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We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growthof an interface of height $h(x,t)$ on the positive half line with boundarycondition $\partial_x h(x,t)|_{x=0}=A$. It is equivalent to a continuumdirected polymer (DP) in a random potential in half-space with a wall at $x=0$either repulsive $A>0$, or attractive $A<0$. We provide an exact solution,using replica Bethe ansatz methods, to two problems which were recently provedto be equivalent [Parekh, arXiv:1901.09449]: the droplet initial condition forarbitrary $A \geqslant -1/2$, and the Brownian initial condition with a driftfor $A=+\infty$ (infinite hard wall). We study the height at $x=0$ and obtain(i) at all time the Laplace transform of the distribution of its exponential(ii) at infinite time, its exact probability distribution function (PDF). Theseare expressed in two equivalent forms, either as a Fredholm Pfaffian with amatrix valued kernel, or as a Fredholm determinant with a scalar kernel. Fordroplet initial conditions and $A> - \frac{1}{2}$ the large time PDF is the GSETracy-Widom distribution. For $A= \frac{1}{2}$, the critical point at which theDP binds to the wall, we obtain the GOE Tracy-Widom distribution. In thecritical region, $A+\frac{1}{2} = \epsilon t^{-1/3} \to 0$ with fixed $\epsilon= \mathcal{O}(1)$, we obtain a transition kernel continuously depending on$\epsilon$. Our work extends the results obtained previously for $A=+\infty$,$A=0$ and $A=- \frac{1}{2}$.

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line