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. © 2000 John Wiley & Sons, Inc.

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationThe dynamics of modulated wave trains A. Doelman, B. Sandstede, A. Scheel, G. Schneider The dynamics of modulated wave trains ABSTRACT We investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg--Landau equation, we establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to Burgers equation over the natural time scale. In addition to the validity of Burgers equation, we show that the viscous shock profiles in Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, we 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, we also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results represented here are then applied to the FitzHugh--Nagumo equation and to hydrodynamic stability problems. REPORT MAS-E0504 JANUARY 2005 AbstractWe investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg-Landau equation, we establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to Burgers equation over the natural time scale. In addition to the validity of Burgers equation, we show that the viscous shock profiles in Burgers equation for the wave number can be found as genuine modulated waves in the underlying reactiondiffusion system. In other words, we 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, we also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results represented here are then applied to the FitzHugh-Nagumo equation and to hydrodynamic stability problems.

SynopsisModulation 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 wuh 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 equatton is shown to describe the modulation of wave packets in the Sine-Gordon equation

We are interested in the long{time behavior of nonlinear parabolic PDEs de ned on unbounded cylindrical domains. For dissipative systems de ned on bounded domains, the long{time behavior can often be described by the dynamics in their nite{dimensional attractors. For systems de ned on the in nite line, very little is known at present, since the lack of compactness prevents application of the standard existence theory for attractors. We develop here an abstract theorem based on the interaction of a uniform and a localizing (weighted) norm which allows us to de ne 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.

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.

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In a previous paper we proved that long-wavelength solutions of the waterwave problem in the case of zero surface tension 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 (KdV) equation. In this paper we examine the effect of surface tension on this scenario. We find that we obtain three different physical regimes depending on the strength of the surface tension. For weak surface tension, the propagation of the wave packets is very similar to that in the zero surface tension case. For strong surface tension, the evolution is again governed by a pair of KdV equations, but the coefficients in these equations have changed in such a way that the KdV soliton now represents a wave of depression on the fluid surface. Finally, at a special, intermediate value of the surface tension (where the Bond number equals 1 3 ) the KdV description breaks down and it is necessary to introduce a new approximating equation, the Kawahara equation, which is a fifth order, nonlinear partial differential equation. In each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water-wave problem.

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