Localized Solitary Wave Solutions Research Articles

Dynamics of the breather and solitary waves in plasma using the generalized variable coefficient Gardner equation

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma.

Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations

The prime objective of this paper is to explore the new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and (2+1)-dimensional breaking soliton equations via a generalized exponential rational function (GERF) method. Many different kinds of exact soliton solution are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact soliton solutions are also demonstrated by a choice of appropriate values of the free constants that aid in understanding the nonlinear complex phenomena of such equations. These exact soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular-form solitons, single solitons, double solitons, triple solitons, bell-shaped solitons, combo singular solitons, breather-type solitons, elastic interactions between triple solitons and kink waves, and elastic interactions between diverse solitons and kink waves. Because of the reduction in symbolic computation work and the additional constructed closed-form solutions, it is observed that the suggested technique is effective, robust, and straightforward. Moreover, several other types of higher-dimensional nonlinear evolution equation can be solved using the powerful GERF technique.

Traveling and localized solitary wave solutions of the nonlinear electrical transmission line model equation

Analytical solutions describing the kink–antikink type, hyperbolic, trigonometric, doubly periodic, localized bright and dark soliton like, breather and rational periodic train solutions to the discrete nonlinear electrical transmission line in $$(2+1)$$ -dimensions have been investigated with the Kudryashov method, the fractional linear transform method, and the simplest equation method with its modified and extended versions. Consequently, many new and more general explicit forms of traveling waves are retrieved under parametric conditions. The 3D, 2D, and density profile of some of the obtained results are numerically simulated by selecting suitable values of the various parameters of the NLTL model equation. Furthermore, all the obtained results were checked and verified by putting back into the considered nonlinear transmission line model equation with Maple and Mathematica software. The acquired results show the validity and reliability of the proposed schemes to the studied nonlinear electrical transmission line model.

A variety of new traveling and localized solitary wave solutions of a nonlinear model describing the nonlinear low- pass electrical transmission lines

Abstract In this work, we focus on investigating the traveling and other localized solitary wave propagation in nonlinear low-pass electrical transmission lines model practicing the new modified sub-ODE method, the unified Riccati equation expansion method, and the fractional linear transform method. A variety of traveling and solitary wave solutions are emerging comprising of bright, dark, kink, anti-kink, hyperbolic function, and doubly periodic Jacobian elliptic function solutions. The applied three integration schemes are reliable and stalwart for acquiring the new kink, bright, dark, periodic and non-singular soliton solutions of the wave propagation in nonlinear low-pass electrical transmission lines. Also, we give the geometric description of some of the obtained solutions for the considered model by computing the most important geometric quantities viz the Gaussian and the mean curvatures. To display the extant physical significance of the considered model equation, some two, three-dimensional figures and density profiles of the acquired solutions are illustrated for the specific choice of arbitrary parameters. The effects of the variation of nonlinear parameters of the nonlinear low-pass electrical transmission lines model on the evolution of soliton solutions are demonstrated. In addition, all derived solutions were verified by substituting back into the considered equation with the aid of Mathematica software. Furthermore, a comparison of our results for the considered model with the obtained solutions in the literature is also provided.

Nanoptera and Stokes curves in the 2-periodic Fermi–Pasta–Ulam–Tsingou equation

This work presents asymptotic solutions to a singularly-perturbed, period-2 FPUT lattice and uses exponential asymptotics to examine ‘nanoptera’, which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains which appear behind the wave front. Using an exponential asymptotic approach, this work isolates the exponentially small oscillations, and demonstrates that they appear as special curves in the analytically-continued solution, known as ‘Stokes curves’ are crossed. By studying the asymptotic form of these isolated oscillations, it is shown that there are special mass ratios which cause the oscillations to vanish, producing localized solitary-wave solutions. The asymptotic predictions are validated through comparison with numerical simulations.

Nanoptera in a Period-2 Toda Chain

We study asymptotic solutions to a singularly-perturbed, period-2 Toda lattice and use exponential asymptotics to examine `nanoptera', which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains. With this approach, we isolate the exponentially small, constant-amplitude waves, and we elucidate the dynamics of these waves in terms of the Stokes phenomenon. We find a simple asymptotic expression for the waves, and we study configurations in which these waves vanish, producing localized solitary-wave solutions. In the limit of small mass ratio, we derive a simple anti-resonance condition for the manifestation these wave-free solutions.

Breaking Waves And Solitary Waves To The Rotation-Two-Component Camassa--Holm System

In this paper, we consider two types of solutions of the rotation-two-component Camassa--Holm (R2CH) system, a model of equatorial water waves that includes the effect of the Coriolis force. The first type of solutions exhibits finite time singularity in the sense of wave-breaking. We perform a refined analysis based on the local structure of the dynamics to provide some criteria that leads to the blow-up of solutions. The other type of solutions we study is the solitary waves. We classify various localized solitary wave solutions for the R2CH system. In addition to those smooth solitary wave solutions, we show that there are solitary waves with singularities, like peakons and cuspons, depending on the values of the rotating parameter $\Omega$ and the balance index $\sigma$. We also prove that horizontally symmetric weak solutions of this model must be traveling waves.

Fully localized solitary waves for the forced Kadomtsev–Petviashvili equation

The Kadomtsev–Petviashvili (KP) equation is a generalized form of the Korteweg–de Vries equation for a three-dimensional channel flow. Three-dimensional fully localized solitary wave solutions can be obtained when the effect of surface tension is significant; we refer to this as the KP-I equation. An exact solution of a three-dimensional fully localized stationary solution for the KP-I equation has been found in the absence of any obstacle on the rigid channel bottom. However, three-dimensional fully localized stationary solutions for the KP-I equation have not been found in the presence of a forcing neither analytically nor numerically. This forcing term comes from some obstacles in the rigid channel bottom. In this work, we focus on the fully localized solitary wave solutions when a positive bump or a negative hole is given as the rigid bottom configuration. The forced KP-I equation is defined in an infinite domain and it is reduced to a finite computational domain by introducing artificial boundary conditions. Interestingly, there are at least two distinct stationary lump-type solutions for a positive bump, while there are at least three distinct stationary lump-type solutions for a negative hole. Furthermore, we investigate their numerical stability by solving the forced time-dependent KP-I equation using them as initial conditions. Our numerical results confirm that there exists a stable stationary lump-type solitary wave solution for both a positive bump and a negative hole when they evolve in time.

Effects of ion mobility and positron fraction on solitary waves in weak relativistic electron-positron-ion plasma

The effects of ion mobility and positron fraction on the solitary waves of the laser field envelope and the potential of the electrostatic field in weak relativistic electron-positron-ion plasma are investigated. The parameter region for the existence of solitary waves is obtained analytically, and a reasonable choice of parameters is clarified. Both cases of mobile and immobile ions are considered. It is found that the amplitudes of solitary waves in the former case are larger compared to the latter case. For small plasma density, the localized solitary wave solutions in terms of the approximate perturbation analytical method are very consistent with those by exact numerical calculations. However, as the plasma density increases the analytical method loses its validity more and more. The influence of the positron fraction on the amplitudes of solitary waves shows a monotonous increasing relation. The implications of our results to particle acceleration are also discussed briefly.

Transverse linear instability of solitary waves for coupled long-wave–short-wave interaction equations

In this paper, we investigate the transverse linear instability of one-dimensional solitary wave solutions of the coupled system of two-dimensional long-wave–short-wave interaction equations. We show that the one-dimensional solitary waves are linearly unstable to perturbations in the transverse direction if the coefficient of the term associated with transverse effects is negative. This transverse instability condition coincides with the non-existence condition identified in the literature for two-dimensional localized solitary wave solutions of the coupled system.

Quantization of spin direction for solitary waves in a uniform magnetic field

It is known that there are nonlinear wave equations with localized solitary wave solutions. Some of these solitary waves are stable (with respect to a small perturbation of initial data) and have nonzero spin (nonzero intrinsic angular momentum in the center of momentum frame). In this paper we consider vector-valued solitary wave solutions to a nonlinear Klein–Gordon equation and investigate the behavior of these spinning solitary waves under the influence of an externally imposed uniform magnetic field. We find that the only stationary spinning solitary wave solutions have spin parallel or anti-parallel to the magnetic field direction.

Generalized neighbor-interaction models induced by nonlinear lattices

It is shown that the tight-binding approximation of the nonlinear Schrödinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present nonstandard possibilities, among which we mention a quasilinear regime, where the pulse dynamics obeys essentially the linear Schrödinger equation. We analyze the properties of such models both in connection to their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions.

Quadratic solitons with gain and loss

Propagation of soliton signals in quadratic nonlinear media under conditions for second-harmonic generation in the presence of multiple-frequency-band absorption and amplification is addressed. The reshaping and energy redistribution of the multiple-frequency solitons under the action of the single-band loss and gain is investigated and shown in detail. New chirped localized solitary-wave solutions that exactly compensate loss and gain are presented. Practical implementation conditions are discussed.

A pair of coupled equations for high frequency Langmuir and dispersive ion-acoustic waves with collisional damping

By using slow space-time variation of the amplitude, a pair of coupled nonlinear Schrodinger equations are derived from the equation of continuity, fluid equation of motion with collisional damping effects for both electron and ion, and the Poisson equation. If the collisional damping effect is neglected then localized solitary wave solutions exist for certain wave numbers of high frequency oscillation with a range of low-frequency wave numbers. Among these localized solitary wave solutions some are found to be stable. The basic balance equations for the solution parameters have been derived from the evolution equations. It is found that the width and wave numbers of the solitary wave remain constant while in motion.

Stability of solitary wave solutions of simultaneous nonlinear Schrödinger equations

The stability of localized solitary wave solutions of simultaneous nonlinear Schrödinger equations describing different types of interacting waves in a plasma has been investigated. It is found that the stability depends on the nature and strength of the interaction potential between the two waves. The possible results of interactions between two colliding solitary waves have been discussed using the conservation laws.