Hamiltonian systems with several space variables: dressing, explicit solutions and energy relations

We construct so-called Darboux transformations and solutions of the dynamical Hamiltonian systems with several space variables $\frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}={\sum }_{k=1}^r{H}_k(t)\frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\zeta }_k}$ $(H_k(t)=H_k(t{)}^{\ast })$. In particular, such systems are analogs of the port-Hamiltonian systems in the important and insufficiently studied case of several space variables. The corresponding energy relations are written down. The method is illustrated by several examples, where explicit solutions are given.