Class Of Nonlinear Evolution Equations Research Articles

DEFORMED KORTEWEG-DE VRIES EQUATION WITH SYMBOLIC COMPUTATION

The Korteweg-de Vries-typed equations are the most important class of nonlinear evolution equations, with numerous applications in physical and engineering sciences. In this paper, for the deformed Korteweg-de Vries equation of the form [Formula: see text], we make use of computerized symbolic computation and obtain several families of new exact analytic solutions, some of which are solitonic.

Bessel (Riesz) potentials on Banach function spaces and their applications II, on existence of solutions for a class of nonlinear evolution equations

Bessel (Riesz) potentials on Banach function spaces and their applications II, on existence of solutions for a class of nonlinear evolution equations

New explicit and exact travelling wave solutions for a class of nonlinear evolution equations

With the help of Mathematica, many travelling for a class of nonlinear evolution equations uu+ auxx+ bu + cu2+ du3 = 0 are obtained by using hyperbola function method and WU-elimination method, which include new travelling wave solutions, periodic solutions and kink soliton solutions. Some equations such as Duffing equation, sin-Gordon equation, Φ4 and Klein-Gordon equation are particular cases of the evolution equations. The method can also be applied to other nonlinear equations.

Congruences of Dislocations in Continuously Dislocated Crystals

The time-dependent congruences of Volterra-type dislocations are investigated based on the generalized formulas of Frenet in a Riemannian space. The analysis is applied to the description of congruences of edge and mixed dislocations consistent with a continuous distribution of dislocations for which its material space is an equidistant Riemannian space. In particular, the principal congruences of dislocations are considered. The kinematics of congruences of mixed dislocations endowed with univocally defined local slip planes is discussed. It is shown that the geometry of such congruences of dislocations admits a class of nonlinear evolution equations describing the curvature and torsion of a congruence of curves in a Riemannian space. Additional conditions imposed on the derived system of equations in order to describe the evolution of curvature and torsion of congruences of edge dislocations are proposed. In the static case, an expression is given for shear stresses required to bend prismatic edge dislocations of torsion zero located on the totally geodesic crystal surfaces. It follows that the congruence of these dislocations is endowed with a finite self-energy function.

Non-classical reductions of initial-value problems for a class of nonlinear evolution equations

We classify initial-value problems for a class of one-dimensional evolution equations, which can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The technique applied relies heavily on higher conditional symmetries of the equations under study, which means, in particular, that the obtained reductions cannot be derived within the framework of the standard Lie approach.

Solution of a discrete inverse scattering problem and of the Cauchy problem of a class of discrete evolution equations

The Cauchy problem of a class of nonlinear evolution equations is solved by finding explicit solutions of a discrete inverse scattering problem that are not restricted to the pure soliton case and implementing appropriate time evolution of the scattering data. This yields operator-valued functions, which are shown to solve a hierarchy of operator evolution equations by applying methods similar to those in Marchenko’s work. In addition the relation to canonical Lax constructions is investigated. Using methods introduced by Aden and Carl and Schiebold, one obtains scalar solutions to corresponding scalar equations, sometimes representable by determinants on operator ideals.

Inertial manifold of the atmospheric equations

For a class of nonlinear evolution equations, their global attractors are studied and the existence of their inertial manifolds is discussed using the truncated method. Then, on the basis of the properties of operators of the atmospheric equations, it is proved that the operator equation of the atmospheric motion with dissipation and external forcing belongs to the class of nonlinear evolution equations. Therefore, it is known that there exists an inertial manifold of the atmospheric equations if the spectral gap condition for the dissipation operator is satisfied. These results furnish a basis for further studying the dynamical properties of global attractor of the atmospheric equations and for designing better numerical scheme.

NEW EXPLICIT AND TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR EVOLUTION EQUATIONS

In this paper, with the help of Mathematica, three travelling wave solutions for a class of nonlinear evolution equations utt+auxx+bu+cu3=0 are obtained by trigonometric function method and Wu-eliminition method which include new trarelling wave solutions, bell soliton solutions and kink soliton solutions. Some equations such as Duffing equation, Klein-Gordon equation, Landau-Ginburg-Higgs equation and 4 equation are particular cases of the evolution equations. The method also can be applied to other nonlinear equations.

A Class of Pattern-Forming Models

A general class of nonlinear evolution equations is described, which support stable spatially oscillatory steady solutions. These equations are composed of an indefinite self-adjoint linear operator acting on the solution plus a nonlinear function, a typical example of the latter being a double-well potential. Thus a Lyapunov functional exists. The linear operator contains a parameter ρ which could be interpreted as a measure of the pattern-forming tendency for the equation. Examples in this class of equations are an integrodifferential equation studied by Goldstein, Muraki, and Petrich and others in an activator-inhibitor context, and a class of fourth-order parabolic PDE's appearing in the literature in various physical connections and investigated rigorously by Coleman, Leizarowitz, Marcus, Mizel, Peletier, Troy, Zaslavskii, and others. The former example reduces to the real Ginzburg-Landau equation when ρ = 0 .

New Exact Solutions for a Generalized Breaking Soliton Equation

Abstract The breaking soliton equations are a class of nonlinear evolution equations of broad interest in physical and mathematical sciences. In this paper, the application of the generalized tanh method with symbolic computation leads to new exact solutions for a generalized breaking soliton equation, of which the previously-obtained solutions are the special cases.

MULTIPLE SOLITON SOLUTIONS FOR THE APPROXIMATE EQUATION OF LONG WATER WAVE

By using the homogeneous balance method,we give multiple soliton solutions of the approximate equations of long water wave.The method used herein can be generalized to the study of a class of nonlinear evolution equations.

Nonlinear evolution equations with gradient coupled noise

We show existence and uniqueness of solutions for a class of nonlinear evolution equations with gradient coupled noise. Our results are obtained by using a simple transformation relating the equation under consideration to an underlying deterministic partial differential equation. Both the Ito and the Stratonovich conventions are treated. Several examples show that the properties of solutions for Ito equations can differ significantly from those of Stratonovich equations.

Multiple Soliton-Like Solutions for (2 + 1)-Dimensional Dispersive Long-Wave Equations

By using a homogeneous balance method,multiple-solitonlike solutions of the (2 +1)-dimensional dispersive long-wave equation areconstructed. The method used here can be generalized toa wide class of nonlinear evolution equations.

Dromion solutions for dispersive long-wave equations in two space dimensions

By using the homogeneous balance method, we show that the dromion solutions exist for the (2+1)-dimensional dispersive long-wave equations. It is conjectured that the method used here can be generalized to a class of nonlinear evolution equation. The method here is very concise and primary.

Universal and integrable nonlinear evolution systems of equations in 2+1 dimensions

Integrable systems of nonlinear partial differential equations (PDEs) are obtained from integrable equations in 2+1 dimensions, by means of a reduction method of broad applicability based on Fourier expansion and spatio–temporal rescalings, which is asymptotically exact in the limit of weak nonlinearity. The integrability by the spectral transform is explicitly demonstrated, because the corresponding Lax pairs have been derived, applying the same reduction method to the Lax pair of the initial equation. These systems of nonlinear PDEs are likely to be of applicative relevance and have a “universal” character, inasmuch as they may be derived from a very large class of nonlinear evolution equations with a linear dispersive part.

Convergence and stability of the finite difference scheme for nonlinear parabolic systems with time delay

In this article, a class of nonlinear evolution equations – reaction–diffusion equations with time delay – is studied. By combining the domain decomposition technique and the finite difference method, the results for the existence, convergence and the stability of the numerical solution are obtained in the case of subdomain overlap and when the time-space is completely discretized.

Nonlinear dynamics of vorticity waves in the coastal zone

Vorticity waves are wave-like motions occurring in various types of shear flows. We study the dynamics of these motions in alongshore shear currents in situations where it can be described within weakly nonlinear asymptotic theory. The principal mechanism of vorticity waves can be interpreted as potential vorticity conservation with the background vorticity gradient provided both by the mean current shear and the variation of depth. Under the assumption that the mean potential vorticity distibution is monotonic in the cross-shore direction, the nonlinear stage of the dynamics of weakly nonlinear vorticity waves, long in comparison with the current cross-shore scale, is found to be governed by an evolution equation of the generalized Benjamin–Ono type. The dispersive terms are given by an integro-differential operator with the kernel determined by the large-scale cross-shore depth and current dependence. The derived equations form a wide new class of nonlinear evolution equations. They all tend to the Benjamin–Ono equation in the short-wave limit, while in the long-wave limit their asymptotics depend on the specific form of the depth and current profiles. For a particular family of model bottom profiles the equations are ‘intermediate’ between Benjamin–Ono and Korteweg–de Vries equations, but are distinct from the Joseph intermediate equation. Solitary-wave solutions to the equations for these depth profiles are found to decay exponentially. Taking into account coastline inhomogeneity or/and alongshore depth variations adds a linear forcing term to the evolution equation, thus providing an effective generation mechanism for vorticity waves.

Global Solution and Regularizing Properties on a Class of Nonlinear Evolution Equation

In this paper we will consider the equation[formula]where[formula]The initial value problem is proved to be locally well posed for initial data taken inD(A2)×D(A3/2) and globally well posed for small data, in this case we also show the exponential decay of the solution as time goes to infinity. The main result of this paper is to prove that the solution has the smooting effect property on the initial data. This means that, if the initial data belongs toD(A2)×D(A3/2) then the solutionubelongs toC∞(]0, +∞[; D(Ak)) ∀k∈N, providedM,N, andRareC∞-function.

Perturbed reversible systems

For a class of nonlinear evolution equations describing reversible processes with several equilibrium solutions, we will demonstrate that the addition of time-dependent disturbances can significantly change the stability properties of the model. In particular, we will show that the introduction of bounded time-dependent forcing can cause singular changes in the basins of attraction for Abel's nonlinear ordinary differential equation.

Uniform Nonlinear Evolution Equations for Pure and Mixed Quantum States

We propose a new class of nonlinear evolution equations, satisfying the homogeneity condition, for pure and mixed quantum states. These equations are generalizations of the nonlinear Schrödinger equation, recently proposed by Doebner and Goldin for the wave function written in positional coordinates. The new model is applied to investigate the relaxation processes in systems with equidistant spectra (spin and harmonic oscillator), both in discrete and continuous representations.