Equations of motion/ Boundary conditions in presence of defect

Cosider the following Lagrangian

$$mathcal{L} = Theta(-x) left( frac{1}{2} (partialphi_1)^2 – V_1(phi_1) right) + Theta(x) left( frac{1}{2} (partial phi_2)^2 – V_2(phi_2) right) + delta(x) left( frac{1}{2} (phi_1 partial_t phi_2 – phi_2 partial_t phi_1) – mathcal{B}(phi_1,phi_2) right) $$

where $mathcal{B}$ is a function of the fields but not derivatives thereof.

Then one should get the following set of equations

$$begin{align}
begin{cases}
partial^2phi_1 &= -frac{partial V_1}{partial phi_1} ,, &&x <0
partial^2phi_2 &= -frac{partial V_2}{partial phi_2} ,, &&x >0
partial_x phi_1 – partial_t phi_2 &= – frac{partial mathcal{B}}{partial phi_1} ,, &&x = 0
partial_x phi_2 – partial_t phi_1 &= frac{partial mathcal{B}}{partial phi_2} ,, &&x = 0
end{cases}
end{align}$$

The first two are clearly just the bulk equations of motion, but I can’t see how to get the last two. In particular I do not see where the $partial_x phi_i$ comes from, since from a naive application of Euler-Lagrange, I just get

$$partial_t phi_2 = frac{partial mathcal{B}}{partial phi_1} quad text{at} x=0$$