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Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0001-6191-7769
2018 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 264, no 5, p. 3445-3499Article in journal (Refereed) Published
Abstract [en]

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions a,b,A,B. The functions a(k) and b(k) are defined via a nonlinear Fourier transform of the initial data, whereas A(k) and B(k) are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.

Place, publisher, year, edition, pages
Academic Press, 2018. Vol. 264, no 5, p. 3445-3499
Keywords [en]
Initial-boundary value problem, Inverse scattering, Sine-Gordon equation, Spectral function
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-220955DOI: 10.1016/j.jde.2017.11.023Scopus ID: 2-s2.0-85039794497OAI: oai:DiVA.org:kth-220955DiVA, id: diva2:1172633
Funder
Swedish Research Council, 2015-05430EU, European Research Council, 682537
Note

QC 20180110

Available from: 2018-01-10 Created: 2018-01-10 Last updated: 2018-01-10Bibliographically approved

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Huang, LinLenells, Jonatan

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  • nn-NB
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