To study Korteweg-de Vries (KdV) equation, whose solution is based on the evolution of the Sturm-Liouville (SL) operator, we have recently developed scattering theory of the SL operator for an arbitrary analytic potential on the line. Here, we create a basis for solutions of the KdV of a different class, considering continuously differentiable potentials using the Gelfand-Levitan theory on the half line. The solution of the spectral problem for this operator is achieved by constructing a special object, invented by Livšic in the late 1970s, and called vessel. The central object of a vessel is a compact perturbation of a self-adjoint operator for which we develop a canonical form. Evolving the constructed vessel, we solve the KdV equation on the half line, coinciding with the given potential for t = 0. It is also shown that the initial value on the t-axis, or equivalently for x = 0, is prescribed by the choice of the spectral parameters, but can be perturbed using an “orthogonal” to the problem measure. The results, presented in this work, are following: (1) We address the KdV equation with initial values, satisfying Gelfand-Levitan theory assumptions, providing a detailed formula of the initial conditions for the x = 0, (2) we show that Nonlinear Shrodinger, canonical systems, and many more equations can be solved using theory of vessels, analogously to the Zacharov-Shabbath scheme, (3) we present a generalized inverse scattering theory on a line for potentials with singularities using prevessels, and (4) we present the tau function and its role in the solution of the problem.