Based on the truncated Dyson–Schwinger equations for fermion and massive boson propagators in QED _3 , the fermion chiral condensate and the mass singularities of the fermion propagator via the Schwinger function are investigated. It is shown that the critical point of the chiral phase transition is apparently different from that of the deconfinement phase transition and in Nambu phase the fermion is confined only for small gauge-boson mass.

The interaction between hydrodynamic nonpropagating solitons in a water tank has been investigated within the framework of Larraza & Putterman and Miles theory. It is shown that these solitons, similar to one-dimensional boats, never pass through each other and exchange places, and thus their interaction can be described in terms of two-body potential. The interaction potential for two solitons with opposite polarity and that for solitons with like polarity are found to be pure repulsive and anharmonic, respectively. Based on the results, it is explained well why two solitons of an opposite pair repel each other as a distance is reached. while that of a like pair can oscillate about each other.

The interaction of solitary wave with the boundary of the parametrically excited water trough is studied numerically. Four cases of the interaction found experimentally are verified by the numerical simulation using the Miles' equation. The physical origin of the interaction is also discussed, and a force between the soliton and the boundary is obtained.

Dynamical behavior of parametrically excited solitary waves in Faraday's water trough has been found, in particular the endless collision of two solitary waves of like polarity, and the periodical reflection and attraction of a solitary wave by the boundary (one end of the water trough). A numerical simulation and calculation have been carried out, which explain the experimental phenomena quite well.

The effect of the boundary on the dynamics of solitary waves has been studied. The inverse scattering transformation for the solitary wave equation with boundary condition is treated by the reduction group theory. The effective potential of interaction between the solitary wave and the boundary is found to be same as the Morse potential. The physical mechanism of the soliton oscillation could be interpreted by our results.