Jalal Shatah

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.

Stable standing waves of nonlinear Klein-Gordon equations

In this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear Klein-Gordon equations.

Global existence of small solutions to nonlinear evolution equations

Abstract We study the global existence and asymptotic behaviour of “small” solutions of a large class of nonlinear partial differential equations. If the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation.

PERSISTENT HOMOCLINIC ORBITS FOR A PERTURBED NONLINEAR SCHRODINGER EQUATION

The persistence of homoclinic orbits for certain perturbations of the integrable nonlinear Schrodinger equation under even periodic boundary conditions is established. More specifically, the existence of a symmetric pair of homoclinic orbits is established for the perturbed NLS equation through an argument that combines Melnikov analysis with a geometric singular perturbation theory for the PDE.

Decay estimates for the critical semilinear wave equation

Abstract In this paper we prove that finite energy solutions (with added regularity) to the critical wave equation □u + u5 = 0 on R 3 decay to zero in time. The proof is based on a global space-time estimate and dilation identity.

The Cauchy Problem for Harmonic Maps on Minkowski Space

A variant of the isoperimetric problem is to classify and study the hypersurfaces in the Euclidean space \({\mathbb{E}^{n + 1}}\) that have critical area subject to the requirement that they enclose a fixed volume. In physical terms this is equivalent to having a soap film in equilibrium under its surface tension and a uniform gas pressure applied to one of its sides; hence, such surfaces are often called soap bubbles. The geometric condition for such a surface is that its mean curvature H is a nonzero constant. The precise value of the constant is not important because it can be changed to any desired value by a homothetic expansion. We will be using the abbreviation “CMC surface” to mean “complete smooth hypersurface properly immersed in \({\mathbb{E}^{n + 1}}\) with H ≡ 1”. Notice that the above definitions do not require embeddedness.

Periodic solutions for Hamiltonian systems under strong constraining forces

Abstract We consider periodic solutions of Hamiltonian systems in Euclidean spaces whose motion is constrained to a submanifold M . We prove that under some nondegeneracy assumptions, periodic solutions persist when the constraint is replaced by a strong restoring potential.

Orbits homoclinic to centre manifolds of conservative PDEs

In this paper, perturbations to orbits homoclinic to saddle-centres for conservative systems are considered. We prove that if the Hessian of the conserved energy at the saddle-centre is positive definite in the centre directions, then either single bump orbits homoclinic to the saddle-centre persist or its centre-unstable and centre-stable manifolds intersect transversally, where the centre manifold is stable. The return map induced by the homoclinic orbits to the centre manifold and applications to sine-Gordon breathers, homoclinic orbits for nonlinear Schrodinger equations, and periodic travelling waves for Klein–Gordon equations are discussed.

Scattering for the Zakharov System in 3 Dimensions

We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t−1, whereas the Schrödinger component decays almost at a rate of t−7/6.

Breathers as homoclinic geometric wave maps

A classical breather solution of the sine-Gordon equation can be viewed as a homoclinic wave map from periodic Minkowski space Mp2 into S2. This wave map is an unstable orbit of a simple steady-state wave map. Homoclinic wave maps are also constructed into non-round spheres.