Linearizable Initial-Boundary Value Problems for the sine-Gordon Equation on the Half-Line
| dc.creator | Fokas, A. S. | |
| dc.date | 2004-12-01 | |
| dc.date.accessioned | 2026-07-07T05:36:08Z | |
| dc.date.available | 2026-07-07T05:36:08Z | |
| dc.description | A rigorous methodology for the analysis of initial boundary value problems on the half-line, $0<x<\infty$, $t>0$, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the solution $q(x,t)$ of the sine-Gordon equation can be obtained in terms of the solution of a $2\times 2$ matrix Riemann-Hilbert problem. This problem is formulated in the complex $k$-plane and is uniquely defined in terms of the so called spectral functions $a(k)$, $b(k)$, and $B(k)/A(k)$. The functions $a(k)$ and $b(k)$ can be constructed in terms of the given initial conditions $q(x,0)$ and $q_t(x,0)$ via the solution of a system of two {\it linear} ODE's, while for \emph{arbitrary} boundary conditions the functions $A(k)$ and $B(k)$ can be constructed in terms of the given boundary condition via the solution of a system of four {\it nonlinear} ODEs. In this paper we analyse two \emph{particular} boundary conditions: the case of constant Dirichlet data, $q(0,t) = χ$, as well as the case that $q_x(0,t)$, $\sin (q(0,t)/2)$, and $\cos(q(0,t)/2)$ are linearly related by two constants $χ_1$ and $χ_2$. We show that for these particular cases, the system of the above nonlinear ODEs can be avoided, and $B(k)/A(k)$ can be computed explicitly in terms of $\{a(k),b(k),χ\}$ and $\{a(k),b(k), χ_1, χ_2\}$ respectively. Thus these ``linearizable'' initial-boundary value problems can be solved with absolutely the same level of efficiency as the classical initial value problem of the line. | |
| dc.identifier | https://arxiv.org/abs/nlin/0412010 | |
| dc.identifier | http://arxiv.org/abs/nlin/0412010 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/80887 | |
| dc.subject | Exactly Solvable and Integrable Systems | |
| dc.subject | Mathematical Physics | |
| dc.subject | Analysis of PDEs | |
| dc.title | Linearizable Initial-Boundary Value Problems for the sine-Gordon Equation on the Half-Line | |
| dc.type | text |