The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L Schrödinger data and wave data in H 1 2 ×H − 1 2 . In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces – previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-...

We consider the Birkhoo normal form for the water wave problem posed in a uid of innnite depth, with the starting point of our analysis a version of the Hamiltonian given by V.E. Zakharov. We verify that in the fourth order normal form, the coeecients vanish for all non-generic resonant terms, and we show that the resulting truncated system is completely integrable. In contrast we show that the...

Abstract. Very steep waves constitute an essentially nonlinear and complicated phenomenon. Inter-related experimental and theoretical efforts are thus required to gain a better understanding of their generation and propagation mechanisms. A nonlinear focusing process in which a single unidirectional steep wave emerges from an initially wide amplitudeand frequency-modulated wave group at a predi...

Consider the Zakharov-Shabat (or Dirac) operator Tzs on L 2(R) L2(R) with real periodic vector potential q = (q1; q2) 2 H = L 2(T) L2(T). The spectrum of Tzs is absolutely continuous and consists of intervals separated by gaps (z n ; z + n ); n 2 Z. >From the Dirichlet eigenvalues mn; n 2 Z of the Zakharov-Shabat equation with Dirichlet boundary conditions at 0; 1, the center of the gap and the...

The potential flow equations which govern the free–surface motion of an ideal fluid (the water wave problem) are notoriously difficult to solve for a number of reasons. First, they are a classical free– boundary problem where the domain shape is one of the unknowns to be found. Additionally, they are strongly nonlinear (with derivatives appearing in the nonlinearity) without a natural dissipati...