Nonlinear evolution equations and their stationary reductions

In recent years there have been many papers on stationary flows of integrable nonlinear evolution equations and their Hamiltonian properties. In particular there have been some results concerning the reversal of the roles of x and t, resulting in PDEs which are Hamiltonian and give the usual stationary Poisson brackets in the reduced case. To date the results have been rather ad hoc and disparate. In this brief report we give a systematic construction of these x−t reversed equations and their Hamiltonian properties, using their isospectral properties. We illustrate our approach with examples from the KdV hierarchy. Bibliography: 5 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1996, pp. 245–159.     

Solution of nonlinear evolution equations

Nonlinear evolution equations are solved by decomposition.     

Linearized stability for nonlinear evolution equations

We present a general principle of linearized stability at an equilibrium point for the Cauchy problem , for an -accretive, possibly multivalued, operator in a Banach space , that has a linear 'resolvent-derivative' . The result is applied to derive linearized stability results for the case of under 'minimal' differentiability assumptions on the operators and at the equilibrium point, as well as for partial differential delay equations. RID="h1" ID="h1"To the memory of Philippe Bénilan     

The EHTA for nonlinear evolution equations

Two types of important nonlinear evolution equations are investigated by using the extended homoclinic test approach (EHTA). Some exact soliton solutions including breather type of soliton, periodic type of soliton and two soliton solutions are obtained. These results show that the extended homoclinic test technique together with the bilinear method is a simple and effective method to seek exact solutions for nonlinear evolution equations.     

Parameter estimation in nonlinear evolution equations

We develop a convergence theory for identifying parameters in a general class of nonau-

tonomous nonlinear evolution equations. An application of this theory to a nonlinear reaction diffusion equation is presented.     

Feynman formulas for nonlinear evolution equations

Transformations of measures, generalized measures, and functions generated by evolution differential equations on a Hilbert space E are studied. In particular, by using Feynman formulas, a procedure for averaging nonlinear random flows is described and an analogue of the law of large number for such flows is established (see [1, 2]).     

Anti-periodic solutions to nonlinear evolution equations

We deal with anti-periodic problems for nonlinear evolution equations with nonmonotone perturbations. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.     

Error control of nonlinear evolution equations

We derive novel a posteriori error estimates for backward Euler approximations of evolution inequalities in Hilbert spaces. The underlying nonlinear (multivalued) monotone operator is subdifferential, or more generally angle-bounded. The estimates depend solely on the discrete solution and data, impose no constraints between consecutive time-steps, exhibit explicit stability factors, and are optimal with respect to both order and regularity.     

Semilinear evolution equations with nonlinear constraints and applications

This paper deals with a general class of evolution problems for semilinear equations coupled with nonlinear constraints. Those constraints may contain compositions of nonlinear operators and unbounded linear operators, and hence the associated operators are not necessarily formulated in the form of continuous perturbations of linear operators. Accordingly, a family of equivalent norms is introduced to discuss 'quasidissipativity' in a local sense of the operators and a generation theory for nonlinear semigroups is employed to construct solution operators on bounded sets. It is a feature of our treatment that the resultant solution operators are obtained as nonlinear semigroups on the whole space which are not 'quasicontractive' but locally equi-Lipschitz continuous.     

Anti-periodic solutions of some nonlinear evolution equations

Following a recent work of H. Okochi in the case of evolution equations generated by subdifferential operators in a real Hilbert space, we point out that many quasi-autonomous evolution equations of non monotone type associated to odd non linear operators have some anti-periodic solutions provided the forcing term is anti-periodic. This comes from the fact that the space of anti-periodic functions is transversal to the kernel of the linear part and stable under the action of odd non linear operators. The proofs of our results combine strong a priori estimates which depend very little on the non-linearities with an application of Schauder's fixed point theorem to some related dissipative equations.     

Linearly implicit methods for nonlinear evolution equations

Summary.  We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations and extend thus recent results concerning the discretization of nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit–explicit multistep schemes as well as the combination of implicit Runge–Kutta schemes and extrapolation. We establish optimal order error estimates. The abstract results are applied to a third–order evolution equation arising in the modelling of flow in a fluidized bed. We discretize this equation in space by a Petrov–Galerkin method. The resulting fully discrete schemes require solving some linear systems to advance in time with coefficient matrices the same for all time levels. Received October 22, 2001 / Revised version received April 22, 2002 / Published online December 13, 2002 Mathematics Subject Classification (1991): Primary 65M60, 65M12; Secondary 65L06 Correspondence to: G. Akrivis     

Generating operators for integrable nonlinear evolution equations

For linear problems which are associated with known, exactly integrable nonlinear evolution equations, one gives the corresponding integrodifferential Λ-operators. Relative to the expansions with respect to the elgenfunctions of Λ-operators, the method of the inverse scattering problem can be considered as the analog of the Fourier transform of linear problems, while the Λ-operators are the analogues of the differentiation operator. One considers the equations: Koteweg-de Vries, the nonlinear Schrödinger equations, the nonlinear Schrödinger equations with a derivative, the system of three waves, the matricial analog of the KdV equation, the Toda chain equation.     

Analyticity of solutions of nonlinear evolution equations

Analyticity in t of solutions u(t) of nonlinear evolution equations of the form $\text{u′ + A(t, u)u = ?(t, u), t > 0, u(0) = u}{}_0$, is established under suitable conditions on $\text{A(t, u), ?(t, u),}\text{and}\text{u}{}_0$. An application is given to quasilinear parabolic equations.     

Parameter identification in nonlocal nonlinear evolution equations

We develop an approximation framework for identifying parameters in a general class of nonautonomous, nonlocal and nonlinear evolution equations. After establishing existence and uniqueness of solutions, we present a convergence theory for Galerkin approximations to inverse problems involving these equations. Our approach relies on the theory of maximal monotone operators in Banach spaces. An application to a nonautonomous nonlinear integral equation arising in heat flow is also discussed.     

非线性发展方程新的周期解

In this paper,some new periodic solutions of nonlinear evolution equations and corresponding travelling wave solutions are obtained by using the double function method and Jacobi elliptic functions.     

Long-time behavior of nonlinear integro-differential evolution equations

We study the long-time behavior as time tends to infinity of globally bounded strong solutions to certain integro-differential equations in Hilbert spaces. Based on an appropriate new Lyapunov function and the Łojasiewicz–Simon inequality, we prove that any globally bounded strong solution converges to a steady state in a real Hilbert space.