Analytical Traveling Wave Solutions Research Articles

Consistent travelling wave characteristic of space–time fractional modified Benjamin–Bona–Mahony and the space–time fractional Duffing models

Study on solitary wave phenomenon are closely related on the dynamics of the plasma and optical fiber system, which carry on broad range of wave propagation. The space–time fractional modified Benjamin–Bona–Mahony equation and Duffing model are important modeling equations in acoustic gravity waves, cold plasma waves, quantum plasma in mechanics, elastic media in nonlinear optics, and the damping of material waves. This study has effectively developed analytical wave solutions to the aforementioned models, which may have significant consequences for characterizing the nonlinear dynamical behavior related to the phenomenon. Conformable derivatives are used to narrate the fractional derivatives. The expanded tanh-function method is used to look into such kinds of resolutions. An ansatz for analytical traveling wave solutions of certain nonlinear evolution equations was originally a power sequence in tanh. The discovered explanations are useful, reliable, and applicable to chaotic vibrations, problems of optimal control, bifurcations to global and local, also resonances, as well as fusion and fission phenomena in solitons, scalar electrodynamics, the relation of relativistic energy–momentum, electromagnetic interactions, theory of one-particle quantum relativistic, and cold plasm. The solutions are drafted in 3D, contour, listpoint, and 2D patterns, and include multiple solitons, bell shape, kink type, single soliton, compaction solitary wave, and additional sorts of solutions. With the aid of Maple and MATHEMATICA, these solutions were verified and discovered that they were correct. The mentioned method applied for solving NLFPDEs has been designed to be practical, straightforward, rapid, and easy to use.

Analytical solutions and mathematical simulation of traveling wave solutions to fractional order nonlinear equations

We investigate the tanh–coth method’s usefulness in understanding space–time fractional nonlinear evolution problems. To use this method and then examine their plots, we take into consideration three significant equations: the space–time fractional Burgers equation, the space–time fractional regularized long wave equation, and the space–time fractional Boussinesq equation. The modified Riemann–Liouville derivatives are used to define the fractional derivative. The approach provides a linearization- or small perturbation-free analytical solution in the form of a convergent series with easily calculable components. For the aforementioned nonlinear fractional equations, we have discovered their precise solutions. We utilize a generalized fractional complex transform to translate these to ordinary differential equations which subsequently resulted into number of exact solutions. The method provides us abundant analytical traveling wave solutions containing fewer number parameters with the aid of Mathematica and MATLAB.

A phase-field model for ferroelectrics with general kinetics, Part I: Model formulation

When subjected to electro-mechanical loading, ferroelectrics see their polarization evolve through the nucleation and evolution of domains. Existing mesoscale phase-field models for ferroelectrics are typically based on a gradient-descent law for the evolution of the order parameter. While this implicitly assumes that domain walls evolve with linear kinetics, experiments instead indicate that domain wall kinetics is nonlinear. This, in turn, is an important feature for the modeling of rate-dependent effects in polarization switching. We propose a new multiple-phase-field model for ferroelectrics, which permits domain wall motion with nonlinear kinetics, with applications in other solid–solid phase transformation problems. By means of analytical traveling wave solutions, we characterize the interfacial properties (energy and width) and the interface kinetics of straight domain walls, as furnished by the general kinetics model, and compare them to those of the classical Allen–Cahn model. We show that the proposed model propagates domain walls with arbitrarily chosen nonlinear kinetic relations, which can be tuned to differ for the different types of domain walls in accordance with experimental evidence.

Lie point symmetry infinitesimals, optimal system, power series solution, and modulational gain spectrum to the mathematical Noyes–Field model of nonlinear homogeneous oscillatory Belousov–Zhabotinsky reaction

Introduction:The chemical oscillators are identified as open system that demonstrate periodic changes in the concentration of some reaction species as a result of intricate physico-chemical mechanisms which can lead to bi-stability, the occurrence of limit cycle attractors, the emergence of spiral waves and turing patterns, and finally, deterministic chaos. Objectives:The main objective of this paper is to analyze the simple Noyes–Field governing system of differential equations for the nonlinear Belousov–Zhabotinsky reaction which delineates the non-linear oscillatory behavior of chemical systems that occurs in the homogeneous media. Methodology:The Lie symmetry invariance analysis performed to extract the symmetries infinitesimal generators and the adjoint representation carried out to develop optimal system for the obtained Lie vectors. The significant power series approach applied to obtain the analytical solution. The modulation instability criteria ensured the stability of nonlinear oscillatory Belousov–Zhabotinsky reaction process. Results:The one-dimensional Lie symmetry generators algebra of the mathematical Noyes–Field governing system for oscillatory reaction is established. Furthermore, similarity reductions are carried out as well as the development of an optimal system of the sub-algebras. The similarity transformation technique converted the controlling system to ordinary differential equations and generates the large quantity of analytical traveling wave solutions. Moreover, the closed-form analytical solution for the proposed homogeneous nonlinear oscillatory chemical process is secured. The (MI) gain spectrum graphically visualized with the suitable choice of arbitrary parameters. Conclusion:The graphical performance of the Noyes–Field model solution at various settings reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields.

Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

Analytical Traveling Wave and Soliton Solutions of the $$(2+1)$$ Dimensional Generalized Burgers–Huxley Equation

Analytical Traveling Wave and Soliton Solutions of the $$(2+1)$$ Dimensional Generalized Burgers–Huxley Equation

Analytical traveling-wave solutions and HYDRUS modeling of wet wedges propagating into dry soils: Barenblatt's regime for Boussinesq's equation generalized

The classical Barenblatt solution of an initial-boundary value problem (IBVP) to the parabolic Boussinesq equation, which gives a rectangular triangle of full saturation, propagating from a reservoir into an adjacent porous bank with a vertical slope, is shown to coincide with a solution of IBVP to the elliptic Laplace equation with a phreatic surface along which both isobaricity and kinematic conditions are exactly met. For an arbitrary bank slope, a saturated wedge, which propagates (translates) into dry soil, is also explicitly found. The analytical solutions favorably compare with the results of HYDRUS-2D modeling, i.e., with the FEM solutions of the same IBVPs to the Richards equation. Applications to geotechnical engineering of dykes subject to the impact of flash floods are discussed by comparisons of phreatic lines, loci of the fronts, isobars, equipotential contours, vector fields of Darcian velocity, isotachs, and streamlines in the three models. For example, it is shown that a rapid drawup of the reservoir level induces hydraulic gradients, which may cause seepage-induced erosion of the porous medium, in particular, lessivage.

Analysis of the Solutions of the Equation Modeled in the Field of Nonlinear Sciences

In this article, the travelling wave solutions of the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) equation are investigated using the modified exponential function method (MEFM). This method is used to find analytical travelling wave solutions of the AKNS equation. The different travelling wave solutions are obtained by determining the appropriate values for the parameters. Two and three dimensional graphics of the different wave solutions found in this way are plotted with the help of Mathematica package program by determining the appropriate parameters.

Perturbative Traveling-Wave Solution for a Flux-Limited Reaction-Diffusion Morphogenesis Equation

In this study, we investigate a porous medium-type flux limited reaction--diffusion equation that arises in morphogenesis modeling. This nonlinear partial differential equation is an extension of the generalized Fisher--Kolmogorov--Petrovsky--Piskunov (Fisher-KPP) equation in one-dimensional space. The approximate analytical traveling wave solution is found by using a perturbation method. We show that the morphogen concentration propagates as a sharp wave front where the wave speed has a saturated value. The numerical solutions of this equation are also provided to compare them with the analytical predictions. Finally, we qualitatively compare our theoretical results with those obtained in experimental studies.

New optical explicit plethora of the resonant Schrodinger’s equation via two recent computational schemes

This research paper investigates the computational solutions of the resonant Schr?dinger?s equation. The modified Khater method and Adomian decomposition method are applyied for construct new analytical traveling and semi-analytical wave solutions. This model describes the pulse phenomena and studied in non-linear optics. For further illustration of our obtained solutions, some distinct types of sketches are given.

Lump soliton wave solutions for the (2+1)-dimensional Konopelchenko–Dubrovsky equation and KdV equation

This paper studies (2+1)-dimensional Konopelchenko–Dubrovsky equation and (2+1)-dimensional KdV equation via a modified auxiliary equation technique. These two systems describe the connection between the nonlinear weaves with a weak scattering and long-range interactions between the tropical, mid-latitude troposphere, the interaction of equatorial and mid-latitude Rossby waves, respectively. We implement a novel technique to these systems to find analytical traveling wave solutions. The performance of this novel method shows its ability for applying on various nonlinear partial differential equations. All solutions obtained are checked by the Maple software system and verified for its high fidelity.

A Riccati–Bernoulli sub-ODE Method for Some Nonlinear Evolution Equations

Abstract This article concerns with the construction of the analytical traveling wave solutions for the model of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave and the higher-order nonlinear Schrödinger equation by Riccati–Bernoulli sub-ODE method. We give the exact solutions for these two equations. The proposed method is effective tool to solve many other nonlinear partial differential equations. Moreover, this method can give a new infinite sequence of solutions. These solutions are expressed by hyperbolic, trigonometric and rational functions. Finally, with the aid of Matlab release 15, some graphical simulations were designed to see the behavior of these solutions.

Travelling wave solutions for fractional Korteweg-de Vries equations via an approximate-analytical method

This paper introduces an approximate-analytical method (AAM) for solving nonlinear fractional partial differential equations (NFPDEs) in full general forms. The main advantage of the paper is to apply the proposed AAM to solve the fractional Korteweg-de Vries (KdV) equations. Moreover, the analytical travelling wave solutions for the fractional KdV equation and the modified fractional KdV equation are successfully obtained. The numerical solutions are also obtained in the forms of tables and graphs. The fractional partial derivatives are considered in Caputo sense.

Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications

AbstractThe aim of this article is to construct some new traveling wave solutions and investigate localized structures for fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) water wave dynamical equation. The simple equation method (SEM) and the modified simple equation method (MSEM) are applied in this paper to construct the analytical traveling wave solutions of AKNS equation. The different waves solutions are derived by assigning special values to the parameters. The obtained results have their importance in the field of physics and other areas of applied sciences. All the solutions are also graphically represented. The constructed results are often helpful for studying several new localized structures and the waves interaction in the high-dimensional models.

A family of analytical solutions of a nonlinear diffusion–convection equation

Despite its popularity in many engineering fields, the nonlinear diffusion–convection equation has no general analytical solutions. This work presents a family of closed-form analytical traveling wave solutions for the nonlinear diffusion–convection equation with power law nonlinearities. This kind of equations typically appears in nonlinear problems of flow and transport in porous media. The solutions that are addressed are simple and fully analytical. Three classes of analytical solutions are presented depending on the type of the nonlinear diffusion coefficient (increasing, decreasing or constant). It has shown that the structure of the traveling wave solution is strongly related to the diffusion term. The main advantage of the proposed solutions is that they are presented in a unified form contrary to existing solutions in the literature where the derivation of each solution depends on the specific values of the diffusion and convection parameters. The proposed closed-form solutions are simple to use, do not require any numerical implementation, and may be implemented in a simple spreadsheet. The analytical expressions are also useful to mathematically analyze the structure and properties of the solutions.

Analytical traveling-wave solutions to a generalized Gross–Pitaevskii equation with some new time and space varying nonlinearity coefficients and external fields

We present analytical matter-wave solutions to a generalized Gross–Pitaevskii (GGP) equation with several new time and space varying nonlinearity coefficients and external fields. This is realized by taking a suitable similarity transformation to the GGP equation which makes the original partial differential equation into a stationary and ordinary differential equation. We report a few families of analytical solutions of the GGP equation with several new time and space varying nonlinearity interactions, in which some physically relevant soliton solutions are found. The profile features of the evolution wave functions depend on the different choices of the composite functions ξ.

General set of traveling-wave solutions for amplitude equations in the phase field crystal model

Fronts dynamics of periodic crystalline state, which invades the homogeneous state (liquid phase), are analysed. These fronts are considered as traveling waves of atomic density amplitudes. The propagation of amplitudes is described by the hyperbolic equation of an extended Allen-Cahn type for which the complete set of analytical traveling-wave solutions are obtained by tanh-method. The set of solutions includes previously known traveling waves for the parabolic Allen-Cahn equation of both extended and standard form.

Explicit solitons in the parabolic law nonlinear negative-index materials

This paper deals with the existence of explicit ultra-short solitons in the negative-index materials with third-order dispersion and higher-order nonlinearities. Four integration algorithms, that are the improved modified extended tanh-function method, the extended trial equation method, the extended Jacobi elliptic function expansion method and the \(\exp \left( -\Phi \left( \eta \right) \right) \)-expansion method, are employed to extract analytical traveling wave solutions. The presented results show that exact solitons can exist in this physical model.

A study on dust acoustic traveling wave solutions and quasiperiodic route to chaos in nonthermal magnetoplasmas

Bifurcations and chaotic behaviors of dust acoustic traveling waves in magnetoplasmas with nonthermal ions featuring Cairns–Tsallis distribution is investigated on the framework of the further modified Kadomtsev–Petviashili (FMKP) equation. The FMKP equation is derived employing the reductive perturbation technique (RPT). Bifurcations of dust acoustic traveling waves of the FMKP equation is presented. Using the bifurcation theory of planar dynamical systems, two new analytical traveling wave solutions for solitary and periodic waves are derived depending on the parameters $$\alpha , \alpha _1, q, l$$ and U. Considering an external periodic perturbation, the chaotic behavior of dust acoustic traveling waves is investigated through quasiperiodic route to chaos. The parameter q significantly affects the chaotic behavior of the perturbed FMKP equation.

Analysis of virotherapy in solid tumor invasion

Cancer treatment is an inexact science despite traditional cancer therapies. The traditional cancer treatments have high levels of toxicity and relatively low efficacy. Current research and clinical trials have indicated that virotherapy, a procedure which uses replication-competent viruses to kill cancer cells, has less toxicity and a high efficacy. However, the interaction dynamics of the tumor host, the virus, and the immune response is poorly understood due to its complexity. We present a mathematical analysis of models that study tumor-immune-virus interactions in the form of differential equations with spatial effects. A stability analysis is presented and we obtained analytical traveling wave solutions. Numerical simulations were obtained using fourth order Runge–Kutta and Crank–Nicholson methods. We show that the use of viruses as a cancer treatment can reduce the tumor cell concentration to a very low cancer dormant steady state or possibly deplete all tumor cells in body tissue. The traveling waves indicated an exponential increase and decrease in the cytotoxic-T-lymphocytes (CTLs) density and tumor load in the long term respectively.