Higher-order Evolution Equations Research Articles

Global existence and estimates for Blackstock's model of thermoviscous flow with second sound phenomena

According to the Lighthill scheme of approximation procedures, we establish the Blackstock-Cattaneo model in this paper to describe nonlinear acoustics' propagations with second sound phenomena in perfect gases under the irrotational flow, which is an approximated higher-order equation for Navier-Stokes-Cattaneo equations. This paper is devoted to the mathematical analysis of the linear and nonlinear higher-order evolution equations in Rn. Concerning the linearized Cauchy problem for the Blackstock-Cattaneo equation, by means of the Fourier analysis and the WKB analysis, we will derive sharp L2 estimates of solutions with additional L1 regularity for initial data. For another, we demonstrate global (in time) well-posedness for the nonlinear Blackstock-Cattaneo equation with small initial data and suitable regularity of Sobolev solutions.

The asymptotic behavior for a binary alloy in energy and material science: The unified method and its applications

The Cahn–Hilliard system was proposed to the first time by Chan and Hilliard in 1958. This model (or system of equations) has intrinsic participation energy and materials sciences and depicts significant characteristics of two phase systems relating to the procedures of phase separation when the temperature is constant. For instance, it can be noticed when a binary alloy (“Aluminum + Zinc” or “Iron + Chromium”) is cooled down adequately. In this case, partially or totally nucleation (nucleation means the appearance of nuclides in the material) is observed: the homogeneous material in the initial state gradually turns into inhomogeneous, giving rise to a very accurate dispersive microstructure. Next, when the time scale is slower the microstructure becomes coarse. In this work, to the first time, the unified method is presented to investigate some physical interpretations for the solutions of the Cahn–Hilliard system when its coefficients varying with time, and to show how phase separation of one or two components and their concentrations occurs dynamically in the system. Finally, 2D and 3D plots are introduced to add more comprehensive study which help to understand the physical phenomena of this model. The technique applied in this analysis is powerful and efficient, as evidenced by the computational work and results. This technique can also solve a large number of higher-order evolution equations.

Solitary wave solutions of the higher-order evolution equations for two ordering parameters in the shallow water waves

By introducing the relation between the orders of the two independent small expansion parameters, the amplitude parameter α and the long wavelength parameter β, which are taken in the form β=0αn and α=0βm, with n and m specified to two particular cases, we derive some new higher order partial differential nonlinear evolution equations for unidirectional shallow water waves under the influence of gravity as well as surface tension. We then show that it is possible to construct one-solution which satisfy these estimates using the Hirota’s bilinear method.

Gray and black optical solitons with quintic nonlinearity

A higher-order nonlinear Schrödinger equation with the third-order dispersion, the cubic–quintic nonlinearities, derivative non-Kerr nonlinear terms, Kerr dispersion, and stimulated inelastic scattering is considered. The model can be used to describe the subpicosecond or femtosecond optical pulse propagation in highly nonlinear optical fibers. We find the exact gray solitary wave solution (i.e., a dark localized structure with nonzero minimum in intensity) on a continuous-wave background for the equation by adopting an appropriate ansatz solution. The propagation characteristics of such solitary pulses are determined in the presence of all higher-order effects. We also find black solitary wave solutions for the higher-order evolution equation under specific conditions for its coefficients. The parametric conditions on fiber parameters for the existence of both types are given.

Existence Results for Some Higher-Order Evolution Equations with Time-Dependent Unbounded Operator Coefficients

Abstract In this paper we establish the existence of C(n)-pseudo almost automorphic mild solutions to some n-order evolution equations involving not only time-dependent operator coefficients but also pseudo almost automorphic forcing terms. To illustrate our abstract results, the existence of C(2)-pseudo almost automorphic mild solutions to some damped second-order nonautonomous differential equations, which among other things model the nonautonomous damped beam equation, will be discussed.

Existence results for some higher-order evolution equations with operator coefficients

In this paper we study and obtain the existence of C(r)-pseudo almost automorphic solutions to some higher-order differential equations with operator coefficients whose forcing is C(s)-pseudo almost automorphic. We next use these abstract results to study the existence of C(r)-pseudo almost automorphic solutions to some second-order differential equations in Hilbert space which, among other things, model the so-called linear beam equation.

WAVE AND MAXWELL'S EQUATIONS IN CARNOT GROUPS

In this paper we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher-order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher-order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations" defined in the intrinsic complex, as well as existence and properties of solutions.

Computational examples of soliton-like classes of solutions of some higher-order evolution equations

A computational approach is used to find new solitary solutions of some nonlinear partial differential equations of higher order. The usual starting point is a special transformation converting the equation under consideration in its two variables x and t into a nonlinear ordinary differential equation in the single variable ξ. When we consider the method of hyperbolic functions, new distinctive classes of solutions of physical relevance result. The new feature of this paper is the fact that we are able to calculate distinctive classes of solitary solutions which cannot be found in the literature. In other words using this method the solution manifold is augmented by further classes of solution functions. Simultaneously we stress the necessity of such sophisticated methods since a general theory of nonlinear partial differential equations does not exist. Otherwise this paper is a natural completion of our recent results using alternative approaches for calculating soliton-like solutions.

On an improved method for solving evolution equations of higher order importantly in science and technology

In this paper we introduce a new algebraic procedure to compute new classes of solutions of (1+1)-nonlinear partial differential equations (nPDEs) both of physical and technical relevance. The basic assumption is that the unknown solution(s) of the nPDE under consideration satisfy an ordinary differential equation (ODE) of the first order that can be integrated completely. This solution manifold of these first-order ODEs play an essential part in solving given nPDEs. A further important aspect however is the fact that we have the freedom in choosing some parameters bearing positively on the algorithm and hence, the solution-manifold of any nPDEs under consideration are therefore augmented naturally. The present algebraic procedure can widely use to study many nPDE and is not only restricted to time-dependent problems. We note that no numerical methods are necessary and so analytical closed-form classes of solutions result. The algorithm works accurately, is clear structured and can be converted in any computer language. On the contrary it is worth to stress the necessity of such sophisticated methods since a general theory of nPDEs does not exist. Keywords: Nonlinear partial differential equations, evolution equations, special function methods.

The application of a novel computational approach to physically important nonlinear evolution equations of higher order

A new algebraic procedure is introduced to find new classes of solutions of (1+1)-nonlinear partial differential equations (nPDEs) of physical relevance. The proposed new computational algebraic approach leads to traveling-wave solutions and moreover a new class of explicit one-soliton-solutions can be obtained. The crucial step of the method is the basic assumption that unknown solutions satisfy an ordinary differential equation (ODE) of first order that can be integrated easily. A further important aspect of this paper, however, is the fact that one is able to calculate a distinctive class of solutions which cannot be found in the current literature. By increasing the number of arbitrary constants the approach allows one to handle problems occurring in solving nonlinear algebraic systems closely related with nPDEs. It is worth mentioning that the benefit of the given approach is surprisingly the easiness in deriving different classes of solutions; so it can therefore be applied in the physical sciences, biology, chemistry and related domains alike. On the contrary it is worth stressing the necessity of such sophisticated methods since a general theory of nPDEs does not exist.

Loss of convexity and embeddedness for geometric evolution equations of higher order

We show that for a large class of geometric evolution equations of immersed surfaces in the Euclidean space, there are compact embedded surfaces that lose their embeddedness and compact strictly convex surfaces that lose their convexity under these evolution equations.

On non-linear evolution equations of higher order – the introduction and application of a novel computational approach

The purpose of the present paper is to introduce a new computational algebraic procedure that can easily be applied to derive class of solutions of non-linear partial differential equations (nPDE) especially of higher order. The crucial step needs an auxiliary variable satisfying some ordinary differential equations (ODE) of first order containing sine, cosine and their hyperbolic varieties introducing to the first time. General transformations are given to determine class of solutions explicitly. The validity and reliability of the method is tested by its application to some important non-linear evolution equations leading to new class of solutions with physical significance. Nevertheless it should be emphasised that this techniques do not need the solution of complicate nODEs as in the case of similarity reduction. Further, the algorithm works efficiently, is clear structured and can be used in any applications independent of the order of the nPDE. For computational purposes the method is appropriate to rewrite it in any computer languages. Therefore, the given novel algebraic approach is suitable for a wider class of nPDE in order to augment the solution manifold by a straightforward alternative approach.

A novel algebraic procedure for solving non-linear evolution equations of higher order

Abstract We report here a systematic approach that can easily be used for solving non-linear partial differential equations (nPDE), especially of higher order. We restrict the analysis to the so called evolution equations describing any wave propagation. The proposed new algebraic approach leads us to traveling wave solutions and moreover, new class of solution can be obtained. The crucial step of our method is the basic assumption that the solutions satisfy an ordinary differential equation (ODE) of first order that can be easily integrated. The validity and reliability of the method is tested by its application to some non-linear evolution equations. The important aspect of this paper however is the fact that we are able to calculate distinctive class of solutions which cannot be found in the current literature. In other words, using this new algebraic method the solution manifold is augmented to new class of solution functions. Simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of nPDE does not exist. Otherwise, for practical use the algebraic construction of new class of solutions is of fundamental interest.

POINCARE-LIGHTHILL-KUO METHOD AND SYMBOLIC COMPUTATION

This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method (PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi-inverse algorithm was proposed by the author, with which the lengthy parts of intermediate expressions are first frozen in the form of symbols till the final stage of seeking perturbation solutions. To discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer-extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of 1 and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head-on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher-order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro-commputers. Finally, it is concluded that with the aid of symbolic computation, the vitality of the PLK method is greatly strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.

Nonparaxial one-dimensional spatial solitons.

Scalar and vector nonlinear nonparaxial evolution equations are developed for propagation in two-dimensions. Using standard soliton scalings, it is found that nonparaxial propagation is accompanied by higher-order linear and nonlinear terms and an effective quintic nonlinear index. The presence of an intrinsic quintic nonlinearity arising from chi((5)) must also be considered at the order of the analysis. These terms represent corrections to the well-known nonlinear Schrodinger equation. Exact and approximate solutions to these higher-order evolution equations are obtained and are shown to exhibit quasi-soliton behavior based on propagation and collision studies. (c) 2000 American Institute of Physics.

Evolution equations of higher order in banach spaces

The present paper deals with a minimal extension of the classical semigroup theory for equations of any order in Banach spaces with closed densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., Semigroups. We present here a generalization of Myadera-Phillips--Feller theorem, of Hille theorem and some other results. The method is quite general. We focus our attention on a particular operator solution (main propagator or abstract initial value Green function) and we assume some properties about it. From this we can obtain all needed information about complementary operator solutions, among others.

Classification of conservation laws for KdV-like equations

We prove that the computation of the conservation laws of (2 n + 1)th-order KdV-like equations (i.e. higher order evolution equations with the same scaling as KdV) can be restricted to polynomials with constant terms, except when the order of the conservation law equals n − 1, in which case the density has linear t dependence. This shows that existing computer algebra programs which assume the conservation law to be of this form are providing the complete answer.

Multiple-time higher-order perturbation analysis of the regularized long-wavelength equation.

By considering the long-wavelength limit of the regularized long wave (RLW) equation, we study its multiple-time higher-order evolution equations. As a first result, the equations of the Korteweg--de Vries hierarchy are shown to play a crucial role in providing a secularity-free perturbation theory in the specific case of a solitary-wave solution. Then, as a consequence, we show that the related perturbative series can be summed and gives exactly the solitary-wave solution of the RLW equation. Finally, some comments and considerations are made on the N-soliton solution, as well as on the limitations of applicability of the multiple-scale method in obtaining uniform perturbative series. \textcopyright{} 1996 The American Physical Society.

On the asymptotic integrability of a higher-order evolution equation describing internal waves in a deep fluid

A higher-order nonlocal evolution equation describing internal waves in a deep fluid is shown to be asymptotically integrable only if the coefficients of the higher-order terms satisfy certain constraints. In this case, the nonlocal equation can be transformed to the integrable Benjamin–Ono equation. The asymptotic integrability of the reductions of the higher-order evolution equation to a complex Burgers equation, to an envelope-wave equation, and to a finite-dimensional dynamical system is also considered.

Hardy Function and Unique Continuation for Evolution Equations

In this paper, unique continuation problems are considered for the first order evolution equation: ut(x, t) = P(t, D) u(x, t), x ∈ Rn, t ∈ R, (∗) where P(t, D) represents a mth order differential operator with time dependent coefficients. In one space dimension case (n = 1), a necessary and sufficient condition is given for any nonzero solution of (∗) in the class of C(R; L2(R)) to have a support on a horizontal half line in the x−t space at two different times. With some assumptions on the coefficients of P(t, D), it is shown that any solution u(x, t) ∈ C(R; L2(R)) is uniquely determined by its part on any two horizontal half lines or any open subset in the x−t space. In higher space dimension case (n > 1), a necessary condition is given for any solution u ∈ C(R; L2(Rn)) of (∗) to be supported on a half hyperplane in the space Rn at two different times. Some higher order evolution equations are also considered for their unique continuation properties.