Solitary Solutions Research Articles

The Evolutionary Properties on Solitary Solutions of Nonlinear Evolution Equations

The evolution process of four class soliton solutions is investigated by basic calculus theory. For any given x, we describe the special curvature evolution following time t for the curve of soliton solution and also study the fluctuation of solution curve.

Thresholds for Existence of Dispersion Management Solitons for General Nonlinearities

We prove a threshold phenomenon for the existence of solitary solutions of the dispersion management equation for positive and zero average dispersion for a large class of nonlinearities. These solutions are found as minimizers of nonlinear and nonlocal variational problems which are invariant under a large noncompact group. There exists a threshold such that minimizers exist when the power of the solitons is bigger than the threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. The existence of dispersion-managed solitons is shown under very mild conditions on the dispersion profile and the nonlinear polarization of an optical active medium, which cover all physically relevant cases for the dispersion profile and a large class of nonlinear polarizations; for example, they are allowed to change sign.

The Classification of the Single Traveling Wave Solutions to the Modified Fornberg–Whitham Equation

The aim of present paper is to obtain the analytical solution of modified Fornberg–Whitham equation by using \(\tan (\phi /2)\)-expansion and \(\tanh (\phi /2)\)-expansion methods. These methods are used to construct solitary and soliton solutions of nonlinear evolution equation. These methods are straightforward and concise, and their applications are promising. These methods are developed for searching exact traveling wave solutions of nonlinear partial differential equations. The exact particular solutions containing five types hyperbolic function solution, trigonometric function solution, exponential solution, logarithmic solution and rational solution. The expansion methods presents a wider applicability for handling nonlinear wave equations. Also, their are shown that the expansion methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations.

Comments on “Soliton solutions to fractional-order nonlinear differential equations based on the exp-function method”

In their recently published article, Guner and Atik apply the exp-function method to derive solitary solutions to fractional-order nonlinear differential equations. We argue that the solutions provided by Guner and Atik do not satisfy the considered equations. Furthermore, we derive correct solitary solutions as well as necessary and sufficient conditions for the existence of these solutions in the space of equation parameters and initial conditions.

Symbolic computation and abundant travelling wave solutions to KdV–mKdV equation

In this article, the novel (G ′/G)-expansion method is successfully applied to construct the abundant travelling wave solutions to the KdV–mKdV equation with the aid of symbolic computation. This equation is one of the most popular equation in soliton physics and appear in many practical scenarios like thermal pulse, wave propagation of bound particle, etc. The method is reliable and useful, and gives more general exact travelling wave solutions than the existing methods. The solutions obtained are in the form of hyperbolic, trigonometric and rational functions including solitary, singular and periodic solutions which have many potential applications in physical science and engineering. Many of these solutions are new and some have already been constructed. Additionally, the constraint conditions, for the existence of the solutions are also listed.

Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability

The higher order nonlinear Schrödinger (NLS) equation describes ultra-short pluse propagation in optical fibres. By using the amplitude ansatz method, we derive the exact bright, dark and bright-dark solitary wave soliton solutions of the generalized higher order nonlinear NLS equation. These solutions for the generalized higher order nonlinear NLS equation are obtained precisely and efficiency of the method can be demonstrated. The stability of these solutions and the movement role of the waves are analyzed by applying the modulation instability analysis and stability analysis solutions. All solutions are exact and stable.

Existence of second order solitary solutions to Riccati differential equations coupled with a multiplicative term

Existence of second order solitary solutions to Riccati differential equations coupled with a multiplicative term

Exact analytical solutions of higher-order nonlinear Schrödinger equation

The higher-order nonlinear Schrödinger equation is solved based on Jacobi elliptic function and other auxiliary equation methods Two new families of exact analytical solutions are reached, i.e., hybrid Jacobi elliptic function solutions and single special function solutions. Some solutions are degenerated to new solitary solutions and new triangular functions solutions in the limit case. A new solitary wave solution family like a single soliton pulse embedded on a pronounced platform is also obtained.

Kink solitary solutions to generalized Riccati equations with polynomial coefficients

Inverse and direct balancing methods for the construction of kink solitary solutions to generalized Riccati equations with polynomial coefficients are presented in this paper. Necessary and sufficient conditions for the existence of kink solitary solutions are derived in terms of the equation's parameters and initial conditions. The proposed technique enables a straightforward derivation of parameters of solitary solutions. Computational experiments are used to demonstrate the efficiency of the proposed analytical approach.

First-ever model simulation of the new subclass of solitons “Supersolitons” in plasma

“Supersolitons,” the structures associated with the stationary solitary solutions with the Mach number greater than those associated with the double layers, were introduced in 2012. Later, many researchers have reported the existence domain of the supersolitons in different plasma constituents. However, their evolutionary dynamical behavior and stability were main concerns and were not yet explored. We performed fluid simulation of ion acoustic supersolitons in a plasma containing two-temperature electrons having kappa distributions in the presence of cold fluid ions. Our simulation shows that a specific form of the initial perturbation in the equilibrium electron and ion densities can evolve into ion acoustic supersolitons, which maintain their shape and size during their propagation. This is first-ever simulation to confirm the stability of the supersolitons that opens a new era in the field of solitary wave structures in space and laboratory plasmas.

SOLITARY SOLUTION OF A CLASS OF NONLINEAR TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

This paper presents a general solitary solution of a class of nonlinear time- fractional partial differential equations by Adomian decomposition method(ADM). This class of nonlinear time-fractional partial differential equations include a lot of standard nonlinear partial differential equations in mathematical physics. The solitary solution obtained by ADM is a general solitary solution and admit you investigate the solution for different initial condi- tions and different �( � is the order of derivative respect to time 0 < � ≤ 1). Also the solution subject to the especial initial conditions and � = 1 reduce to the solution of standard partial differential equation. Additionally, it use the fractional derivative of Caputo sense.

Solitary-Wave and Periodic Solutions of the Kuramoto-Velarde Dispersive Equation

Abstract In the present paper, solitary solutions of the Kuramoto- Velarde (K-V) dispersive equation have been found, using the deformation and mapping approach. These exact solutions show the dynamics and the evolution of dispersive solitary waves. In the case α2 = α3, three families of exact periodic solutions have been obtained by employing the bilinear transformation method.

Exact solution of CKP equation and formation and interaction of two solitons in pair-ion-electron plasma

In the present investigation, cylindrical Kadomstev-Petviashvili (CKP) equation is derived in pair-ion-electron plasmas to study the propagation and interaction of two solitons. Using a novel gauge transformation, two soliton solutions of CKP equation are found analytically by using Hirota's method and to the best of our knowledge have been used in plasma physics for the first time. Interestingly, it is observed that unlike the planar Kadomstev-Petviashvili (KP) equation, the CKP equation admits horseshoe-like solitary structures. Another non-trivial feature of CKP solitary solution is that the interaction parameter gets modified by the plasma parameters contrary to the one obtained for Korteweg–de Vries equation. The importance of the present investigation to understand the formation and interaction of solitons in laboratory produced pair plasmas is also highlighted.

Localized traveling wave solution for a logarithmic nonlinear Schrödinger equation

We investigate localized traveling wave solutions for a Schrödinger equation with two logarithmic nonlinear terms under no external potential. It is shown that it can have solitary wave type solutions whose envelope profile depends on the two types of nonlinearity. Remarkably, the profile has cutoffs in the coordinate of propagation. We argue also some fundamental properties that discriminate it from power law type nonlinear Schrödinger equations.

Solitary solutions for a class of Schrödinger equations in $${\mathbb{R}^3}$$ R 3

In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain $$-\Delta u + (\lambda + \varepsilon') u \mp \Delta \sqrt{1-u^{2}}\frac{u}{\sqrt{1- u^{2}}} - \varepsilon'\frac{u}{\sqrt{1 - u^{2}}} = 0, \, x \in \mathbb{R}^{3},$$ where \({\lambda}\) and \({\varepsilon'}\)are real constants. By variational methods and perturbation arguments, we study the existence of positive classical solutions. Our results generalize the previous results in one-dimensional space given by Brull et al. [4].

Solitary solutions to a relativistic two-body problem

Necessary and sufficient conditions for the existence of solitary solutions to a generalized model of a two-body problem perturbed by small post-Newtonian relativistic term are derived in this paper. It is demonstrated that kink, bright and dark solitary solutions exist in the model, when the relativistic effects are treated as higher order perturbations. Numerical experiments are used to verify theoretical results.

New solitary solutions in FPU-β atom chain

Solitons in one-dimensional FPU-[Formula: see text] lattice chains were studied by the Jacobi elliptic function expansion method under the condition of quasi-continuum approximation. It is found that there exhibits a novel nonlinear elementary excitation, i.e. exact solitary solutions. On the basis of the above, the influence of the nonlinear intensity on the solitons was analyzed. It is found that the modulation amplitude of soliton is related closely to the nonlinear parameter also.

Travelling wave solutions for some two-component shallow water models

In the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyze the celebrated Green–Naghdi equations, the integrable two-component Camassa–Holm equations and a new two-component system of Green–Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models. In particular, we show the existence of new type of solutions.

Generalized solitary waves in nonintegrable KdV equations

The generalization of the classical Korteweg-de-Vries (KdV) solitary wave solution is presented in this paper. The amplitude and the propagation speed of generalized KdV solitary waves vary in time. Generating partial differential equations and conditions of existence of the generalized KdV solitary waves are derived using the inverse balancing method. Computational experiments illustrate the variety of new solitary solutions and their generating equations.

Spatiotemporal soliton solution to generalized nonlinear Schrödinger equation with a parabolic potential in Kerr media

A class of new spatiotemporal solitary solution to nonlinear Schrödinger equation with a parabolic potential is investigated analytically and numerically using the F-expansion method and homogeneous balance principle. The propagation characteristics of soliton wave solutions are analyzed with/without spatial-temporal chirp. It is noteworthy that, by calculating spatial and temporal second-order intensity moment, several novel features of optical beam propagations are obtained, such as stable, oscillating, decaying and blowing up. Additionally, controllability of these solutions with the modulation depth of the parabolic potential is demonstrated.