2004/12/01 by A. S. Fokas, Fokas, A. S.
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math-ph #math.AP #math.MP #nlin.SI
paper · pdf · doi:10.48550/arxiv.nlin/0412010
arxiv created 2004/12/01 · arxiv updated 2009/12/01
A rigorous methodology for the analysis of initial boundary value problems on the half-line, 0<x<∞, 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× 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 qt(x,0) via the solution of a system of two \it linear ODE's, while for 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 particular boundary conditions: the case of constant Dirichlet data, q(0,t) = χ, as well as the case that qx(0,t), sin (q(0,t)/2), and cos(q(0,t)/2) are linearly related by two constants χ1 and χ2. 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), χ1, χ2\ 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.