Higher Order Nonlinear Evolution Equations Research Articles

Global existence of weak solutions to a class of higher-order nonlinear evolution equations

<p>This paper deals with the initial boundary value problem for a class of <italic>n</italic>-dimensional higher-order nonlinear evolution equations that come from the viscoelastic mechanics and have no positive definite energy. Through the analysis of functionals containing higher-order energy of motion, a modified potential well with positive depth is constructed. Then, using the potential well method, and Galerkin method, it has been shown that when the initial data starts from the stable set, there exists a global weak solution to such an evolution problem.</p>

Analysis of new soliton type solutions to generalized extended (2 + 1)-dimensional Kadomtsev-Petviashvili equation via two techniques

In this paper, we examine the solitary wave solutions of the generalised (2 + 1) extended Kadomtsev–Petviashvili (eKP) equation, which is used as model for the surface waves and internal waves in straits or channels and describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. We secure distinct soliton solutions by employing two amelioration methodologies, namely the improved F-expansion method and the new Kudryashov method. These methods explore several unique forms of solitary wave solutions, including dark, rational, singular, periodic, and exponential ones. In order to illustrate propagation of some reported wave solutions, the graphs associated with these solutions are presented by choosing appropriate parametric values in 2D, 3D, contour, and density plots with the use of the symbolic software Mathematica 13.0. The presented solutions realize very significant fact for investigation the qualitative interpretations for various phenomena in our nature. The computed solutions revealed that the applied methods are influential, efficious and skilful and can be a best way to handle other higher order nonlinear evolution equations.

Solitons Solution of Riemann Wave Equation via Modified Exp Function Method

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp(−Φ(η))-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions.

Competent closed form soliton solutions to the Riemann wave equation and the Novikov-Veselov equation

The Riemann wave equation and the Novikov-Veselov equation are interesting nonlinear equations in the sphere of tidal and tsunami waves in ocean, river, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media etc. In this article, the generalized Kudryashov method is executed to demonstrate the applicability and effectiveness to extract travelling and solitary wave solutions of higher order nonlinear evolution equations (NLEEs) via the earlier stated equations. The technique is enucleated to extract solitary wave solutions in terms of trigonometric, hyperbolic and exponential function. We acquire bell shape soliton, consolidated bell shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton and other types of soliton solutions by setting particular values of the embodied parameters. For the precision of the result, the solutions are graphically illustrated in 3D and 2D. The analytic solutions greatly facilitate the verification of numerical solvers on the stability analysis of the solution.

Finite time blow-up for some nonlinear evolution equations

Nonexistence of global solutions to a class of high-order nonlinear evolution equations is investigated. Indeed, under sufficient conditions on the datum and the source term, solutions with arbitrary initial energy blow-up in a finite time.

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

Novel soliton breathers for the higher-order Ablowitz–Kaup–Newell–Segur hierarchy

In the framework of the nonautonomous exactly integrable higher-order equations belonging to the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy, the method for systematic construction of the bound soliton states (localized breathers on the zero background) is presented, and general and specific conditions for their appearance are formulated. We show that both all parameters of solitons forming the breather, and the varying dispersion and nonlinearities should be appropriately chosen in accordance with the exact integrability conditions of the models. Novel breather solutions of the higher-order nonlinear evolution equations with vanishing boundary conditions generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. We demonstrate that novel soliton breather solutions substantially extend the breather concept known so far. The revealed specific conditions for the breather formation can be generalized and applied to the evolution equations of the next, more and more higher orders of the AKNS hierarchy.

Elliptic function and solitary wave solutions of the higher-order nonlinear Schrödinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability

The higher-order nonlinear Schrodinger equation (NLSE) with fourth-order dispersion, cubic-quintic terms, self-steepening and nonlinear dispersive terms describes the propagation of extremely short pulses in optical fibers. In this paper, the elliptic function, bright and dark solitons and solitary wave solutions of higher-order NLSE are constructed by employing a modified extended direct algebraic method, which has important applications in applied mathematics and physics. Furthermore, we also present the formation conditions of the bright and dark solitons for this equation. The modulation instability is utilized to discuss the stability of these solutions, which shows that all solutions are exact and stable. Many other higher-order nonlinear evolution equations arising in applied sciences can also be solved by this powerful, effective and reliable method.

Exact bright–dark solitary wave solutions of the higher-order cubic–quintic nonlinear Schrödinger equation and its stability

The higher-order nonlinear Schrödinger equation with fourth-order dispersion, cubic–quintic terms, nonlinearity, self-steepening and nonlinear dispersive terms describes the propagation of extremely short pulses in optical fibers. The extended form of simple equation method is proposed to construct exact soliton and solitary wave solutions of higher-order nonlinear Schrödinger equation with fourth-order dispersion and cubic–quintic nonlinearity. These new exact solutions are expressed in the forms of trigonometric, hyperbolic, exponential and rational functions. These solutions are also presented graphically. Furthermore, many other higher-order nonlinear evolution equations arising in mathematical physics and other areas of applied sciences can also be solved by this powerful, reliable and capable method.

New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations

In this paper, we construct many new types of Jacobi elliptic function solutions of nonlinear evolution equations using the so-called new extended auxiliary equation method. The effectiveness of this method is demonstrated by applications to three higher order nonlinear evolution equations, namely, the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, the higher order dispersive nonlinear Schrödinger equation and the generalized nonlinear Schrödinger equation. The solitary wave solutions and periodic solutions are obtained from the Jacobi elliptic function solutions. Comparing our new results and the well-known results are given.

Exact traveling wave solutions for system of nonlinear evolution equations.

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

On symmetry-preserving difference scheme to a generalized Benjamin equation and third-order Burgers equation

In this paper, an exposition of a method is presented for discretizing a generalized Benjamin equation and third-order Burgers equation while preserving their Lie point symmetries. By using local conservation laws, the potential systems of original equation are obtained, which can be used to construct the invariant difference models and symmetry-preserving difference models of original equation, respectively. Furthermore, this method is very effective and can be applied to discrete high-order nonlinear evolution equations.

The(G'/G,1/G)-Expansion Method and Its Applications for Solving Two Higher Order Nonlinear Evolution Equations

The two variable(G'/G,1/G)-expansion method is employed to construct exact traveling wave solutions with parameters of two higher order nonlinear evolution equations, namely, the nonlinear Klein-Gordon equations and the nonlinear Pochhammer-Chree equations. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations are rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original(G'/G)-expansion method proposed by Wang et al. It is shown that the two variable(G'/G,1/G)-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.

New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation

In this article, new (G′/G)-expansion method and new generalized (G′/G)-expansion method is proposed to generate more general and abundant new exact traveling wave solutions of nonlinear evolution equations. The novelty and advantages of these methods is exemplified by its implementation to the KdV equation. The results emphasize the power of proposed methods in providing distinct solutions of different physical structures in nonlinear science. Moreover, these methods could be more effectively used to deal with higher dimensional and higher order nonlinear evolution equations which frequently arise in many scientific real time application fields.

Semigroup-theoretical approach to higher order nonlinear evolution equations

Semigroup-theoretical approach to higher order nonlinear evolution equations

Note on an Open Problem of Higher Order Nonlinear Evolution Equations

In view of a new idea on initial conditions, an open problem of nonlinear evolution equations with higher order, which was given by J. L. Lions, is solved. Effect of our results is shown on an example.

Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations

The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.

Constructing rational and multi-wave solutions to higher order NEEs via the Exp-function method

In this paper, we present an application of some known generalizations of the Exp-function method to the fifth-order Burgers and to the seventh-order Korteweg de Vries equations for the first time. The two examples show that the Exp-function method can be an effective alternative tool for explicitly constructing rational and multi-wave solutions with arbitrary parameters to higher order nonlinear evolution equations. Being straightforward and concise, as pointed out previously, this procedure does not require the bilinear representation of the equation considered. Copyright © 2011 John Wiley & Sons, Ltd.

Solitary wave solutions and periodic solutions for higher-order nonlinear evolution equations

The sine–cosine and the tanh methods combined with the homogeneous balance method are used for analytic study for higher-order nonlinear evolution equations. Exact solitary wave solutions and periodic solutions are developed. A comparison between the two used methods is presented.

Solitary Waves for Cubic-Quintic Nonlinear Schrödinger Equation withVariable Coefficients

The cubic-quintic nonlinearSchrödinger equation (CQNLS) plays importantparts in the optical fiber and the nuclear hydrodynamics. By using thehomogeneous balance principle, the bell type, kink type, algebraic solitarywaves, and trigonometric traveling waves for the cubic-quintic nonlinearSchrödinger equation with variable coefficients (vCQNLS) are derivedwith the aid of a set of subsidiary high-order ordinary differentialequations (sub-equations for short). The method used in this paper mighthelp one to derive the exact solutions for the other high-order nonlinearevolution equations, and shows the new application of the homogeneousbalance principle.