wave solutions for the nonlinear-disturbed time delay long-wave systems are constructed.

OVERVIEW OF IONOSPHERIC MODIFICATION BY HIGH FREQUENCY (HF) HEATERS-THEORY

The scenarios of achieving efiective ionospheric modiflcation are summarized and the likely physical processes engaging linear and nonlinear mode conversions through plasma inhomogeneity and nonlinearity are revealed. Parametric instabilities, which are the directly relevant processes to achieve efiective heating of the ionospheric F region, are formulated and analyzed. The threshold flelds and growth rates of instabilities are obtained. The nonlinear Schrodinger equation governing the nonlinear evolution of Langmuir waves is derived and analyzed. The nonlinear periodic and solitary solutions of the equation are obtained. The analyses illustrate the conditions for the generation of Langmuir soliton and nonlinear periodic Langmuir waves.

Why calcium channel blockers could be an elite choice in the treatment of Alzheimer’s disease: a comprehensive review of evidences

The universality and vitality of calcium ions are implicit from its diverse physiological functions, from regulation of enzymes to synaptic plasticity and memory. However, overloading of these ions could result in life-threatening degenerative disorders. Calcium channels, which are involved in the transport of calcium ions, are targeted and blocked to prevent its overload, favoring vascular relaxation. Besides this primary action, calcium channel blockers (CCBs) also genuinely exhibit cognitive-enhancing abilities and reduce the risk of dementia, especially of Alzheimer's type. Alzheimer's disease (AD) is triggered by the disruption of calcium homeostasis, which underlies the observed progressive cognitive decline that occurs in this neurodegenerative disorder. Fortunately, CCB is expected to offer neuroprotection and additionally demonstrate antiamyloid, antitau, antiphospholipase, antiplatelet, antioxidant, and anti-inflammatory activity, a solitary solution to all the subcellular physiological complications that are observed in AD. Therefore, the aim of this review was to unearth the prospective of CCB against cognitive frailty with a sole purpose of elucidating CCB as cognitive enhancers, which could find its use as a drug in prevention or treatment of AD.

Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations

The coupled nonlinear Schrödinger equations are highly used in modeling the various phenomena in nonlinear fiber optics, like propagation of pulses. Efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system. In this paper, we introduce a new version of the Cox and Matthews third order exponential time differencing Runge–Kutta (ETD3RK) scheme based on the (1,2)-Padé approximation to the exponential function. In addition, we present its local extrapolation form to improve temporal accuracy to the fourth order. The developed scheme and its extrapolation are seen to be strongly stable, which have ability to damp spurious oscillations caused by high frequency components in the solution. A computationally efficient algorithm of the new scheme, based on a partial fraction splitting technique is presented. In order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear Schrödinger equations and performed several numerical experiments on them. The numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long-range solitary solutions given by coupled nonlinear Schrödinger equations and conserved the conserved quantities mass and energy exactly, to at least five decimal places.

Obliquely propagating solitary structures in a heavier ion Fermi plasma

By employing quantum hydrodynamic formulation, oblique propagation of electrostatic ion waves and nonlinear structures are investigated in a magnetized dense Fermi plasma. Constituents are quantum (degenerate) electrons and non-degenerate mobile ions in presence of stationary massive ions (either positive or negative) in background. To reveal features of low frequency ion waves, where effects of Fermi degeneracy and quantum diffraction are significant, linear dispersion equation is derived by Fourier analysis of model equations. Nonlinear solitary pulse solution in low-amplitude limit is obtained via Korteweg de Vries equation. It is shown that wave dispersion due to electron Fermi pressure is important at very short wavelength regime. Effects of concentration of heavy ions and angle of propagation on wave are also studied. Results are discussed numerically with relevance of superdense plasmas mainly found in astrophysical regimes.

Ultrashort optical solitons in transparent nonlinear media with arbitrary dispersion

We consider the propagation of ultrashort optical pulses in nonlinear fibers and suggest a new theoretical framework for the description of pulse dynamics and exact characterization of solitary solutions. Our approach deals with a proper complex generalization of the nonlinear Maxwell equations and completely avoids the use of the slowly varying envelope approximation. The only essential restriction is that fiber dispersion does not favor both the so-called Cherenkov radiation, as well as the resonant generation of the third harmonics, as these effects destroy ultrashort solitons. Assuming that it is not the case, we derive a continuous family of solitary solutions connecting fundamental solitons to nearly single-cycle ultrashort ones for arbitrary anomalous dispersion and cubic nonlinearity.

Electron acoustic soliton energy of the Kadomtsev-Petviashvili equation in the Earth’s magnetotail region at critical ion density

The nonlinear properties of small amplitude electron-acoustic solitary waves (EAWs) in a homogeneous system of unmagnetized collisionless plasma consisted of a cold electron fluid and isothermal ions with two different temperatures obeying Boltzmann type distributions have been investigated. A reductive perturbation method was employed to obtain the Kadomstev-Petviashvili (KP) equation. At the critical ion density, the KP equation is not appropriate for describing the system. Hence, a new set of stretched coordinates is considered to derive the modified KP equation. Moreover, the solitary solution, soliton energy and the associated electric field at the critical ion density were computed. The present investigation can be of relevance to the electrostatic solitary structures observed in various space plasma environments, such as Earth’s magnetotail region.

Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity

By a perturbation approach, we construct traveling solitary solutions with various vortex structures(vortex pairs, vortex rings) for Klein-Gordon equation with Ginzburg-Landau nonlinearities.

Dissipative ion-acoustic solitary and shock waves in a plasma with superthermal electrons

The excitation of a nonlinear ion-acoustic (IA) wave packet in a viscous plasma consisting of superthermal electrons and cold ions is investigated. It is assumed that the superthermal components obey the Kappa distribution. The standard reductive perturbation technique is employed to obtain a cubic complex Ginzburg–Landau type equation, which governs the dynamics of the amplitude of IA wave packets in a viscous plasma. In a weakly dissipative regime, by means of a perturbation method adopted from non-integrable dynamical systems theory, explicit soliton solutions are obtained of the dissipative dynamical system, determined as fixed points of a two-dimensional system of evolution equations describing the soliton amplitude and velocity. The stability of bright solitons is addressed using adiabatic perturbation theory in a weakly dissipative case. In a general dissipation regime, a novel type of travelling solitary (dissipative solitons) and shock-type solutions are derived for an envelope equation. Moreover, the dependence of the solitary and shock wave characteristics on relevant physical parameters of the problem is studied. It is shown that dissipative solitons do exist, sustained by a mutual balance between loss and gain terms, in addition to the interplay between dispersion and nonlinearity. The dependence of solitary and shock wave velocities on the parameters of the system are investigated. To feature distinctive characteristics of a dissipative soliton, it is compared with the bright soliton solution of the nonlinear Schrödinger equation.