Existence Of Solitary Waves Research Articles

Effect of ion temperature on oblique propagation of large amplitude solitary kinetic Alfvén waves

The effect of ion temperature on the existence of large amplitude solitary kinetic Alfvén waves and double layers are investigated in a two-fluid model using Sagdeev’s pseudopotential technique. The ion temperature, obliqueness parameter, and the ratio of cyclotron frequency to ion frequency are found to play significant roles in the formation and the shape of solitary kinetic Alfvén waves and double layers. Hump-type solitary waves and dip-type double layers are also observed. Conditions for the existence of solitary waves and double layers are discussed.

Solitary waves of the two-dimensional Benjamin equation

In this paper, we study the existence of solitary waves associated to the two-dimensional Benjamin equation. This equation governs the evolution of waves at the interface of a two-fluid system in which surface-tension effects cannot be ignored. We classify the existence and nonexistence cases according to the sign of the transverse dispersion coefficients. Moreover, we show that the solitary waves of the 2D Benjamin equation, when they exist, converge to those of the KPI equation as the parameter preceding the nonlocal operator $H\partial^2_x$ goes to zero. We also prove the regularity of solitary waves, as well as their symmetry with respect to the transverse variable and their algebraic decay at infinity.

Arbitrary amplitude dust acoustic solitary waves and double layers in nonthermal plasma including the effect of dust temperature

A computational scheme has been developed to study the arbitrary amplitude dust acoustic solitary waves and double layers in nonthermal plasma consisting of negatively charged dust grains, nonthermal ions, and isothermal electrons including the effect of dust temperature. The Sagdeev potential approach, which is valid to study the arbitrary amplitude solitary waves and double layers, has been employed. The computation has been carried out over the entire interval of β1:0≤β1<βM. This β1 is a parameter associated with the nonthermal distribution of ions and βM is the upper bound of β1. Depending on the nature of existence of solitary waves and double layers, the interval for β1 can be broken up into four disjoint subintervals holding the other parameters fixed. By nature of existence of solitary waves and double layers, it is meant that in some subinterval only negative potential solitary waves can exist, whereas in another both negative and positive potential solitary waves can coexist along with a double layer, etc. Corresponding to every β1 lying within a subinterval of β1, there is a definite interval for the Mach number (definite value of the Mach number) for which there exists solitary waves (double layer) specific for that subinterval of β1. The role of dust temperature on the subintervals of β1 and on amplitude of solitary waves and double layers has been explored.

Bifurcations of travelling wave solutions for the combined [formula omitted]–[formula omitted] equation

By using the theory of bifurcations of dynamical systems to the combined kdv–mkdv equation, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Existence of solitary waves for the discrete Schrödinger equation coupled to a nonlinear oscillator

The discrete Schrodinger equation with a nonlinearity concentrated at a single point is an interesting and important model to study the long-time behavior of solutions, including the asymptotic stability of solitary waves and properties of global attractors. In this note, the global well-posedness of this equation and the existence of solitary waves is proved and the properties of these waves are studied.

Effect of electron inertia on large amplitude solitary waves in presence of kinematic viscosity in dusty plasma

Nonlinear dust acoustic solitary waves in a dusty plasma are studied for nonzero kinematic viscosity. Sagdeev’s potential can be obtain upto any order in ϕ. The existence of soliton solution is determined by pseudopotential approach. It is seen that the electron inertia has a significant effect on the existence of solitary waves in presence of kinematic viscosity.

Regular and embedded solitons in a generalized Pochammer PDE

Variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochammer PDE that is currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely.

Solitary waves in twist-opening models of DNA dynamics

We analyze traveling solitary wave solutions in the Barbi-Cocco-Peyrard twist-opening model of nonlinear DNA dynamics. We identify conditions, involving an interplay of physical parameters and asymptotic behavior, for such solutions to exist, and provide first-order ordinary differential equations whose solutions give the required solitary waves; these are not solvable in analytical terms, but are easily integrated numerically. The conditions for existence of solitary waves are not satisfied for trivial asymptotic behavior and physical values of the parameters, i.e., the Barbi-Cocco-Peyrard model admits only solitary wave solutions that entail a global modification of the molecule; this is compared with the situation met in another recently formulated class of DNA models with two degrees of freedom per site.

Solitary Waves in Abelian Gauge Theories

Abstract Abelian gauge theories consist of a class of field equations which provide a model for the interaction between matter and electromagnetic fields. In this paper we analyze the existence of solitary waves for these theories. We assume that the lower order term W is positive and we prove the existence of solitary waves if the coupling between matter and electromagnetic field is small. We point out that the positiveness assumption on W implies that the energy is positive: this fact makes these theories more suitable to model physical phenomena.

Remarks on NLS with higher order anisotropic dispersion

We show that the introduction of third and fourth order anisotropic dispersion terms in the nonlinear Schrödinger equation enables to solve the global Cauchy problem for nonlinearities which are (super) critical in the standard case. Some blow-up results are proved in the supercritical case. Then, we prove the existence of solitary waves and discuss their regularity, their exponential decay rate at infinity, and their stability.

ON A CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS

The existence of solitary wave, kink wave and periodic wave solutions of a class of singular reaction–diffusion equations is obtained using some effective methods from the dynamical systems theory. Specially, for a class of nonlinear wave equations, fundamental properties of profiles of traveling wave solutions determined by some bounded orbits of the traveling wave systems are rigorously proved. Parametric conditions that guarantee the existence of the aforementioned solutions are derived and given explicitly.

Potential methods for nonlinear slow magnetoacoustic waves in pressure anisotropic high-β plasmas

We perform a parametric study of highly oblique solitary and periodic nonlinear stationary slow magnetoacoustic waves in the model of McKenzie and Doyle, and Stasiewicz, taking plasma pressure anisotropy into account. The model was proposed for the interpretation of linear magnetic holes, where there is almost no variation in direction of the ambient magnetic field direction. In particular, we focus on magnetic structures in high-β (ratio of kinetic to magnetic pressures) plasmas. We find that gas pressure anisotropy is crucial for the existence of solitary structures in this model for the parameter regime considered. Pressure anisotropy leads to the existence of field enhancing bright solitary structures and field depleting dark solitary structures. To explain the parametric range of the existence of solitary and periodic wave solutions of the model, a pseudo-(Sagdeev) potential is used. We obtain an analytic form for this potential in the case of low Alfvén mach waves. We find that the potential method accurately predicts the parametric range of the existence of solitary waves as calculated by direct numerical simulations.

A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

We study existence and asymptotic properties of solutions to a semilinear elliptic equation in the whole space. The equation has a cylindrical symmetry and we find cylindrical solutions. The main features of the problem are that the potential has a large set of singularities (i.e. a subspace), and that the nonlinearity has a double power-like behaviour, subcritical at infinity and supercritical near the origin. We also show that our results imply the existence of solitary waves with nonvanishing angular momentum for nonlinear evolution equations of Schrodinger and Klein-Gordon type.

The sine-Gordon equation in toroidal magnetic-fusion experiments

This paper presents a new nonlinear model which describes localized magnetohydrodynamic (MHD) modes in reversed-field pinch (RFP) experiments. To date, nearly all experimental and theoretical work in this area has relied on the use of Fourier decomposition of spatial variations as a function of time. Moreover, due to the complexity of this nonlinear problem, previous work has been restricted to the analysis of a relatively small number of modes. In contrast, the model studied in this paper, based on the damped-driven sine-Gordon (DDSG) equation, addresses the full nonlinearity, does not rely on Fourier decomposition, and does not require the range of the nonlinearity to be small. A specific consequence of working with the full nonlinearity is the existence of solitary waves in dispersive media. These solitary waves, a key part of the model, are used to describe the so-called slinky mode propagating in the plasma. A remarkable resemblance is seen between the waveforms obtained from experiments and the mathematical predictions of the new model.

Nonlinear deformation waves in solids and dispersion

In contemporary technology, high-speed loading, high frequencies and high-amplitude excitations are widely used. For proper analysis, the microstructure of solids must then be taken into account. Together with nonlinear effects, the dispersive effects due to the microstructure are of importance. In this paper, the Mindlin-type model based on the theory of continua is used for describing the microstructure. On the other hand, it is shown how straight-forward numerical methods, like the finite-volume methods, permit to describe the material properties for every discrete element used in numerical simulation. Several phenomena like the existence of solitary waves, the emergence of solitary wave trains, and waves in piece-wise nonlinear laminated materials are briefly discussed and further problems indicated.

Magnetoacoustic solitary waves in a pressure anisotropic high-β plasma

Oblique stationary wave solutions of one-dimensional Hall-MHD with explicit pressure anisotropy are investigated using asymptotic analysis, and the results are compared to numerics. An analytic condition is found the existence of solitary waves. Pressure anisotropy is found to be a crucial for their existence.

Ion-acoustic solitary waves in a relativistic plasma

The possibility of an ion-acoustic solitary wave in a relativistic plasma comprised of hot electrons and cold ions is investigated by Sagdeev’s pseudopotential method using a set of fully relativistic two-fluid equations. In the nonrelativistic limit, the general relativistic conditions for the existence of solitary waves found in the present study reduces exactly to the well known expressions for the nonrelativistic conditions. It is found by a graphical method that as the flow velocity of the plasma increases, the shape of the solitary wave becomes sharper and more prominent if the flow velocity is larger than the solitary-wave velocity. In the case in which the flow velocity is smaller than the solitary-wave velocity, however, the solitary wave is more pronounced for smaller flow velocity. It is also found that, for a fixed flow velocity, the sharpness and the height of the solitary wave decrease as the temperature increases as in nonrelativistic plasmas. Unlike in nonrelativistic plasmas, the present study predicts the existence of nonmoving soliton-like electrostatic structures.

Shear driven solitary waves on a liquid film

Long nonlinear two-dimensional traveling waves on a film driven by laminar gas flow are investigated numerically via solving Navier-Stokes equations. The evolution of their shape, amplitude, and speed with increasing Reynolds number is studied. The existence of solitary waves is demonstrated. A comparison between shear driven and gravity-capillary waves is made and discussed. It is shown that shear driven waves as compared to gravity driven waves are much higher for equal film Reynolds numbers and much slower for equal wave amplitudes.

On the existence of ion-acoustic solitary waves in a gravitating dusty plasma having charge fluctuation

Theoretical investigation on the propagation of ion-acoustic waves in an unmagnetized self-gravitating plasma has been made for the existence of solitary waves using the reductive perturbation method. It is observed that nonlinear excitations follow a coupled third-order partial differential equation which is slightly different from the usual case of coupled Korteweg-de Vries (K-dV) system. It appears that the system so deduced is a two-component generalization of the previous one derived by Paul et al. (1999) in which it was shown that ion-acoustic solitary waves can not exist in such system.

Solitary Waves for Nonconvex FPU Lattices

Solitary waves in a one-dimensional chain of atoms $\{q_j\}_{j\in{\Bbb Z}}$ are investigated. The potential energy is required to be monotone and grow super-quadratically. The existence of solitary waves with a prescribed asymptotic strain is shown under certain assumptions on the asymptotic strain and the wave speed. It is demonstrated that the invariance of the equations allows one to transform a system with nonconvex potential energy density to the situation under consideration.