Abstract: | We consider the sine-Gordon equation in laboratory coordinates with both x and t in 0, ). We assume that u(x, 0), ut(x, 0), u(0, t) are given, and that they satisfy u(x, 0) 2 q, ut(x, 0) 0, for large x, u(0, t) 2 p for large t, where q, p are integers. We also assume that ux(x, 0), ut(x, 0), ut(0, t), u(0, t)-2 p, u(x, 0)-2 q L2. 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.The authors dedicate this paper to the memory of M. C. PolivanovDepartment of Mathematics and Computer Science; Institute for Nonlinear Studies, Clarkson University, Postdam, New York. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 387–403, September, 1992. |