Linearizable initial boundary value problems for the sine-Gordon equation on the half-line

Abstract

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 linear ODEs, 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 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 when 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.