Hamiltonian formalism of weakly nonlinear hydrodynamic systems

Systems of quasilinear equations are considered which are diagonalizable and Hamiltonian, with the condition $\partial_iv^i=0$ where $u_t^i=v^i(u)u_x^i$, $i=1,\dots,N$. Conservation laws of such systems are found as well as metrics and Poisson brackets. By concrete examples the procedure of finding the solutions is demonstrated. Conditions of the existence of solutions and continuity of commuting flows are pointed out.