On the other hand, in the theory of brane inflation, cosmic strings could be formed due to the attraction between D-branes, while the transverse motion of D-branes is described by DBI action.

The relations of the resistivity of the metal conductor versus specific action, pressure, energy density, expansion speed of explosive products, density of conductor current and dielectric etc.

To give the integral invariants of Birkhoff system, including the Poincaré Cartan integral invariant and the Poincaré linear integral invariant, the formula of the asynchronous variation of Pfaffian action and the Birkhoff equations were used.

Firstly, the parametric equations for the Birkhoffian systems are established; secondly, the Noether s theorem and its inverse theorem for the systems are given, which are based upon the invariant properties of the Pfaffian action with respect to the action of the infinitesimal transformation; and finally, an examples is given to illustrate the application of the results.

In the model, theirrotationality of wave number vector and the wave action conservation equationare employed to deal with the wave refraction and diffraction under the effect ofcurrents.