Integrable variational equations of non-integrable systems

Paper is devoted to the solvability analysis of variational equations obtained by linearization of the Euler-Poisson equations for the symmetric rigid body with a fixed point on the equatorial plain. In this case Euler-Poisson equations have two pendulum like particular solutions. Symmetric heavy top is integrable only in four famous cases. In this paper is shown that a family of cases can be distinguished such that Euler-Poisson equations are not integrable but variational equations along particular solutions are solvable. The connection of this result with analysis made in XIX century by R. Liouville is also discussed.     

Lepage forms,closed 2-forms and second-order ordinary differential equations

Lepage 2-forms appear in the variational sequence as representatives of the classes of 2-forms. In the theory of ordinary differential equations on jet bundles they are used to construct exterior differential systems associated with the equations and to study solutions, and help to solve the inverse problem of the calculus of variations: since variational equations are characterized by Lepage 2-forms that are closed. In this paper, a general setting for Lepage forms in the variational sequence is presented, and Lepage 2-forms in the theory of second-order differential equations in general and of variational equations in particular, are investigated in detail. The text was submitted by the authors in English.     

Variational method in wiener-hopf equations and its applications

Further discussions are made on equivalence of variational inequalities with Wiener—Hopf equations especially with Toeplitz equations. equivalent propositions are given. Starting from the results, variational method for solving Toeplitz equation is considered and several applications of Toeplitz equation on space crack and contact problems are also discussed.     

Wiener-hopf equations and variational inequalities

In this paper, we show that the general variational inequality problem is equivalent to solving the Wiener-Hopf equations. We use this equivalence to suggest and analyze a number of iterative algorithms for solving general variational inequalities. We also discuss the convergence criteria for these algorithms.     

Numerical solutions of nonlinear evolution equations using variational iteration method

The variational iteration method is used to solve three kinds of nonlinear partial differential equations, coupled nonlinear reaction diffusion equations, Hirota–Satsuma coupled KdV system and Drinefel’d–Sokolov–Wilson equations. Numerical solutions obtained by the variational iteration method are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. He's variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomial in Adomian method. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

We derive an asymptotic solution of the vacuum Einstein equations that describe the propagation and diffraction of a localized, large‐amplitude, rapidly varying gravitational wave. We compare and contrast the resulting theory of strongly nonlinear geometrical optics for the Einstein equations with nonlinear geometrical optics theories for variational wave equations. © 2007 Wiley Periodicals, Inc.     

Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples

This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order k variational equations, for arbitrary values of k, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear spring-pendulum problem for the values of the parameter that can not be decided using first order variational equations.      

Applications of He’s principles to partial differential equations

This paper applies the variational iteration method (VIM) and semi-inverse variational principle to obtain solutions of linear and nonlinear partial differential equations. The nonlinear model is considered from gas dynamics, fluid dynamics and Burgers equation. The linear model is the heat transfer (diffusion) equation. Results show that variational iteration method is a powerful mathematical tool for solving linear and nonlinear partial differential equations, and therefore, can be widely applied to engineering problems.     

Resolvent equations technique for variational inequalities

In this paper, we establish the equivalence between the general resolvent equations and variational inequalities. This equivalence is used to suggest and analyze a number of iterative algorithms for solving variational inclusions. We also study the convergence criteria of the iterative algorithms. Our results include several previously known results as special cases.     

System of generalized resolvent equations with corresponding system of variational inclusions

The purpose of this paper is to introduce a new system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between system of generalized resolvent equations and system of variational inclusions. The iterative algorithms for finding the approximate solutions of system of generalized resolvent equations are proposed. The convergence of approximate solutions of system of generalized resolvent equations obtained by the proposed iterative algorithm is also studied.      

A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.     

Multi-component bi-Hamiltonian Dirac integrable equations

A specific matrix iso-spectral problem of arbitrary order is introduced and an associated hierarchy of multi-component Dirac integrable equations is constructed within the framework of zero curvature equations. The bi-Hamiltonian structure of the obtained Dirac hierarchy is presented be means of the variational trace identity. Two examples in the cases of lower order are computed.     

Integral equations with non integrable kernels

We study here some integral equations linked to the Laplace or the Helmholtz equation, or to the system of elasticity equations. These equations lead to non integrable kernels only defined as finite parts, so that they are quite difficult to approximate. In each case, we introduce a variational formulation which avoids this difficulty and allow us to use stable finite element approximations for these problems     

Variational methods to the fourth-order linear and nonlinear differential equations with non-instantaneous impulses

In this paper, the existence of solutions for the fourth-order linear and nonlinear differential equations with non-instantaneous impulses is studied by applying variational methods. The interesting point lies in that the variational structures corresponding to the fourth-order linear and nonlinear differential equations with non-instantaneous impulses are established for the first time.     

Conservative solutions to a system of variational wave equations

We investigate a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. The main difficulty arises from the possible concentration of energy. We construct the solution by introducing a new set of variables depending on the energy, whereby all singularities are resolved.     

Solitary waves of the generalized Kadomtsev-Petviashvili equations

In this paper, we consider the existence of solitary waves of the generalized Kadomtsev-Petviashvili equations by using variational methods.     

Anti-self-dual Lagrangians: Variational resolutions of non-self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler–Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.     

Variational Iteration Method for Delay Differential Equations

Since1930'sand40's,theexamplesofdelaydifferentialequationsarisinginpracticalapplicationshavebeenescalatedrapidly,andhavebeenstudiedextensively(fordetails,see[1]).Inthispaperwewillproposeanovelmethodcalledvariationaliterationmethod[2]tosolvesuchproblems.Considerfollowingpopulationgrowthmodel[1]x′(t)+cθ(t-1)x(t)+cθ(t-1)=0(1a)x(0)=θ(0),-1≤t≤0(1b)　　Accordingtovariationaliterationmethod[2],thecorrectionfunctionalcanbeconstructedasfollowsxn+1(t)=xn(t)+∫t0λ[x′nτ+cθ(τ-1)xn(τ)+cθ(τ-1…     

Global energy conservative solutions to a system of variational wave equations

This paper is concerned with a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. The global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy is established by using the method of energy-dependent coordinates and the Young measure theory.     

Integro-differential equations for the characteristic functional that are generated by a system of random integral equations

We consider a boundary value problem for a special system of integro-differential equations with variational derivatives. We establish the relationship between this problem and a system of integral equations with a power-law nonlinearity whose kernels and right-hand sides are random functions. We study the solvability of the boundary value problem. Special cases and examples are considered.