Jean Claude Saut

Short-wave instabilities and ill-posed initial-value problems

We characterize ill-posed problems as catastrophically (Hadamard) unstable to short waves. The growth rate tends to infinity as the wavelength tends to zero. The mathematical description of ill-posed problems is framed in terms of instability. These problems cannot be integrated numerically; the finer the mesh, the worse is the result. The instability must be regularized. Ill-posed problems which arise in problems involving interfaces, oil recovery, granular media, and viscoelastic fluids are regularized in different ways, by adding effects of surface tension or viscosity or compressibility or by weakening the initial discontinuity. Problems which are stables t → ∞ for any fixed wavelength λ, no matter how small, can be Hadamard unstable with catastrophic instability as λ → 0 for a fixed t, no matter how large. We stress the utility of freezing coefficients in nonlinear and quasilinear systems and prove that in general ill-posed problems cannot be solved unless the initial data is analytic. We show why the shock up of first-order systems which are nonlinear in first derivatives can be expected to lead to discontinuities in second, rather than first, derivatives.

Model equations for waves in stratified fluids

Model equations for gravity waves in horizontally stratified fluids are considered. The theories to be addressed focus on stratifications featuring either a single pycnocline or neighbouring pairs of pycnoclines. Particular models investigated include the general version of the intermediate long-wave equation derived by Kubota, Ko and Dobbs to simulate waves in a model system consisting of two homogeneous layers separated by a narrow region of variable density, and the related system of equations derived by Liu, Ko and Pereira for the transfer of energy between waves running along neighbouring pycnoclines. Issues given rigorous mathematical treatment herein include the well-posedness of the initial value problem for these models, the question of existence of solitary wave solutions, and theoretical results about the stability of these solitary waves.

Non Classical Solitary Waves

This lecture will survey some recent results, mainly due to Anne de Bouard and the author, on solitary waves solutions to nonlinear dispersive equations with weak transverse effects. Typical examples are the generalized Kadomtsev Petviashvili equation or the equation for Langmuir waves in a weakly magnetized plasma. Those solitary waves are not “classical” in the sense that (contrarily to the solitary waves of the Korteweg de Vries or the usual nonlinear Schrodinger equations), they are not radial and do not decay rapidly at infinity. This is due to the breaking of radial symmetry which leads to the anisotropy of the underlying partial differential equation.