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

摘要: A rigorous methodology for the analysis of initial boundary value problems on half-line, 0 0, integrable nonlinear evolution PDEs has recently appeared in literature. As an application this solution q(x, t) sine-Gordon equation can be obtained terms a 2 × matrix Riemann–Hilbert problem. This problem is formulated complex k-plane and uniquely defined so-called spectral functions a(k), b(k), B(k)/A(k). The a(k) b(k) constructed given conditions 0) qt(x, via system two linear ODEs, while arbitrary A(k) B(k) condition four ODEs. In paper, we analyse particular conditions: case constant Dirichlet data, q(0, = χ, as well when qx(0, t), sin (q(0, t)/2), cos(q(0, t)/2) are linearly related by constants χ1 χ2. We show that these cases, above ODEs avoided, B(k)/A(k) computed explicitly { χ} {a(k), χ1, χ2}, respectively. Thus, 'linearizable' solved with absolutely same level efficiency classical line.