Final state problem for the cubic nonlinear Klein-Gordon equation

Journal Article · Thu Oct 15 04:00:00 EDT 2009 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3215980· OSTI ID:21294399

Hayashi, Nakao ^[1]; Naumkin, Pavel I ^[2]

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1. Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043 (Japan)
2. Instituto de Matematicas, UNAM, Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacan (Mexico)

We study the final state problem for the nonlinear Klein-Gordon equation, u{sub tt}+u-u{sub xx}={mu}u{sup 3}, t is an element of R,x is an element of R, where {mu} is an element of R. We prove the existence of solutions in the neighborhood of the approximate solutions 2 Re U(t)w{sub +}(t), where U(t) is the free evolution group defined by U(t)=F{sup -1}e{sup -it<{xi}}{sup >}F, <x>={radical}(1+x{sup 2}), F and F{sup -1} are the direct and inverse Fourier transformations, respectively, and w{sub +}(t,x)=F{sup -1}(u{sub +}({xi})e{sup (3/2)i{mu}}{sup <{xi}}{sup >{sup 2}}{sup |u{sub +}{sup ({xi})|{sup 2}}{sup log t}), with a given final data u{sub +} is a real-valued function and parallel <{xi}>{sup 3}<i{partial_derivative}{sub {xi}}>u{sub +}({xi}) parallel {sub L{sup {infinity}}} is small.

• OSTI ID: 21294399
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 10 Vol. 50; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
APPROXIMATIONS
FOURIER TRANSFORMATION
GROUP THEORY
KLEIN-GORDON EQUATION
MATHEMATICAL EVOLUTION
MATHEMATICAL SOLUTIONS
NONLINEAR PROBLEMS
SCHROEDINGER EQUATION