In this excellently answered question 'How are gates implemented in a continuous-variable quantum computer?' the typical gates for CV quantum computing were listed and described. In particular for almost all the gates there is a description of their resulting effects on the quadrature position ($X$) and momentum ($P$).
The exception is with the action of the Kerr hamiltonian (whose action is also not described explicitely on the quadratures in the original paper by Lloyd either). There is likely a reason for this but all the same my question is:
What is the action of the Kerr hamiltonian, or a non linear equivalent like the cubic gate, on the quadratures (in the sense that the translation with P sends $x$ to $x+t$ or the squeeze gate $x$ to $xe^t$ etc)?