Displaying similar documents to “Scattering for Semilinear Wave Equations with Small Data in Three Space Dimensions.”

Scattering theory for a nonlinear system of wave equations with critical growth

Changxing Miao, Youbin Zhu (2006)

Colloquium Mathematicae

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We consider scattering properties of the critical nonlinear system of wave equations with Hamilton structure ⎧uₜₜ - Δu = -F₁(|u|²,|v|²)u, ⎨ ⎩vₜₜ - Δv = -F₂(|u|²,|v|²)v, for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). By using the energy-conservation law over the exterior of a truncated forward light cone and a dilation identity, we get a decay estimate for...

Wave Operators for Defocusing Matrix Zakharov-Shabat Systems with Pnonvanishing at Infinity

Demontis, Francesco, der Mee, Cornelis van (2010)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: Primary: 34L25; secondary: 47A40, 81Q10. In this article we prove that the wave operators describing the direct scattering of the defocusing matrix Zakharov-Shabat system with potentials having distinct nonzero values with the same modulus at ± ∞ exist, are asymptotically complete, and lead to a unitary scattering operator. We also prove that the free Hamiltonian operator is absolutely continuous.

Scattering of small solutions of a symmetric regularized-long-wave equation

Sevdzhan Hakkaev (2004)

Applicationes Mathematicae

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We study the decay in time of solutions of a symmetric regularized-long-wave equation and we show that under some restriction on the form of nonlinearity, the solutions of the nonlinear equation have the same long time behavior as those of the linear equation. This behavior allows us to establish a nonlinear scattering result for small perturbations.

Long range scattering and modified wave operators for Hartree equations

Jean Ginibre, Giorgio Velo (1999)

Journées équations aux dérivées partielles

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We study the theory of scattering for the Hartree equation with long range potentials. We prove the existence of modified wave operators with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.

Global Waves with Non-Positive Energy in General Relativity

Bachelot, Alain (2008)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 35Lxx, 35Pxx, 81Uxx, 83Cxx. The theory of the waves equations has a long history since M. Riesz and J. Hadamard. It is impossible to cite all the important results in the area, but we mention the authors related with our work: J. Leray [34] and Y. Choquet-Bruhat [9] (Cauchy problem), P. Lax and R. Phillips [33] (scattering theory for a compactly supported perturbation), L. H¨ ormander [27] and J-M. Bony [7] (microlocal analysis). In...

Invisible obstacles

A. G. Ramm (2007)

Annales Polonici Mathematici

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It is proved that one can choose a control function on an arbitrarilly small open subset of the boundary of an obstacle so that the total radiation from this obstacle for a fixed direction of the incident plane wave and for a fixed wave number will be as small as one wishes. The obstacle is called "invisible" in this case.