Nonlinear Coupled Evolution Equations Research Articles

Multiple and Singular Soliton Solutions for Space–Time Fractional Coupled Modified Korteweg–De Vries Equations

The focus of this paper is on the nonlinear coupled evolution equations, specifically within the context of the fractional coupled modified Korteweg–de Vries (mKdV) equation, employing the conformable fractional derivative (CFD) approach. The primary objective of this paper is to thoroughly investigate the applicability of the Hirota bilinear method for deriving analytical solutions to the fractional mKdV equations. A range of exact analytical solutions for the fractional coupled mKdV equations is obtained. The findings in general indicate that the Hirota bilinear method is an effective approach for resolving the complexities associated with the fractional coupled mKdV equations.

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

Oblique resonance wave phenomena for nonlinear coupled evolution equations with fractional temporal evolution

This work investigates the obliquely propagating resonance wave phenomena described by coupled model equations with fractional temporal evolution arising in many branches of physics. The (2 + 1) -dimensional coupled nonlinear Schrodinger equation (NLSE), long wave-short wave resonance interaction (LSRI) equation and Maccari system (MS) having fractional time evolution are considered to study such physical phenomena. The modified Kudryashov method (mKM) with the properties of Khalil's conformable fractional derivatives is implemented to divulge scattered oblique wave solutions of considered equations. The effects of obliqueness and fractional parameter on obtained results are demonstrated graphically along with the physical descriptions. It is found that the resonance wave phenomena are changed with the increase of obliqueness as well as fractional parameter.

Nonlinear longwave stability of two liquid layers coating both sides of a thick wall in presence of gravity

The linear and nonlinear thermocapillary instability of two liquid layers coating both sides of a horizontal solid wall in the presence of gravity was investigated in the absence of thermal buoyancy. A set of two coupled nonlinear evolution equations was calculated under the small wavenumber approximation. The two liquid layers were horizontal and had thermal interaction through the wall by means of a temperature gradient perpendicular to the two layers and the wall. The linear growth rate of the system instability was investigated with respect to a variety of parameters. In the same way as in a previous report in the absence of gravity, here the second stationary mode was always stable. In the presence of gravity it was possible to have stationary convection alone, stationary and oscillatory convection and oscillatory convection alone, depending on the magnitudes and signs of the parameters. It was also shown that the linear stability can be described by means of three parameters alone, different to those of a previous paper. Numerical analysis of the nonlinear coupled evolution equations of the free surfaces deformations was performed to find out which conditions lead to the sinuous or to the varicose mode of instability.

Resonance nonlinear wave phenomena with obliqueness and fractional time evolution via the novel auxiliary ordinary differential equation method

In this article, the oblique resonance wave phenomena are investigated by considering nonlinear coupled evolution equations with fractional time evolution. In order to investigate such physical phenomena arising in many branches of physics, the time fractional coupled (2 + 1)-dimensional nonlinear Schrodinger and long-short wave resonance interaction evolution equations are considered. The analytical solutions of considered equations are achieved by implementing the proposed auxiliary ordinary differential equation method along with the properties of Khali’s fractional derivatives. The obtained outcomes may be useful for better understanding the basic properties of internal oblique propagating wave dynamics in many branches of science and engineering.

Advance Exp (-Φ(ξ)) Expansion Method and Its Application to Find the Exact Solutions for Some Important Coupled Nonlinear Physical Models

The theoretical investigations of resonance physical phenomena by nonlinear coupled evolution equations are become important in currently. Hence, the purpose of this paper is to represent an advance exp (-Φ(ξ))-expansion method with nonlinear ordinary differential equation for finding exact solutions of some nonlinear coupled physical models. The present method is capable of evaluating all branches of solutions simultaneously and this difficult to distinguish with numerical technique. To verify its computational efficiency, the coupled classical Boussineq equation and (2+1)-dimensional Boussinesq and Kadomtsev-Petviashili equation are considered. The obtained solutions in this paper reveal that the method is a very effective and easily applicable of formulating the exact traveling wave solutions of the nonlinear coupled evolution equations arising in mathematical physics and engineering.

Application of Tanh Method to Complex Coupled Nonlinear Evolution Equations

M.A. Abdelkawy, A.H. Bhrawy, E. Zerrad and A. Biswas Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University, Riyadh-11566, Saudi Arabia Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Department of Physics and Engineering, Delaware State University, Dover, DE 19901-2277, USA Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah-21589, Saudi Arabia

Combined Effects of Thermal Diffusion and Diffusion-Thermo Effects on Transient MHD Natural Convection and Mass Transfer Flow in a Vertical Channel with Thermal Radiation

We study the nonlinear coupled evolution equations which model the transient MHD natural convection and mass transfer flow of viscous, incompressible and electrically conducting fluid between two infinite vertical plates in the presence of the transversal magnetic field, thermal radiation, thermal diffusion and diffusion-thermo effects. Both analytical and numerical methods are used for this study.

Numerical Solutions of Coupled Nonlinear Evolution Equations via El-gendi Nodal Galerkin Method

In this research the solution of coupl ed modified Korteweg -de Vries equation (mKdV) and the generalized Hirota–Satsuma coupled KdV equation by using El-gendi nodal Galerkin (EGG) approaches are presented. El-gendi nodal Galerkin (EGG) (EGG) approaches consist of two approaches, the first is El-gendi Chebyshev nodal Galerkin (ECG) and the second approach is called El-gendi Legendre nodal Galerkin (ELG). In these new approaches spaces of the solution and the weak form to the system are presented. The resulted systems of ODES are solved by the fourth order Runge-Kutta solver. The convergence and the stability of these new methods are analyzed numerically. Numerical results are presented and compared with the results obtained by pseudo-spectral method. Original Research Article

Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations

In this article, the modified simple equation (MSE) method has been executed to find the traveling wave solutions of the coupled (1+1)-dimensional Broer-Kaup (BK) equations and the dispersive long wave (DLW) equations. The efficiency of the method for finding exact solutions has been demonstrated. It has been shown that the method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations. Keywords: MSE method; NLEE; BK equations; DLW equations; Solitary wave solutions. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.16671 J. Sci. Res. 6 (2), 273-284 (2014)

A Meshless Method of Lines for Numerical Solution of Some Coupled Nonlinear Evolution Equations

Abstract Method of lines (MOL) together with radial basis functions (RBFs) is proposed to find the numerical solution of coupled Korteweg-de Vries (KdV) equations, coupled modified Korteweg-de Vries (mKdV) equations and coupled KdV system. The initial-boudary-value problem is reduced to a system of ordinary differential equations. The system is then solved by the fourth order Runge-Kutta method. Numerical examples are given to illustrate practical usefulness of the present approach. The method is efficient and all the examples take very low CPU time. Accuracy of the method is assessed in terms of L∞ error norm, number of nodal points and time step size. Superiority of the suggested method is shown through comparison with the spectral collocation method.

On the justification of the quasistationary approximation of several parabolic moving boundary problems—Part I

We consider nonlinear coupled evolution equations evolving according to different timescales and study the behavior of solutions as their ratio becomes singular. We derive an abstract result and use it to justify rigorously the quasistationary approximation of a moving boundary problem modeling the growth of an avascular tumor. Another application is a quasilinear formulation of the Keller–Segel model on a bounded domain in RN.

Traveling Wave Solutions of Some Coupled Nonlinear Evolution Equations

The modified simple equation (MSE) method is executed to find the traveling wave solutions for the coupled Konno-Oono equations and the variant Boussinesq equations. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It has been shown that the proposed method is direct, effective, and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations.

Nonlinear dynamics and chaos of magnetized resonant surface waves in a rectangular container

In this paper the nonlinear evolution of the three-dimensional instability of resonated standing surface waves along the interface of a weakly viscous, incompressible magnetic liquid within a rectangular basin is investigated. A combination of the Rosensweig instability with Faraday instability is created, where the system is assumed to be stressed by a normal alternating magnetic field together with an external vertical oscillating force. First, it is assumed that the liquid is inviscid and thereby the motion is irrotational, a system of nonlinear coupled evolution equations governing the complex amplitudes of the different modes is derived. Second, a system of linear equations, derived via solving the linearized Navier–Stokes equations, is obtained. Consequently, the nonlinear equations of the complex amplitudes that correspond to the ideal fluid case are modified by adding the linear damping. This system is exploited to determine the steady-state solutions and then studying their stability both analytically and numerically. The results show that the liquid viscosity rather than the magnetic field affects the qualitative behavior of the wave motion and the system response alternates between the regular periodic and chaotic behavior depending on the specific values of some parameters.

The Stability of Evaporating Thin Liquid Film Containing Insoluble Surfactant on Heated Substrate

Flow of evaporating thin liquid film containing insoluble surfactant on a uniformally heated substrate is considered in this paper. Coupled nonlinear evolution equations for the film thickness and surfactant concentration are derived on the base of lubrication theory and reasonable boundary conditions. The flow stability of the thin liquid film has been studied using normal mode method according to the linear stability theory. The results show that the film stability is promoted by increasing the Capillary number and the surfactant Peclet number, while increasing the Marangoni number, the interface resistance number, the vapor recoil number and the evaporation number can reduce the stability of the system.

Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method

This paper studies two nonlinear coupled evolution equations. They are the Zakharov equation and the Davey–Stewartson equation. These equations are studied by the aid of Jacobi’s elliptic function expansion method and exact periodic solutions are extracted. In addition, the Zakharov equation with power law nonlinearity is solved by traveling wave hypothesis.

Reductions and Solutions of Two Types of Coupled Nonlinear Evolution Equations in Optical Fibers and Fluid Dynamics

Abstract Under investigation in this paper are the coupled nonlinear Schrödinger equations (CNLSEs) and coupled Burgers-type equations (CBEs), which are, respectively, a model for certain birefringent optical fibers Raman-scattering, Kerr and gain/loss effects, and a generalized model in fluid dynamics. Special attention should be paid to the existing claim that the solitons for the CNLSEs do not exist. Through certain dependent-variable transformations, the CNLSEs are reduced to a Manakov system and the CBEs are linearized. In that way, some new solutions of the CNLSEs and CBEs are obtained via symbolic computation. Especially the one-dark-soliton-like solutions for the CNLSEs have been found, against the existing claim.

Rupture of thin liquid films: Generalization of weakly nonlinear theory

In this paper, we investigate the rupture dynamics of thin liquid films driven by intermolecular forces via weakly nonlinear bifurcation analysis. The dynamic equations governing slow dynamics of the perturbation amplitude of the near-critical mode corresponding to several models describing the evolution of thin liquid films in different physical situations appear to have the same structure. When antagonistic (attractive and repulsive) molecular forces are considered, nonlinear saturation of the instability becomes possible, while the boundary of this supercritical bifurcation is determined solely by the form of the intermolecular potential. The rupture time estimate obtained in closed form shows an excellent agreement with the results of the previously reported numerical simulations of the strongly nonlinear coupled evolution equations upon fitting the amplitude of the small initial perturbation. We further extend the weakly nonlinear analysis of the film dynamics and apply the Galerkin approximation to derive the amplitude equation(s) governing the dynamics of the fastest growing linear mode far from the instability threshold. The comparison of the rupture time derived from this theory with the results of numerical simulations of the original nonlinear evolution equations shows a very good agreement without any adjustable parameters.

A Maple package for finding exact solitary wave solutions of coupled nonlinear evolution equations

The tanh function expansion method for finding traveling solitary wave solutions to coupled nonlinear evolution equations is described. A complete implementation RATHS written in Maple is presented, in which the operator mains can output exact solitary wave solutions entirely automatically. Furthermore, RATHS can handle any number of dependent variables u i as well as any number of independent variables x j contained in the input system. This package can also be applied to ODEs. The effectiveness of RATHS is illustrated by applying it to a variety of equations. Program summary Title of program: RATHS Catalogue identifier: ADSD (also ADQK) Program Summary URL: http://cpc.cs.qub.ac.uk/summaries/ADRY (also ADQR) Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Computers: PC Pentium IV Installations: Copy Operating systems: Windows 98/2000/XP Program language used: Maple V R6 Memory required to execute with typical data: depends on the problem, minimum about 8M words No. of bits in a word: 8 No. of bytes in distributed program, including the test data, etc.: 16 608 Distribution format:tar gzip file Keywords: Coupled nonlinear evolution equations, traveling solitary wave solutions, dependent variable, independent variable Nature of physical problem: Our program give out exact solitary wave solutions, which can describe various phenomena in nature, and thus can give more insight into the physical aspects of problems and may be easily used in further applications. Restriction on the complexity of the problem: The program can handle coupled nonlinear evolution equations, in which every equation is a polynomial (or can be converted to a polynomial) in the unknown functions and their derivatives. Typical running time: It depends on the input equations as well as the degrees of the desired polynomial solutions. For most of the coupled equations which we have computed, the running time is no more than 20 seconds.

Analysis of tear film rupture: effect of non-Newtonian rheology

We investigate the rupture mechanism of a precorneal thin mucus coating sandwiched between the aqueous tear film and the corneal epithelial surface with a monolayer of surfactant overlying the aqueous layer. The Ostwald constitutive relation is employed to model mucus and a linear equation of state describing the relationship between surface tension and surfactant concentration is adopted. Three nonlinear coupled evolution equations governing the transport of surfactant, mucus, and total liquid layer thicknesses, based on lubrication theory and a perturbation expansion technique, have been derived. The resulting equations are solved numerically in order to explore the influence of the rheological properties of mucus, aqueous–mucus thickness ratio, aqueous–mucus interfacial tension, Marangoni number, and surfactant concentration on both the onset of instability and tear film evolution in the presence of van der Waals interactions, which could rupture the tear film. Our results reveal that the influence of rheological properties, aqueous–mucus thickness ratio, and interfacial tension on the time required for film rupture can be significant and varies considerably, depending on the magnitude of the Hamaker constants governing the strength of the van der Waals forces.