On the Nonlinear Stability of Symplectic Integrators

The modified Hamiltonian is used to study the nonlinear stability of symplectic integrators, especially for nonlinear oscillators. We give conditions under which an initial condition on a compact energy surface will remain bounded for exponentially long times for sufficiently small time steps. While this is easy to achieve for non-critical energy surfaces, in some cases it can also be achieved for critical energy surfaces (those containing critical points of the Hamiltonian). For example, the implicit midpoint rule achieves this for the critical energy surface of the Hénon–Heiles system, while the leapfrog method does not. We construct explicit methods which are nonlinearly stable for all simple mechanical systems for exponentially long times. We also address questions of topological stability, finding conditions under which the original and modified energy surfaces are topologically equivalent. This revised version was published online in July 2006 with corrections to the Cover Date.     

Sensitivity of solutions of linear DAE to perturbations of the system matrices

This paper studies the effect of perturbations in the system matrices of linear Differential Algebraic Equations (DAE) onto the solutions. It turns out that these may result in a more complicated perturbation pattern for higher index problems than in the case for (standard) additive perturbations. We give an analysis here for linear index-1 and index-2 problems, which, however, has clear ramifications in nonlinear problems (e.g., via the Newton process). This analysis is sustained by a number of examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonlinear Theory of a Caustic Region

The caustic formed when a water wave is propagating at incidence into increasing depth is considered first in the linear approximation. A scheme for a nonlinear approach is indicated by this analysis, and the nonlinear equations valid in the caustic region are obtained. Comparison with the gas-dynamics case shows differences from the equations adopted for the sonic-boom caustic.     

Remarques sur les mesures de Wigner

We compute the Wigner measures associated to solutions of semi-classical nonlinear Schrödinger equations. These solutions focus at a point. Outside the caustic, the measures “smooth” the nonlinearity. In the critical cases, a scattering operator describes the jump of Wigner measures at the focus. We show that the problem is ill-posed in terms of Wigner measures.     

Nonlinear oscillations beyond caustics

This paper studies the focusing of high-frequency solutions of semilinear hyperbolic equations. In previous papers, we studied two opposite phenomena. First, the focusing of nonlinear waves can force the solutions to blow up, even before reaching the caustics. Second, for strongly dissipative equations, nonlinear oscillations can be completely absorbed when they reach the caustic set. In this paper, we study the intermediate case of equations with globally Lipschitz nonlinearities. The nonlinear oscillations persist after crossing the caustic set. The solutions are described using oscillatory integrals, which are associated with Lagrangian manifolds in the cotangent bundle. The equations of nonlinear geometric optics lift to these manifolds. In contrast to the linear case, the transport equations for amplitudes living above the same points of spacetime are coupled. © 1996 John Wiley & Sons, Inc.     

The Nonlinear Critical Layer in a Slightly Stratified Shear Flow

An attempt is made in this paper to extend the nonlinear critical layer analysis, as developed for homogeneous shear flows by Benney and Bergeron [1] and Davis [2], to the case of a stratified shear flow. Although the analysis is restricted to small values of the Richardson number evaluated at the edge of the critical layer, it is definitely shown that buoyancy leads to the formation within the critical layer region of thin velocity and thermal boundary layers which tend to reduce the local Richardson number. We suggest that this result has considerable relevance to the phenomenon of clear air turbulence. As in the homogeneous case, no phase change of the disturbance takes place across the nonlinear critical layer.     

Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities

We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcritical Caffarelli–Kohn–Nirenberg inequalities. By extremal functions we mean functions that realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli–Kohn–Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition that determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden–Fowler transformation have to be modified.     

Résolution numérique par différences finies d'un système couplé issu de la dynamique des plasmas

In this Note, a nonlinear system governing nonlinear optics in a crystal is solved with a finite-differences scheme using absorbing boundary conditions. To do so, a changing of unknown functions enables to deal with a standard nonlinear Schrödinger equation. Computations performed with realistic values of various parameters give critical regimes which have been previously detected in experimentations, in the case of the second-harmonic generation and parametric amplification physical background.     

Fractional Hamiltonian Monodromy

We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy–momentum map of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors before. We prove that the 1:( − 2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss manifestations of this monodromy in quantum systems. Communicated by Eduard Zehnder Submitted: February 25, 2005; Accepted: November 17, 2005     

Stability with respect to the initial data and monotonicity of an implicit difference scheme for a homogeneous porous medium equation with a quadratic nonlinearity

We study the stability and monotonicity of a conservative difference scheme approximating an initial-boundary value problem for a porous medium equation with a quadratic nonlinearity under certain conditions imposed only on the input data of the problem. We prove a grid analog of the Bihari lemma, which is used to obtain a priori estimates for higher derivatives; these estimates are needed both in the proof of the continuous dependence of the solution on small perturbations in the input data and for the analysis of monotonicity in the nonlinear case. We show that, regardless of the smoothness of the initial condition, the higher derivatives can become infinite in finite critical time. We give an example in which there arises a runningwave solution, which justifies the theoretical conclusions.     

On perfect nonlinear functions

An Erratum has been published for this article in Journal of Combinatorial Designs 14: 82–82, 2006 . We give the equivalence between perfect nonlinear functions and appropriate splitting semi‐regular relative difference sets, construct a class of splitting relative difference sets by using Galois rings and bent functions, and prove that there exists a 4‐phase perfect nonlinear function if and only if the number of input variables is at least twice the number of output variables. © 2005 Wiley Periodicals, Inc.     

Schrödinger–Poisson systems in the 3-sphere

We investigate nonlinear Schrödinger–Poisson systems in the 3-sphere. We prove existence results for these systems and discuss the question of the stability of the systems with respect to their phases. While, in the subcritical case, we prove that all phases are stable, we prove in the critical case that there exists a sharp explicit threshold below which all phases are stable and above which resonant frequencies and multi-spikes blowing-up solutions can be constructed. Solutions of the Schrödinger–Poisson systems are standing waves solutions of the electrostatic Maxwell–Schrödinger system. Stable phases imply the existence of a priori bounds on the amplitudes of standing waves solutions. Unstable phases give rise to resonant states.     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

Algebraic Representation of Correlation Functions in Integrable Spin Chains

Taking the XXZ chain as the main example, we give a review of an algebraic representation of correlation functions in integrable spin chains obtained recently. We rewrite the previous formulas in a form which works equally well for the physically interesting homogeneous chains. We discuss also the case of quantum group invariant operators and generalization to the XYZ chain. Communicated by Vincent Rivasseau Dedicated to the memory of Daniel Arnaudon Submitted: January 18, 2006; Accepted: February 28, 2006     

Nonlinear uzawa methods for solving nonsymmetric saddle point problems

In [A new nonlinear Uzawa algorithm for generalized saddle point problems, Appl. Math. Comput., 175(2006), 1432–1454], a nonlinear Uzawa algorithm for solving symmetric saddle point problems iteratively, which was defined by two nonlinear approximate inverses, was considered. In this paper, we extend it to the nonsymmetric case. For the nonsymmetric case, its convergence result is deduced. Moreover, we compare the convergence rates of three nonlinear Uzawa methods and show that our method is more efficient than other nonlinear Uzawa methods in some cases. The results of numerical experiments are presented when we apply them to Navier-Stokes equations discretized by mixed finite elements.     

Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases

We consider problems for the nonlinear Boltzmann equation in the framework of two models: a new nonlinear model and the Bhatnagar-Gross-Krook model. The corresponding transformations reduce these problems to nonlinear systems of integral equations. In the framework of the new nonlinear model, we prove the existence of a positive bounded solution of the nonlinear system of integral equations and present examples of functions describing the nonlinearity in this model. The obtained form of the Boltzmann equation in the framework of the Bhatnagar-Gross-Krook model allows analyzing the problem and indicates a method for solving it. We show that there is a qualitative difference between the solutions in the linear and nonlinear cases: the temperature is a bounded function in the nonlinear case, while it increases linearly at infinity in the linear approximation. We establish that in the framework of the new nonlinear model, equations describing the distributions of temperature, concentration, and mean-mass velocity are mutually consistent, which cannot be asserted in the case of the Bhatnagar-Gross-Krook model.     

Critical point theory for perturbations of symmetric functionals

Functionals which are invariant under the action of a compact transformation groupG often have many critical values. Here we consider functionals which are notG-invariant and give conditions for them to have infinitely many critical values; including a mountain pass theorem. We apply it to prove the existence of infinitely many solutions of a nonlinear Dirichlet problem with perturbedG-symmetries.     

Stability of Critical Values and Isolated Critical Continua

We extend the study of critical points in [4]. We show that isolated components of critical points lying on a levelset can be described by an integer which is a lower bound to the “number” of critical points of any function near to the original one in C¹-sup-norm. We also derive a global theorem about continua of critical values similar to that given by Rabinowitz for continua of solutions of certain nonlinear eigenvalue problems. We give a simple application of our abstract results to the problem of bifurcation for gradient systems when the linearization is not completely continuous.