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

This article is cited in 14 scientific papers (total in 14 papers)

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.

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

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Received: 30.06.1992
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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
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\paper An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1225785}
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\transl
\jour Theoret. and Math. Phys.
\yr 1992
\vol 92
\issue 3
\pages 964--978
\crossref{https://doi.org/10.1007/BF01017074}
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath  isi
    3. I. T. Habibullin, “KdV equation on a half-line with the zero boundary condition”, Theoret. and Math. Phys., 119:3 (1999), 712–718  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. Fokas, AS, “Integrable Nonlinear evolution equations on the half-line”, Communications in Mathematical Physics, 230:1 (2002), 1  crossref  mathscinet  zmath  adsnasa  isi
    7. de Monvel, AB, “Generation of asymptotic solitons of the nonlinear Schrodinger equation by boundary data”, Journal of Mathematical Physics, 44:8 (2003), 3185  crossref  mathscinet  zmath  adsnasa  isi
    8. O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230  mathnet  crossref  mathscinet  zmath  elib
    9. Fokas, AS, “Linearizable initial boundary value problems for the sine-Gordon equation on the half-line”, Nonlinearity, 17:4 (2004), 1521  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  adsnasa  isi
    11. de Monvel, AB, “Integrable nonlinear evolution equations on a finite interval”, Communications in Mathematical Physics, 263:1 (2006), 133  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref  isi
    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  mathnet  crossref  mathscinet
    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  mathnet  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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