Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field. Oblatum: 6-XI-1998 & 12-VI-1998 / Published online: 14 January 1999     

Asymptotic stability and smooth equivalence op ordinary equations : PMM vol.40, n 6, 1976, pp. 1048–1057

Under certain specified conditions the asymptotic stability is a coarse property [1],(i.e. addition of fairly smooth functions to the right-hand sides of equations, does not disturb the asymptotic stability). It is shown below that in this cage the unperturbed system is coarse in a more general sense, namely, any smooth system acted upon by fairly small smooth perturbations, can be returned to its unperturbed state by a smooth reversible transformation. The value and order of the perturbations and the domain of existence of the transformation are all estimated explicitly. The condition required for the above assertion to hold, is that of the existence of a Liapunov function admitting, together with its derivative, specified estimates. This requirement holds, in particular, in the case when the right-hand sides of the unperturbed system are homogeneous functions, the position of equilibrium is asymptotically stable, and its neighborhood contains no solutions bounded when −∞ <t < ∞ (see [1]). If the system is analytic, the requirement will hold in at least all critical cases investigated in which the asymptotic stability with t → ∞ or t → −∞ is fixed, since in these cases the Liapunov function will be analytic, or simply polynomial. It follows therefore from the theorem which we prove, that in all the cases in question, the system is reduced by a smooth transformation, to the polynomial form. If the unperturbed system is linear, then from the theorem proved follows a theorem on linearization appearing in [2]; if the system is nonlinear but of second order, a theorem from [3] ensues. The results obtained in this paper for the nonlinear autonomous systems are extended to the case when the perturbations are continuous and bounded functions of time. This makes possible the investigation of the dynamics of the process in the neighborhood of asymptotically stable equilibria and of periodic modes, ignoring a wide range of external perturbations.     

Integral Equations of Fredholm Type with Rapidly Varying Kernels and Their Relationship to Dynamic Systems

The relationship between the eigenfunctions of a Fredholm-type integral equation with rapidly oscillating kernel and the dynamic mapping is analyzed. Differential operators commuting with the Fourier operator are constructed. These operators are closely related to nontrivial solutions of the unperturbed nonlinear functional equation related to the dynamic mapping. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 245–249.     

A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-BACKLUND SYMMETRY METHOD

A new method based on Lie-Backlund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained. This method is a generalization of Burde＇s Lie point symmetry technique.     

解扰动非线性发展方程的Lie-B(a)cklund方法

A new method based on Lie-B(a)cklund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained.This method is a generalization of Burde's Lie point symmetry technique.     

Stability of autoresonance models subject to random perturbations for systems of nonlinear oscillation equations

Systems of differential equations arising in the theory of nonlinear oscillations in resonance-related problems are considered. Of special interest are solutions whose amplitude increases without bound with time. Specifically, such solutions correspond to autoresonance. The stability of autoresonance solutions with respect to random perturbations is analyzed. The classes of admissible perturbations are described. The results rely on information on Lyapunov functions for the unperturbed equations.     

GENERALIZED STOCHASTIC PERTURBATION-DEPENDING DIFFERENTIAL EQUATIONS

The paper is devoted to the generalized stochastic differential equations of the Ito? type whose coefficients are additionally perturbed and dependent on a small parameter. Their solutions are compared with the solutions of the corresponding unperturbed equations. We give conditions under which the solutions of these equations are close in the (2m)-th moment sense on finite intervals or on intervals whose length tends to infinity as the small parameter tends to zero. We also give the degree of the closeness of these solutions.     

Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach spaces

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \mathbb Rⁿ{\mathbb R^n} . The solutions take values in ordered Banach spaces E, e.g. E=\mathbb R^N{E=\mathbb R^N} ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.     

Doubly nonlinear evolution equations with non-monotone perturbations

The local (in time) existence of strong solutions to Cauchy problems for doubly nonlinear abstract evolution equations with non-monotone perturbations in reflexive Banach spaces is proved under appropriate assumptions, which allow the case where solutions of the corresponding unperturbed problem may not be unique. To prove the existence, a couple of approximate problems are introduced and delicate limiting procedures are discussed by using various tools from convex analysis and the Kakutani-Ky Fan fixed point theorem. Furthermore, an application of the preceding abstract theory to a nonlinear PDE is also given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n‐th order nonlinear differential equations, equations with a generalized ?‐Laplacian, and nonlinear partial differential systems.     

Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the L ¹-estimate between the entropy solution and the geometric optics expansion function is bounded by O(?²), independent of the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.     

Solutions for systems of nonlinear elliptic equations with nonlinear boundary conditions

We look for solutions of systems of nonlinear elliptic equations with nonlinear boundary conditions and values in some compact convex set M. If the nonlinear terms satisfy a sign condition on the boundary of M and the inhomogeneous terms assume their values in this set existence of solutions is proved. The proof is based on the homotopy invariance of the Leray-Schauder degree and Weinberger's strong maximum principle.     

Parallel nonlinear multisplitting methods

Summary Linear multisplitting methods are known as parallel iterative methods for solving a linear systemAx=b. We extend the idea of multisplittings to the problem of solving a nonlinear system of equationsF(x)=0. Our nonlinear multisplittings are based on several nonlinear splittings of the functionF. In a parallel computing environment, each processor would have to calculate the exact solution of an individual nonlinear system belonging to his nonlinear multisplitting and these solutions are combined to yield the next iterate. Although the individual systems are usually much less involved than the original system, the exact solutions will in general not be available. Therefore, we consider important variants where the exact solutions of the individual systems are approximated by some standard method such as Newton's method. Several methods proposed in literature may be regarded as special nonlinear multisplitting methods. As an application of our systematic approach we present a local convergence analysis of the nonlinear multisplitting methods and their variants. One result is that the local convergence of these methods is determined by an induced linear multisplitting of the Jacobian ofF.Dedicated to the memory of Peter Henrici     

Stability and long time evolution of the periodic solutions to the two coupled nonlinear Schrödinger equations

The system of two coupled nonlinear Schrödinger equations has wide applications in physics. In the past, the main attention has been their solitary waves. Here we turn our attention to their periodic wave solutions. In this paper, the stability of the periodic solutions is studied analytically and the criteria for the stability are obtained. The long time evolution of the solutions to the coupled system is studied numerically for the unstable case emphasizing wave–wave interactions in nonlinear optics. Different kinds of evolution are observed depending on the coefficients of the system and the parameters of the unperturbed waves and perturbation. For certain ranges of parameters, the evolution appears to be periodic, while for some other ranges of parameters, solitary wave or solitary wave pairs can be excited among the irregular background although often the evolution is completely chaotic.     

Solutions of Weakly-Perturbed Linear Systems Bounded on the Entire Axis

We establish conditions under which solutions of weakly-perturbed systems of linear ordinary differential equations bounded on the entire axis R emerge from the point = 0 in the case where the corresponding unperturbed homogeneous linear differential system is exponentially dichotomous on the semiaxes R ₊ and R _–.     

Stability by the linear approximation for discrete equations

This paper is concerned with exponential stability of solutions of perturbed discrete equations. For a given m>1 we will provide necessary and sufficient conditions for exponential stability of all perturbed systems with perturbation of order m under the assumption that the unperturbed linear system is exponentially stable. Basing on this result we obtained necessary and sufficient conditions for exponential stability of the perturbed system for all perturbations of order m>1 for regular systems. Our results are expressed in terms of regular coefficients of the unperturbed system.     

Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.     

Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on ℝN driven by nonlinear noise

This article is concerned with the well-posedness as well as long-term dynamics of a wide class of non-autonomous, non-local, fractional, stochastic FitzHugh-Nagumo systems driven by nonlinear noise defined on the entire space?R^N. The well-posedness is proved for the systems with polynomial drift terms of arbitrary order as well as locally Lipschitz nonlinear diffusion terms by utilizing the pathwise and mean square uniform estimates. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. The existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion terms. The idea of uniform tail-estimates of the solutions in the appropriate spaces is employed to derive the tightness of a family of probability distributions of the solutions in order to overcome the non-compactness of the standard Sobolev embeddings on ?^N as well as the lack of smoothing effect on one component of the solutions. The results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian.

     

Derivatives of the Evans function and (modified) Fredholm determinants for first order systems

The Evans function is a Wronskian type determinant used to detect point spectrum of differential operators obtained by linearizing PDEs about special solutions such as traveling waves, etc. This work is a sequel to the paper “Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves”, published by F. Gesztesy, K. Zumbrun and the second author in J. Math. Pures Appl. 90 , 160–200 (2008), where the Evans and Jost functions for the Schrödinger equations have been considered. In the current work, we study the Evans function for the general case of linear ODE systems, and choose it to agree with the modified Fredholm determinant of the respective Birman‐Schwinger type integral operator. The Evans function is thus the determinant of the matrix composed of the so‐called generalized Jost solutions. These are the solutions of the homogeneous perturbed differential equation which are asymptotic to some reference solutions of the unperturbed equation. One of the main results of the current paper is a formula for the derivative of the Evans function for the first order systems. Its proof uses a matrix composed of the newly introduced modified Jost solutions. These are the solutions of an inhomogeneous perturbed differential equation with the inhomogeneous term constructed by means of the above‐mentioned generalized Jost solutions.     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.