AN INITIAL-BOUNDARY VALUE-PROBLEM FOR THE SINE-GORDON EQUATION IN LABORATORY COORDINATES

We consider the sine-Gordon equation in laboratory coordinates with both x and t in [0, infinity). We assume that u(x, 0), u(t)(x, 0), u(0, t) are given, and that they satisfy u(x, 0) --> 2piq, u(t)(x, 0) --> 0, for large x, u(0, t) --> 2pip for large t, where q, p are integers. We also assume that u(x)(x, 0), u(t)(x, 0), u(t)(0, t), u(0, t) - 2pip, u(x, 0) - 2piq is-an-element-of 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.