Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions

We prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous results of Bourgain to more general nonlinearities of class C ^k and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash–Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the inverse linearized operators. Our approach works also in presence of very large “clusters of small divisors”.     

非线性电报方程组的3个双周期正解问题

This paper studies existence of at least three positive doubly periodic solutions of a coupled nonlinear telegraph system with doubly periodic boundary conditions. First, by using the Green function and maximum principle, existence of solutions of a nonlinear telegraph system is equivalent to existence of fixed points of an operator. By imposing growth conditions on the nonlinearities, existence of at least three fixed points in cone is obtained by using the Leggett-Williams fixed point theorem to cones in ordered Banach spaces. In other words, there exist at least three positive doubly periodic solutions of nonlinear telegraph system.     

Differential characteristic set algorithm for the complete symmetry classification of partial differential equations

In this paper, we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations （PDEs） with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm, i.e., Wu＇s method, in differential equations.     

Treatment of nonlinear terms in the numerical solution of partial differential equations for multiphase flow in porous media

Simultaneous flow of two or three immiscible fluids in porous media is modelled by a system of coupled, nonlinear partial differential equations. These equations are reduced to a system of nonlinear algebraic equations through the use of finite-difference approximations for derivatives. Several types of nonlinearities requiring careful analysis exist in this model. Here, we present a systematic study of all available, and some new, methods for the treatment of nonlinearitics in this model. It is believed that the solution techniques presented here may also prove useful for other strongly nonlinear partial differential equations.     

A Permutation Related to Non-compact Global Attractors for Slowly Non-dissipative Systems

We consider scalar reaction-diffusion equations with non-dissipative nonlinearities generating global semiflows which exhibit blow-up in infinite time. This type of equations was only recently approached and the corresponding dynamical systems are known as slowly non-dissipative systems. The existence of unbounded solutions, referred to as grow-up solutions, requires the introduction of some objects interpreted as equilibria at infinity. By extending known results, we are able to obtain a complete decomposition of the associated non-compact global attractor. The connecting orbit structure is determined based on the Sturm permutation method, which yields a simple criterion for the existence of heteroclinic connections.     

Gevrey Regularity for Nonlinear Analytic Parabolic Equations on the Sphere

The regularity of solutions to a large class of analytic nonlinear parabolic equations on the two-dimensional sphere is considered. In particular, it is shown that these solutions belong to a certain Gevrey class of functions, which is a subset of the set of real analytic functions. As a consequence it can be shown that the Galerkin schemes, based on the spherical harmonics, converge exponentially fast to the exact solutions, as the number of modes involved in the approximation tends to infinity. Furthermore, in the case that the underlying evolution equation has a global attractor, then this global attractor is contained in the space of spatially real analytic functions whose radii of analyticity are bounded uniformly from below.     

A Smooth Center Manifold Theorem which Applies to Some Ill-Posed Partial Differential Equations with Unbounded Nonlinearities

We prove the existence of a smooth center manifold for several partial differential equations, including ill posed equations with unbounded nonlinearities. We also prove smooth dependence on parameters with respect to some perturbations, including unbounded ones. More concretely, we prove an abstract theorem and present applications to several concrete equations: ill posed Boussinesq, equation and system and nonlinear Laplace equations in cylindrical domains. We also consider the effect of some geometric structures.     

Attractors of Parabolic Equations Without Uniqueness

In this paper we study the existence of global compact attractors for nonlinear parabolic equations of the reaction-diffusion type and variational inequalities. The studied equations are generated by a difference of subdifferential maps and are not assumed to have a unique solution for each initial state. Applications are given to inclusions modeling combustion in porous media and processes of transmission of electrical impulses in nerve axons.     

Propagation of surface (seismic) waves: Ordinary differential equations with strongly nonlinear damping

Second-order ordinary differential equations (ODEs) with strongly nonlinear damping (cubic nonlinearities) govern surface wave motions that entail nonlinear surface seismic motions. They apply to dynamic crack propagation and nonlinear oscillation problems in physics and nonlinear mechanics. It is shown that the nonlinear surface seismic wave equation (Rayleigh equation) admits several functional transformations and it is possible to reduce it to an equivalent first-order Abel ODE of the second kind in normal form. Based on a recently developed methodology concerning the construction of exact analytic solutions for the type of Abel equations under consideration, exact solutions are obtained for the nonlinear seismic wave (NLSW) equation for initial conditions of the physical problem. The method employed is general and can be applied to a large class of relevant ODEs in mathematical physics and nonlinear mechanics.     

Pulses in binary wave guide arrays and long wave PDE approximations

Binary waveguide arrays are linear arrays of optical waveguides with binary alternation of parameters, and have been of recent interest. They can be modeled by systems of nonlinear ODEs with forms related to the discrete nonlinear Schrödinger equation. Such equations can also arise in semi-classical molecular models of polymers with excitable states in each monomer, and coupling between these.An important class of solutions arises from an initially highly localized signal, such as input to a single element of the array. Simulations show that for a wide array of parameter values and of such initial data, a pulse is generated that travels approximately as a traveling wave. After a suitable phase shift in the variables, this pulse quickly develops a slow spatial variation, leading to a long-wave approximation by a system of coupled third order PDEs; one each for nodes of even and odd indices.This system of PDEs is presented, and verified to quite accurately reproduce the pulse propagation seen in the ODE system; further there is often a strong tendency for the behavior of the two PDE components to converge, with a corresponding convergence of the even and odd index parts of the ODE system solution. The PDE model gives some indication of why this occurs.     

Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model

We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H ¹. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.     

Long-time Instability and Unbounded Sobolev Orbits for Some Periodic Nonlinear Schrödinger Equations

We study the energy cascade problematic for some nonlinear Schrödinger equations on ${\mathbb{T}^2}$ in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one ${|u|^2u}$ . Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the first Birkhoff normal form).     

Solutions to general forward-backward doubly stochastic differential equations

A general type of forward-backward doubly stochastic differential equations （FBDSDEs） is studied. It extends many important equations that have been well studied, including stochastic Hamiltonian systems. Under some much weaker monotonicity assumptions, the existence and uniqueness of measurable solutions are established with a incthod of continuation. Furthermore, the continuity and differentiability of the solutions to FBDSDEs depending on parameters is discussed.     

Attractor for a Degenerate Nonlinear Diffusion Problem with Nonlinear Boundary Condition

In this paper we prove the existence of a compact attractor in L () for a degenerate nonlinear diffusion problem with nonlinear flux on the boundary. In order to formulate the equation as a dynamical system, some existence and uniqueness results for weak solutions are proved.     

A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater

Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs. The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE^2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number. The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the arithmetic average of log-concentrations of boundary and initial waters.     

A sharp condition for existence of an inertial manifold

It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applies also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semi-linear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherel's theorem for Banach valued functions are used.     

The Vadasz–Olek Model Regarded as a System of Coupled Oscillators

The Vadasz–Olek model of chaotic convection in a porous layer is revisited in this article. The first-order differential equations of this Lorenz-type model are transformed in the governing equations of a damped nonlinear oscillator, modulated by a linear degenerated overdamped oscillator (relaxator) which in turn is coupled to former one by a nonlinear cross force. The benefit of this mechanical analogy is an intuitive picture of the regular and chaotic dynamics described by the Vadasz–Olek model. Thus, there turns out that the “eyes” of the chaotic attractor correspond to the minima of the potential energy of the modulated nonlinear oscillator having a double-well shape. Several new aspects of the subcritical and supercritical dynamic regimes are discussed in some detail.