Abstract.We derive an asymptotic equation that describes the propagation of weakly nonlinear surface waves on a tangential discontinuity in incompressible magnetohydrodynamics. The equation is similar to, but simpler than, previously derived asymptotic equations for weakly nonlinear Rayleigh waves in elasticity, and is identical to a model equation for nonlinear Rayleigh waves proposed by Hamilton et al. The most interesting feature of the surface waves is that their nonlinear self-interaction is nonlocal. As a result of this nonlocal nonlinearity, smooth solutions break down in finite time, and appear to form cusps.

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We introduce a degenerate nonlinear parabolic-elliptic system, which describes the chemical aggression of limestones under the attack of SO 2 , in high permeability regime. By means of a dimensional scaling, the qualitative behavior of the solutions in the fast reaction limit is investigated. Explicit asymptotic conditions for the front formation are derived.

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We study the propagation of orientation waves in a director field with rotational inertia and potential energy given by the Oseen-Frank energy functional from the continuum theory of nematic liquid crystals. There are two types of waves, which we call splay and twist waves. Weakly nonlinear splay waves are described by the quadratically nonlinear Hunter-Saxton equation. Here, we show that weakly nonlinear twist waves are described by a new cubically nonlinear, completely integrable asymptotic equation. This equation provides a surprising representation of the Hunter-Saxton equation for u as an advection equation for v, where u xx = v 2x . There is an analogous representation of the Camassa-Holm equation. We use the asymptotic equation to analyze a one-dimensional initial value problem for the director-field equations with twist-wave initial data.Proof. Writing (1.4) in terms of characteristic coordinates (ξ, τ ) in which τ = t and x τ = u, we find that the PDE becomeswhere F , G are functions of integration, andThe elimination of U from these equations yields a PDE for J,

The Euler-Poisson system consists of the balance laws for electron density and current density coupled to the Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well-and ill-prepared initial data on the whole space case is discussed in this paper.Although it has some relations to the incompressible limit of the Euler equations, i.e. the limit velocity satisfies the incompressible Euler equations with damping, things are more complicated due to the linear singular perturbation including the coupling with the Poisson equation. A careful analysis on the structure of the linear perturbation has been done so that we are able to show the convergence for well-prepared initial data and ill-prepared initial data where the convergence occurs away from time t = 0.