In the strip Π = (-1, 0) × ℝ, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u t + u xxx + uu x = 0 with initial condition either 1) u(-1, x) = -xθ(x), or 2) u(-1, x) = -xθ(-x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (-1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity. © 2008 Pleiades Publishing, Ltd.