RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 TMF: Year: Volume: Issue: Page: Find

 TMF, 1992, Volume 92, Number 3, Pages 387–403 (Mi tmf1510)

An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates

A. S. Fokas, A. R. Its

Clarkson University

Abstract: We consider the sine-Gordon equation in laboratory coordinates with both $x$ and $t$ in $[0,\infty)$. We assume that $u(x,0)$, $u_t(x,0)$, $u(0,t)$ are given, and that they satisfy $u(x,0) \to 2\pi q$, $u_t(x,0)\to 0$, for large $x$, $u(0,t) \to 2\pi p$ for large $t$, where $q$$p$ are integers. We also assume that $u_x(x,0)$, $u_t(x,0)$, $u_t(0,t)$, $u(0,t)-2\pi p$, $u(x,0)-2\pi q \in L_2$. We show that the solution of this initial-boundary value problem can be reduced to solving a linear integral equation which is always solvable. The asymptotic analysis of this integral equation for large $t$, shows how the boundary conditions can generate solitons.

Full text: PDF file (1345 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 1992, 92:3, 964–978

Bibliographic databases:

Language:

Citation: A. S. Fokas, A. R. Its, “An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates”, TMF, 92:3 (1992), 387–403; Theoret. and Math. Phys., 92:3 (1992), 964–978

Citation in format AMSBIB
\Bibitem{FokIts92} \by A.~S.~Fokas, A.~R.~Its \paper An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates \jour TMF \yr 1992 \vol 92 \issue 3 \pages 387--403 \mathnet{http://mi.mathnet.ru/tmf1510} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1225785} \zmath{https://zbmath.org/?q=an:0802.35133} \transl \jour Theoret. and Math. Phys. \yr 1992 \vol 92 \issue 3 \pages 964--978 \crossref{https://doi.org/10.1007/BF01017074} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1992LC29200003} 

• http://mi.mathnet.ru/eng/tmf1510
• http://mi.mathnet.ru/eng/tmf/v92/i3/p387

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. M. Kiselev, “Solution of Goursat problem for Maxwell–Bloch equations”, Theoret. and Math. Phys., 98:1 (1994), 20–26
2. E. D. Belokolos, “General formulae for solutions of initial and boundary value problems for the sine-Gordon equation”, Theoret. and Math. Phys., 103:3 (1995), 613–620
3. I. T. Habibullin, “KdV equation on a half-line with the zero boundary condition”, Theoret. and Math. Phys., 119:3 (1999), 712–718
4. I. T. Habibullin, “An Initial-Boundary Value Problem on the Half-Line for the MKdV Equation”, Funct. Anal. Appl., 34:1 (2000), 52–59
5. A. B. Borisov, “Asymptotic behavior of singular solitons and the inverse scattering method for solving boundary value problems”, Theoret. and Math. Phys., 124:2 (2000), 1094–1104
6. Fokas, AS, “Integrable Nonlinear evolution equations on the half-line”, Communications in Mathematical Physics, 230:1 (2002), 1
7. de Monvel, AB, “Generation of asymptotic solitons of the nonlinear Schrodinger equation by boundary data”, Journal of Mathematical Physics, 44:8 (2003), 3185
8. O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230
9. Fokas, AS, “Linearizable initial boundary value problems for the sine-Gordon equation on the half-line”, Nonlinearity, 17:4 (2004), 1521
10. de Monvel, AB, “Characteristic properties of the scattering data for the mKdV equation on the half-line”, Communications in Mathematical Physics, 253:1 (2005), 51
11. de Monvel, AB, “Integrable nonlinear evolution equations on a finite interval”, Communications in Mathematical Physics, 263:1 (2006), 133
12. Pham Loi Vu, “An Initial-Boundary Value Problem for the Korteweg-de Vries Equation with Dominant Surface Tension”, Acta Appl. Math., 129:1 (2014), 41–59
13. V. P. Kotlyarov, E. A. Moskovchenko, “Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations without Spectral Broadening”, Zhurn. matem. fiz., anal., geom., 10:3 (2014), 328–349
14. M. S. Filipkovska, V. P. Kotlyarov, E. A. Melamedova (Moskovchenko), “Maxwell–Bloch equations without spectral broadening: gauge equivalence, transformation operators and matrix Riemann–Hilbert problems”, Zhurn. matem. fiz., anal., geom., 13:2 (2017), 119–153
•  Number of views: This page: 247 Full text: 108 References: 36 First page: 1