Class Of Nonlinear Evolution Equations Research Articles

Similarity Analysis of Lax Pairs for a Class of Nonlinear Evolution Equations.

Similarity Analysis of Lax Pairs for a Class of Nonlinear Evolution Equations.

The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3D representation and a 2D representation, as well as a 2D contour plot.

Orbital instability of periodic waves for scalar viscous balance laws

The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital (nonlinear) instability in appropriate periodic Sobolev spaces. The analysis is based on the well-posedness theory, the smoothness of the data-solution map, and an abstract result of instability of equilibria under nonlinear iterations. The resulting instability criterion is applied to two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a local Hopf bifurcation around a critical value of the velocity. The second family comprises arbitrarily large period waves which arise from a homoclinic (global) bifurcation and tend to a limiting traveling pulse when their fundamental period tends to infinity. In the case of both families, the criterion is applied to conclude their orbital instability under the flow of the nonlinear viscous balance law in periodic Sobolev spaces with same period as the fundamental period of the wave.

Periodic solutions to a class of quasi-variational evolution equations

We study a time-periodicity problem for a class of abstract nonlinear evolution equations associated with subdifferential operators depending on both time and the unknown. Assuming time-periodicity for the subdifferential operator, we prove the existence of a periodic solution using the abstract theory of time-dependent subdifferential evolution equations and its generalization. We provide application examples for quasilinear parabolic variational inequalities with time-periodic constraints of non-local time-delay or memory type depending on the unknown.

Generalisation of splitting methods based on modified potentials to nonlinear evolution equations of parabolic and Schrödinger type

The present work is concerned with the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schrödinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, an invariance principle is deduced that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schrödinger case.Numerical illustrations for the time-dependent Gross–Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods.The presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross–Pitaevskii systems in real and imaginary time.

Existence and feedback control for a class of nonlinear evolutionary equations

In the paper we provide systematic approaches to study existence and feedback control for a new evolutionary equation involving pseudomonotone operators. We first establish several existence results for the evolutionary equation by exploiting the Rothe method and using a surjectivity result for multivalued pseudomonotone operators. Then we show the existence of feasible pairs for the feedback control problem by assuming some sufficient conditions. Moreover, the existence of solutions to an evolutionary hemivariational inequality is presented.

Dynamical behaviors with various exact solutions to a (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation using two efficient integral approaches

The main goal of this research work is to explore novel soliton solutions and dynamics of solitonic structures to asymmetric Nizhnik–Novikov–Veselov (ANNV) equations by two methods, namely, the generalized exponential rational function (GERF) method and the modified extended tanh expansion (METE) method. These techniques are the most effective and reliable tools for solving nonlinear equations with partial derivatives. The used methods provide a wide range of soliton solutions that can motivate applied scientists and researchers for these specific structures. These acquired solutions are in the form of exponential, trigonometric and hyperbolic functions including, [Formula: see text] [Formula: see text] and of their combinations. It has also been observed that the obtained soliton solutions of the ANNV equations are bell-shape, anti-bell-shape, periodic solution, multisoliton and different types of soliton solutions. To illustrate the physical features and dynamics characteristics of some obtained solutions, three-dimensional (3D) and two-dimensional (2D) figures are exhibited through the Mathematica software. The methods applied in this paper are suitable to study the soliton solutions for the ANNV equation without producing the complexities in some other known analytical method. Finally, the employed methods can be earmarked easily for solving various classes of nonlinear evolution equations appearing in mathematical physics, fluid dynamics, plasma physics, nonlinear waves and nonlinear sciences.

Extended Jacobi elliptic function solutions for general boussinesq systems

In this research paper, we have utilized the Jacobi elliptic function expansion method to obtain the exact solutions of (1+1)- dimensional Boussinesq System (GBQS). The most important difference that distinguishes this method from other methods is the parameters included in the auxiliary equation F’ (ξ) = Ö P F4(ξ) + QF2(ξ) + R. As far as the authors know, there is no other study in which such a variety of solutions has been given. Depending on P, Q and R, nineteen the solitary wave and periodic wave solutions are obtained at their limit conditions. In addition, 3D and contour plot graphics for the constructed waves are investigated with the computer package program by giving special values to the parameters involved. The validity and reliability of the method is examined by its applications on a class of nonlinear evolution equations of special interest in nonlinear mathematical physics. The results were acquired to verify that the recommended method is applicable and reliable for the analytic treatment of a wide application of nonlinear phenomena

Properties of the support of solutions of a class of nonlinear evolution equations

AbstractIn this work we consider equations of the form where P is any polynomial without constant term, and G is any polynomial without constant or linear terms. We prove that if u is a sufficiently smooth solution of the equation, such that for some , then there exists such that for every . Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation, and for the generalized KdV hierarchy

Solving a Class of Nonlinear Evolution Equations Using Jacobi Elliptic Functions

Nonlinear evolution equations play a crucial role in many applied scientific disciplines. Finding the exact solutions to such equations can contribute to a wide variety of applied disciplines. To be able to calculate exact periodic solutions of nonlinear evolution equations, this paper discusses the expansion of the Jacobi elliptic function and finds the proper expansion of the function. Additionally, by analyzing the applicable conditions of the function according to its specific properties, we find the exact periodic solutions to nonlinear evolution equations with constant coefficients and variable coefficients that satisfy the applicable conditions of the Jacobi elliptic function.

Application of Generalized Logistic Function to Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

The generalized Logistic function that solves a first-order nonlinear ODE with an arbitrary positive power term of the dependent variable is introduced in this paper, by means of which the traveling wave solutions of a class of nonlinear evolution equations, including the generalized Fisher equation, the generalized Nagumo equation, the generalized Burgers-Fisher equation, the generalized Gardner equation, the generalized KdV-Burgers equation, and the generalized Benney equation, are obtained successfully. In these particular cases, traveling wave solutions of several important model PDEs in mathematical physics are also discovered.

Airy function solution of two classes of nonlinear evolution equations

In this paper , we study the problem of constructing the new solutions of the mKdV equation and STO equation. By using the Airy equation and its solution, the new solutions of mKdV equation and STO equation are constructed. First of all, through function transformation, mKdV equation and STO equation can be turned into Airy function to solve. Then, construct the solution of mKdV equation and STO equation by using the solution of Airy function. Finally, analyze the properties of these solutions with the help of symbolic computation system: Mathematica.

The travelling wave solutions of nonlinear evolution equations with both a dissipative term and a positive integer power term

<abstract> <p>The formula of solution to a nonlinear ODE with an undetermined coefficient and a positive integer power term of dependent variable have been obtained by the transformation of dependent variable and $(\frac{{G'}}{G})$-expansion method. The travelling wave reduction ODEs (perhaps, after integration and identical deformation) of a class of nonlinear evolution equations with a dissipative term and a positive integer power term of dependent variable that includes GKdV-Burgers equation, GKP-Burgers equation, GZK-Burgers equation, GBoussinesq equation and GKlein-Gordon equation, are all attributed to the same type of ODEs as the nonlinear ODE considered. The kink type of travelling wave solutions for these nonlinear evolution equations are obtained in terms of the formula of solution to the nonlinear ODE considered.</p> </abstract>

New general interaction solutions to the KPI equation via an optional decoupling condition approach

As a kind of analytical exact solutions to the nonlinear evolution equations, the interaction solutions are of great value in the study of the interacting mechanism in nonlinear science. In this paper, an optional decoupling condition approach is proposed for deriving the lump-stripe solutions and lump-soliton solutions to the KPI equation. We derive new and more general solutions to the KPI equation and discuss the link between the two kinds of interaction solutions, which has not been reported before. The interaction solutions to the KPI equation are analyzed and simulated numerically, which show that all the interaction phenomena are completely inelastic. Although we are concerned on the KPI equation in this paper, this approach can be applied to a wide class of nonlinear evolution equations and lays out the framework of deriving the lump-multi-stripe and/or lump-multi-soliton solutions.

The pointwise error estimates of two energy-preserving fourth-order compact schemes for viscous Burgers’ equation

A novel fourth-order three-point compact operator for the nonlinear convection term uux is provided in this paper. The operator makes the numerical analysis of higher-order difference schemes become possible for a wide class of nonlinear evolutionary equations under the unified framework. We take the classical viscous Burgers’ equation as an example and establish a new conservative fourth-order implicit compact difference scheme based on the method of order reduction. A detailed theoretical analysis is carried out by the discrete energy argument and mathematical induction. It is rigorously proved that the difference scheme is conservative, uniquely solvable, stable, and unconditionally convergent in discrete $L^{\infty }$ -norm. The convergence order is two in time and four in space, respectively. Furthermore, we derive a three-level linearized compact difference scheme for viscous Burgers’ equation based on the proposed operator. All numerically theoretical results similar to that of the nonlinear numerical scheme are inherited completely; meanwhile, it is more time saving. Applying the compact operator to other more complex and higher-order nonlinear evolutionary equations is feasible, including Benjamin-Bona-Mahony-Burgers’ equation, Korteweg-de Vries equation, Kuramoto-Sivashinsky equation, and classification to name a few. Numerical results demonstrate that the presented schemes for Burgers’ equation can achieve second-order accuracy in time and fourth-order accuracy in space in $L^{\infty }$ -norm.

Discrete solution for the nonlinear parabolic equations with diffusion terms in Museilak-spaces

In this paper, a class of nonlinear evolution equations with damping arising in fluid dynamics and rheology is studied. The nonlinear term is monotone and possesses a convex potential but exhibits non-standard growth. The appropriate functional framework for such equations is the modularly Museilak–spaces. The existence and uniqueness of a weak solution are proved using an approximation approach by combining an internal approximation with the backward Euler scheme, also a priori error estimate for the temporal semi-discretization is given.

Nonexistence of global solutions for a class of viscoelastic wave equations

<p style='text-indent:20px;'>We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.</p>

Multiple Soliton Solutions for Nonlinear Differential Equations with a New Version of Extended F-Expansion Method

In this article, a new version of extended F-expansion method is suggested. A multiple Jacobi elliptic functions are presented in the solution function. We achieve analytical solutions of the (1 + 1)-dimensional dispersive long wave (DLW) equation and (2 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation using the new version of extended F-expansion method. Then, the single and the combined non-degenerative Jacobi elliptic function solutions and their degenerate solutions to the above-mentioned class of nonlinear evolution equations are obtained.

On some nonlinear parabolic equations with variable exponents and measure data

Abstract We prove the existence of renormalized solutions to a class of nonlinear evolution equations, supplemented with initial and Dirichlet condition in the framework of generalized Sobolev spaces. The data are assumed merely integrable.

Analytical study on the balancing principle for the nonlinear Klein–Gordon equation with a fractional power potential

In the literature, although many methods depend on the ansatz solution, there is not any specific rule to determine the degree of the ansatz i.e. balancing principle. The problem is especially seen with the fractional and the generalized nonlinear evolution equations. But the main thing is how many terms are needed to determine explicit solution of these type equations. As the previous work, in this work the balancing principle is generalized for large classes of nonlinear evolution equations, also includes fractional nonlinear models.