C. J. Amick

Decay of solutions of some nonlinear wave equations

Abstract We study the large-time behaviour of solutions to the initial-value problem for the Korteweg-de Vries equation and for the regularized long-wave equation, with a dissipative term appended. Using energy estimates, a maximum principle, and a transformation of Cole-Hopf type, sharp rates of temporal decay of certain norms of the solution are obtained.

Small internal waves in two-fluid systems

This paper treats travelling waves in a heterogeneous, inviscid, non-diffusive fluid bounded between two horizontal boundaries. The fluid has two incompressible components of different, but constant density and is acted on by gravity. The flow is steady when viewed in a moving reference frame and gives rise to a quasilinear elliptic problem with an eigenvalue parameter related to the wave speed. The small amplitude solutions are analyzed using a dynamical systems approach. A center manifold reduction in combination with a conserved quantity for the flow is used to parametrise all ‘small’ solutions of the full elliptic system in terms of solutions of an autonomous first order ordinary differential equation for a principal component of the wave amplitude. The result is a characterization of all small waves, irrotational in each fluid, near the critical speed for the system. They are: solitary waves; surges connecting distinct conjugate flows at extreme ends of the channel; conjugate flows; and periodic waves.

Beyond all orders: Singular perturbations in a mapping

SummaryWe consider a family ofq-dimensional (q>1), volume-preserving maps depending on a small parameterε. Asε → 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for smallε the heteroclinic connection breaks up and that the splitting between its components scales withε likeεγexp[-β/ε]. We estimateβ using the singularities of theε → 0+ heteroclinic orbit in the complex plane. We then estimateγ using linearization about orbits in the complex plane. These estimates, as well as the assertions regarding the behavior of the functions in the complex plane, are supported by our numerical calculations.

The break-up of a heteroclinic connection in a volume preserving mapping

Abstract We consider a family of three-dimensional, volume preserving maps depending on a small parameter e. As e→0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for small e the heteroclinic connection breaks up and that the splitting between its components scales with e like eγ exp(-β/e). We estimate β using the singularities of the e→0+ heteroclinic orbit in the complex plane. We then estimate γ using linearization about orbits in the complex plane. While these estimates are not proven, they are well supported by our numerical calculations. The work described here is a special case of the theory derived by Amick et al. which applies to q-dimensional volume preserving mappings.

Center manifolds in equations from hydrodynamics

Elliptic partial differential equations on tubular domains arise in hydrodynamics and in other applied settings. The center manifold approach to the analysis of small solutions has as one of its steps the study of a reduced equation of differential-integral type leading to a system of ordinary differential equations, generally of finite order. Here we give existence and regularity results for reduced equations which are adapted to the Hölder space setting and which are suitable for application to quasi-linear problems. Applications to flow problems are given.