Weak Wave Operators for the Nonlinear Wave Equation

On the wave operators for the critical nonlinear Schrodinger equation

We prove that for the L-critical nonlinear Schrödinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the onedimensional case, we show a precise similarity between the L-critical nonlinear Schrödinger equation and a nonlinear Schrödinger equation of derivative type.

Forced Vibrations for a Nonlinear Wave Equation

Nonlinear wave evolution equation for critical layers.

Recent studies of the evolution of weakly nonlinear long waves in shear flows have revealed that when the wave field contains a critical layer, a new nonlinear wave equation is needed to describe the wave evolution. This equation is of the same type as the well-known Korteweg-de Vries equation but has a more complicated nonlinear structure. Our main interest is in the steady solitary wave solut...

Nonlinear progressive wave equation for stratified atmospheres.

The nonlinear progressive wave equation (NPE) [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406-1417 (1987)] is expressed in a form to accommodate changes in the ambient atmospheric density, pressure, and sound speed as the time-stepping computational window moves along a path possibly traversing significant altitude differences (in pressure scale heights). The modification is accomplished b...

Sharp Estimates for Maximal Operators Associated to the Wave Equation

The wave equation, ∂ttu = ∆u, in R, considered with initial data u(x, 0) = f ∈ H(R) and u′(x, 0) = 0, has a solution which we denote by 1 2 (e √ −∆f + e−it √ −∆f). We give almost sharp conditions under which sup0<t<1 |e ±it √ −∆f | and supt∈R |e ±it √ −∆f | are bounded from H(R) to L(R).