Solitary Solutions Research Articles

The existence and decay of solitary waves for the Fornberg–Whitham equation

In this paper, we consider the Fornberg-Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in our proof. Besides, we also prove that the family of solitary solutions is orbital stable and decay exponentially when speed wave c is bigger than 1.

A solitary wave solution to the generalized Burgers-Fisher's equation using an improved differential transform method: A hybrid scheme approach

In this research, an unrivalled hybrid scheme which involves the coupling of the new Elzaki integral transform (an improved version of Laplace transform) and a modified differential transform called the projected differential transform (PDTM) have been implemented to solve the generalized Burgers-Fisher's equation; which springs up due to the fusion of the Burgers' and the Fisher's equation; describing convective effects, diffusion transport or interaction between reaction mechanisms, traffic flows; and turbulence; consequently finding meaningful applicability in the applied sciences viz: gas dynamics, fluid dynamics, turbulence theory, reaction-diffusion theory, shock-wave formation, traffic flows, financial mathematics, and so on.Using the proposed Elzaki projected differential transform method (EPDTM), a generalized exact solution (Solitary solution) in form of a Taylor multivariate series has been obtained; of which the highly nonlinear terms and derivatives handled by PDTM have been decomposed without expansion, computation of Adomian or He's polynomials, discretization, restriction of parameters, and with less computational work whilst achieving a highly convergent results when compared to other existing analytical/exact methods in the literature, via comparison tables, 3D plots, convergence plots and fluid-like plots. Thus showing the distinction, novelty and huge advantage of the proposed method as an asymptotic alternative, in providing generalized or solitary wave solution to a wider class of differential equations.

Stable soliton solutions to the nonlinear low-pass electrical transmission lines and the Cahn-Allen equation

The low-pass nonlinear electrical transmission lines and the Cahn-Allen equation are important nonlinear model equations to figure out different tangible systems, namely, electrical engineering, fluid dynamics etc. The contrivance of this study is to introduce advanced Bernoulli sub-equation function method to search for stable and effective solitary solutions of the described wave equations. Stable solitary solutions are reported as an integration of exponential functions, hyperbolic functions, etc., and the graphical implications for specific values of the corresponding parameters are explained in the solutions obtained in order to uncover the inmost structure of the tangible phenomena. It is establish that the IBSEF method is reliable, contented and might be used in further works to found ample novel soliton solutions for other types of NLEEs arising in physical science and engineering.

Accurate sets of solitary solutions for the quadratic–cubic fractional nonlinear Schrödinger equation

The analytical and semi-analytical solutions to the quadratic–cubic fractional nonlinear Schrödinger equation are discussed in this research article. The model’s fractional formula is transformed into an integer-order model by using a new fractional operator. The theoretical and computational approaches can now be applied to fractional models, thanks to this transition. The application of two separate computing schemes yields a large number of novel analytical strategies. The obtained solutions secure the original and boundary conditions, which are used to create semi-analytical solutions using the Adomian decomposition process, which is often used to verify the precision of the two computational methods. All the solutions obtained are used to describe the shifts in a physical structure over time in cases where the quantum effect is present, such as wave-particle duality. The precision of all analytical results is tested by re-entering them into the initial model using Mathematica software 12.

Novel solitary and resonant multi-soliton solutions to the (3 + 1)-dimensional potential-YTSF equation

In this study, the (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama equation arising from the (3 + 1)-dimensional Kadomtsev–Petviashvili equation is investigated in detail by using two powerful approaches. First, the generalized resonant multi-soliton solution is generated via the simplified linear superposition principle. Second, after applying the simplest equation method, the generalized single solitary solution is extracted. The results show that the obtained solutions are perfect. The physical explanation of the obtained solutions is depicted in various 3D and 2D figures, which are used to illustrate that the interactions of resonant multi-soliton waves are inelastic. Ultimately, the study reveals that the inelastic interactions can be determined by the sign of the wave related number [Formula: see text].

On the Complex Simulations With Dark–Bright to the Hirota–Maccari System

AbstractThis paper investigates the coupled nonlinear Hirota–Maccari system with the help of using an analytical approach, which is the extended sinh-Gordon equation expansion method (ShGEEM). Complex, solitary, and singular periodic traveling solutions are successfully gained to the nonlinear model considered. The constraint conditions that validate the existence of the reported soliton solutions are also given in a detailed manner. The two-dimensional (2D), three-dimensional, and contour graphs to some of the obtained solutions are presented via several computational programs. These simulations present deeper investigations about the wave distributions of the coupled nonlinear Hirota–Maccari system.

New multiple-different impressive perceptions for the solitary solution to the magneto-optic waveguides with anti-cubic nonlinearity

In this study, we will establish multiple-impressive different perceptions for the solitary wave solution to the magneto-optic waveguides with anti-cubic nonlinearity (MOWWACN) in the framework of three distinct manners. The three manners are implemented in the same vein and parallel to extract multiple new perceptions for the solitary wave solutions to this model. The first one is the extended simple equation method (ESEM), while the second one is the Paul-Painlevé approach method (PPAM) and the third one is the Riccati-Bernolli Sub ODE method (RBSODM). A good comparison has been constructed not only between our extracted solitary wave solutions which are achieved by these three methods but also with that extracted previously by other authors.

Awareness of Teledentistry Amongst Dental Professionals in India Amid COVID-19 Pandemic. A Cross-Sectional Study

Background: COVID-19 has become a crisis across the country these days, which likewise changed living of individual to entity contact and its spread. There the presentation of Teledentistry will be the solitary solution that will have the option to treat patients quickly and expertly. Aim: The current study aims to assess the awareness of dental professionals towards teledentistry during COVID-19 crisis in India. Methods: A cross-sectional descriptive survey was conceded on over all 142 dentists. A self-organized, close-ended questionnaire containing 20 questions was appropriately assembled and distributed. Data collected from the survey was examined in terms of frequency (yes/no). One Way Anova test was used to exemplify statistical variances and result came as significant at p < 0.05. Results: The responses from specialist dental professionals (with or without practicing) responded (57%) and dentist working in a private sector (68%) having clinical experience of about 5-10 years (36%). Conclusion: Dentists were well aware of this advanced technique and due to feasibility, most of them were ready to practice it in future.

New perception of the exact solutions of the 3D-fractional Wazwaz-Benjamin-Bona-Mahony (3D-FWBBM) equation

In this article we will utilize new perception of the Benjamin-Bona-Mahony equation that represents developed stretch for the Korteweg-de Varies equation which denotes to the unidirectional propagation of small amplitude long waves on the surface of hydro magnetic and acoustic waves in channel significantly for shallow water. The main idea of this research realizing new exact and hence solitary solutions of this new perception with its spatial and temporal variables and fractional order whose called 3D-fractional Wazwaz-Benjamin-Bona-Mahony model (3D -FWBBM). Two important different techniques are invited for the first time to proposed this new perception of the exact solutions, the first one is the modified simple equation method (MSEM) and the second is the Riccati-Bernolli Sub-ODE method (RBSOM) whose are used effectively to obtain accurate description of the wave solutions to this equation.

Diverse solitary and Jacobian solutions in a continually laminated fluid with respect to shear flows through the Ostrovsky equation

In this paper, the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods are employed to find the soliton, breather, kink, periodic kink, and lump wave solutions of the Ostrovsky equation. This model is considered as a mathematical modification model of the Korteweg-de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions emerge the localized wave packet as a persistent and dominant feature. Many distinct solutions are obtained through the employed computational schemes. Moreover, some solutions are sketched in 2D, 3D, and contour plots. The effective and powerful of the two used computational schemes are tested. Furthermore, the accuracy of the obtained solutions is examined through a comparison between them and that had been obtained in previously published research.

Novel solitary wave solution of the nonlinear fractal Schrödinger equation and its fractal variational principle

PurposeThe nonlinear Schrödinger equation plays a vital role in wave mechanics and nonlinear optics. The purpose of this paper is the fractal paradigm of the nonlinear Schrödinger equation for the calculation of novel solitary solutions through the variational principle.Design/methodology/approachAppropriate traveling wave transform is used to convert a partial differential equation into a dimensionless nonlinear ordinary differential equation that is handled by a semi-inverse variational technique.FindingsThis paper sets out the Schrödinger equation fractal model and its variational principle. The results of the solitary solutions have shown that the proposed approach is very accurate and effective and is almost suitable for use in such problems.Practical implicationsNonlinear Schrödinger equation is an important application of a variety of various situations in nonlinear science and physics, such as photonics, the theory of superfluidity, quantum gravity, quantum mechanics, plasma physics, neutron diffraction, nonlinear optics, fiber-optic communication, capillary fluids, Bose–Einstein condensation, magma transport and open quantum systems.Originality/valueThe variational principle of the Schrödinger equation without the Lagrange multiplier method in the sense of the fractal calculus is developed for the first time in the literature to the best of the author's understanding.

On the soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission line

This paper focuses on finding soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission lattice. Our investigation is based on the fact that for a realistic system, the electrical characteristics of a capacitor (and an inductor via skin effect) should include a fractional-order time derivative. In this respect for the model under consideration, we derive a fractional nonlinear partial differential equation for the voltage dynamics by applying the Kirchhoff’s laws. It is realized that the behavior of new soliton solutions obtained is influenced by the fractional-order time derivative as well as the coupling values. The fractional order also modifies the propagation velocity of the voltage wave notwithstanding their structure and tends to set up localized structure for low coupling parameter values. However, for a high value of the coupling parameter, the fractional order is less seen on the shapes of the new solitary solutions that are analytically derived. Several methods such as the Kudryashov method, the $$(G'/G)$$ -expansion method, the Jacobi elliptical functions method and the Weierstrass elliptic function expansion method led us to derive these solitary solutions while using the modified Riemann–Liouville derivatives in addition to the fractional complex transform. An insight into the overall dynamics of our network is provided through the analysis of the phase portraits.

Higher order solitary solutions to the meta-model of diffusively coupled Lotka\u2013Volterra systems

A meta-model of diffusively coupled Lotka–Volterra systems used to model various biomedical phenomena is considered in this paper. Necessary and sufficient conditions for the existence of nth order solitary solutions are derived via a modified inverse balancing technique. It is shown that as the highest possible solitary solution order n is increased, the number of nonzero solution parameter values remains constant for solitary solutions of order n>3. Analytical and computational experiments are used to illustrate the obtained results.

Linear and Nonlinear Plasma Processes in Ionospheric HF Heating

Featured observations of high frequency (HF) heating experiments are first introduced; the uniqueness of each observation is presented; the likely cause and physical process of each observed phenomenon instigated by the HF heating are discussed. A special point in the observations, revealed through the ionograms, is the competition between the Langmuir parametric instability and upper hybrid parametric instability excited in the heating experiments and the impact of the natural cusp at foE (the peak plasma frequency of the ionospheric E region) on the competition. The ionograms also infer the generation of Langmuir and upper hybrid cavitons. Ray tracing theory is formulated. With and without the appearance of large-scale field-aligned density irregularities in the background ionosphere, ray trajectories of the ordinary mode (O-mode) and extraordinary mode (X-mode) sounding pulses are calculated numerically. The results explain the artificial Spread-F recorded by the digisondes in the heating experiments. Parametric instabilities, which are the directly relevant processes to achieve effective heating of the ionospheric F region, are formulated and analyzed. The threshold fields and growth rates of Langmuir and upper hybrid parametric instabilities are derived as the theoretical basis of many radar observations and electron-plasma wave interactions. Harmonic cyclotron resonance interaction processes between electrons and upper hybrid waves are introduced. Formulation and analysis are presented. The numerical results show that ultra-energetic electrons are generated. These electrons enhance airglow at 777.4 nm as well as cause ionization. Physical processes leading to the generation of artificial ionization layers are discussed. The nonlinear Schrodinger equation governing the nonlinear evolution of Langmuir waves and upper hybrid waves are derived and solved. The nonlinear periodic and solitary solutions of the equations are obtained. The localized Langmuir and upper hybrid waves generated by the HF heater form cavitons near the HF reflection layer and near the upper hybrid resonance layer, which induce bumps in the virtual height spread of the ionogram trace similar to that induced by the density cusp at E-F1 transition layer; the down-going Langmuir waves and upper hybrid waves evolve into nonlinear periodic waves propagating along the magnetic field, which backscatter incoherently the sounding pulses to cause downward virtual height spread.

He's variational method for the time–space fractional nonlinear Drinfeld–Sokolov–Wilson system

The well‐known Drinfeld–Sokolov–Wilson system is widely used to describe the flow of the shallow water. This paper considers the time–space fractional Drinfeld–Sokolov–Wilson system, where He's variational method together with the two‐scale transform is applied to find its solitary solutions. The numerical results have been shown through graphs to demonstrate the applicability and efficiency of our approach. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations.

Multiple rogue wave, dark, bright, and solitary wave solutions to the KP–BBM equation

Under investigation in this paper is the (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. Based on bilinear method, the multiple rogue wave solutions and the novel multiple soliton solutions are constructed by giving some specific activation functions in the considered model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained, via Maple 18. By utilizing improved tan(Φ(ρ)∕2)-expansion technique the series of novel exact solutions in terms of rational, periodic and hyperbolic functions for the fractional cases are derived. Also, the semi-inverse variational principle is offered to get the solitary solutions. We construct the exact lump and rogue wave solutions, by solving the under-determined nonlinear system of algebraic equations for the specified parameters. Via various three-dimensional plots, curve plots, density plots and contour plots, dynamical characteristics of these waves are represented.

Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach

It is known that the Laplace transform method is used to solve only a finite class of linear differential equations. In this paper, we suggest a new method that relies on a new fractional expansion in the Laplace transform space and residual power series method to construct exact solitary solutions to the nonlinear time-fractional dispersive PDEs. The main idea of this method is to convert the original fractional differential equation into the Laplace space, and then, solve the newly obtained equation using a new approach of the residual power series method. Moreover, four important and attractive applications are also presented and discussed based on our proposed method to reveal the efficiency, effectiveness, and applicability of this method for constructing solitary solutions to the target equations. The results confirmed that the proposed method is a simple, understandable, and very fast method to obtain solitary solutions compared with other methods such as homotopy perturbation method, variation iteration method, sine–cosine method, and residual power series method.

Effect of streaming velocity, magnetic field, and higher-order correction on the nature of ion acoustic solitons in the Venusian ionosphere

Propagation properties of weakly nonlinear ion acoustic waves are investigated in a plasma at the Venusian ionosphere. The plasma model consists of two positive cold ions (oxygen O + and hydrogen H +), as well as isothermal electrons. The basic set of fluid equations is reduced to Zakharov-Kuznetsov (ZK) equation and linear inhomogeneous ZK-type equation (LIZKT) equation. The renormalization method is adopted to obtain solitary solutions of both equations. The effects of plasma parameters and higher-order correction on the nature of the solitary waves are investigated. It is found that the wave phase velocity is supersonic, which is in agreement with the observations. Furthermore, the higher-order correction enlarges the soliton amplitude, which is suitable for describing the solitary waves when the wave amplitude grows.

Periodic solution of fractal Phi-4 equation

This paper focuses on a fractal Phi-4 equation with time-space fractal derivatives, though its solitary solutions have been deeply studied, its periodic solution was rarely revealed due to its strong non-linearity. Now the condition is completely changed, He?s frequency formulation provides with a universal tool to having a deep insight into the periodic property of the fractal Phi-4 equation. The two-scale transform is used to convert approximately the fractal Phi-4 equation a differential model, and a criterion is suggested for the existence of a periodic solution of the equation, the effect of fractal orders on the periodic property is also elucidated.

Using a New Auxiliary Equation to Construct Abundant Solutions for Nonlinear Evolution Equations

In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As m → 0 and m → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.