Guido Schneider

The long‐wave limit for the water wave problem I. The case of zero surface tension

The Korteweg-de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two-dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long-wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves.

Attractors for modulation equations on unbounded domains-existence and comparison

We are interested in the long-time behaviour of nonlinear parabolic PDEs defined on unbounded cylindrical domains. For dissipative systems defined on bounded domains, the longtime behaviour can often be described by the dynamics in their finite-dimensional attractors. For systems defined on the infinite line, very little is known at present, since the lack of compactness prevents application of the standard existence theory for attractors. We develop an abstract theorem based on the interaction of a uniform and a localizing (weighted) norm which allows us to define global attractors for some dissipative problems on unbounded domains such as the Swift-Hohenberg and the Ginzburg-Landau equation. The second aim of this paper is the comparison of attractors. The so-called Ginzburg-Landau formalism allows us to approximate solutions of weakly unstable systems which exhibit modulated periodic patterns. Here we show that the attractor of the Swift-Hohenberg equation is upper semicontinuous in a particular limit to the attractor of the associated Ginzburg-Landau equation.

The validity of modulation equations for extended systems with cubic nonlinearities

Modulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems with no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.

The dynamics of modulated wave trains.

The authors of this title investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg - Landau equation, they establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to the Burgers equation over the natural time scale. In addition to the validity of the Burgers equation, they show that the viscous shock profiles in the Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, they establish the existence and stability of waves that are time-periodic in appropriately moving coordinate frames which separate regions in physical space that are occupied by wave trains of different, but almost identical, wave number. The speed of these shocks is determined by the Rankine - Hugoniot condition where the flux is given by the nonlinear dispersion relation of the wave trains. The group velocities of the wave trains in a frame moving with the interface are directed toward the interface. Using pulse-interaction theory, the authors also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results presented here are applied to the FitzHugh - Nagumo equation and to hydrodynamic stability problems.

Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation

We are interested in the nonlinear stability of the Eckhaus-stable equilibria of the Swift-Hohenberg equation on the infinite line with respect to small integrable perturbations. The difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero. The nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant, i.e. that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime.

A new estimate for the Ginzburg-Landau approximation on the real axis

SummaryModulation equations play an essential role in the understanding of complicated systems near the threshold of instability. For scalar parabolic equations for which instability occurs at nonzero wavelength, we show that the associated Ginzburg-Landau equation dominates the dynamics of the nonlinear problem locally, at least over a long timescale. We develop a method which is simpler than previous ones and allows initial conditions of lower regularity. It involves a careful handling of the critical modes in the Fourier-transformed problem and an estimate of Gronwalls type. As an example, we treat the Kuramoto-Shivashinsky equation. Moreover, the method enables us to handle vector-valued problems [see G. Schneider (1992)].

Kawahara dynamics in dispersive media

Abstract In a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled KdV equations. Here, we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes, the long wave limit can be rigorously approximated by two decoupled Kawahara equations.

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross–Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the 1D stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov–Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.

KP description of unidirectional long waves. The model case

The Kadomtsev–Petviashvili (KP) equation can be formally derived as an envelope equation for three-dimensional unidirectional water waves in the limit of long waves. As a first step towards a mathematical justification, we consider here a two-dimensional Boussinesq equation, which is a realistic model for three-dimensional water waves. Using rigorous estimates, we show that part of the dynamics of the KP equation can be found approximately in the two-dimensional Boussinesq equation. On the other hand, there exist initial data for the KP equation such that the corresponding solutions of the two-dimensional Boussinesq equation behave in no way according to the KP prediction. We expect that similar results hold for the three-dimensional water wave problem too.

Global existence results for pattern forming processes in infinite cylindrical domains: Applications to 3D Navier-Stokes problems

Abstract We consider translational invariant systems on unbounded cylindrical domains in R 3 which are described by the Navier–Stokes equations. The examples which we have in mind are the Taylor–Couette problem and Benards problem. For certain parameter ranges these systems exhibit pattern of almost spatial periodic nature. Although classical energy methods fail on unbounded domains the so called Ginzburg–Landau formalism allows us to show the global existence of strong solutions to all initial conditions in a neighborhood U of the weakly unstable ground state. For all times the bifurcating solutions of the original system can be shadowed by pseudo-orbits of the associated formally derived Ginzburg–Landau equation. This allows us to control the size of the solutions in the original system in terms of the bifurcation parameter for t→∞.