Walter A. Strauss

Existence of solitary waves in higher dimensions

The elliptic equation Δu=F(u) possesses non-trivial solutions inRn which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.

Stability of peakons

The peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods. We give a very simple proof of the orbital stability of the peakons in the H1 norm.

Nonlinear scattering theory at low energy

Abstract We study the scattering theory of a conservative nonlinear one-parameter group of operators on a Hilbert space X relative to a group of linear unitary operators. Under certain hypotheses, the scattering operator carries a neighborhood of 0 in X into X . The theory is designed to apply to the semilinear Schrodinger and Klein-Gordon equations.

Instability of nonlinear bound states

We establish a sharp instability theorem for the bound states of lowest energy of the nonlinear Klein-Gordon equation,utt−◃u+f(u)=0, and the nonlinear Schrödinger equation, −iut−◃u+f(u)=0.

Stability of a class of solitary waves in compressible elastic rods

Abstract We prove that the solitary waves of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods are orbitally stable. This establishes that the shape of the wave is stable.

Numerical solution of a nonlinear Klein-Gordon equation

Abstract We compute the solutions of the equation u tt − Δu + m 2 u + gu p = 0 for p odd and m , g > 0. Our computations show that (i) the solutions remain bounded as t → ∞, (ii) the amplitude decreases as p increases, and (iii) the number of oscillations increases as p increases. Because of (i), theoretical results imply that the amplitude goes to zero like O(t −3 2) as t → ∞.

Decay and scattering of solutions of a nonlinear Schrödinger equation

Abstract The scattering operator is well defined for a Schrodinger equation in three dimensions with a cubic self-interaction term.

Nonlinear instability in an ideal fluid

Linearized instability implies nonlinear instability under certain rather general conditions. This abstract theorem is applied to the Euler equations governing the motion of an inviscid fluid. In particular this theorem applies to all 2D space periodic flows without stagnation points as well as 2D space-periodic shear flows.

Instability of a class of dispersive solitary waves

This paper studies the stability and instability properties of solitary wave solutions φ(x-ct) of a general class of evolution equations of the form Mu t +f(u) x =0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves may be stable or unstable. Sharp conditions to that effect are given

Nonlinear scattering theory at low energy: Sequel

Abstract We extend the nonlinear scattering theory and apply it to Schrodinger and Klein-Gordon equations with various kinds of nonlinear terms, including some nonlocal ones.