In the context of the Floquet theory, using a variation of parameter argument, we show that the logarithm of the monodromy of a real periodic Lie system with appropriate properties admits a splitting into two parts, called dynamic and geometric phases. The dynamic phase is intrinsic and linked to the Hamiltonian of a periodic linear Euler system on the co-algebra. The geometric phase is represented as a surface integral of the symplectic form of a co-adjoint orbit.

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We construct a uniform in time asymptotics describing the interaction of two isothermal shock waves with opposite directions of motion. We show that any smooth regularization of the problem implies the realization of the stable scenario of interaction

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