Investigation of a great number of physical systems shows that a Landau free energy of the form F(φ) = Hφ + (A/2) φ2 + (B/3)φ3 + (C/4) φ4 describes a first-order phase transition in an internal or external field H. To study the formation of static domain walls in these systems we include a spatial gradient (Ginzburg) term of the scalar order parameter φ. From the variational derivative of the total free energy we obtain a static equilibrium condition. By solving this equation exactly for different physical parameters and boundary conditions, we obtained different quasi-one-dimensional soliton-like solutions. These solutions correspond to three different types of domain walls between the two different phases which are created in the system. In addition, we obtain soliton lattice (domain wall array) solutions, calculate their formation energy and the asymptotic interaction between the solitons. By introducing certain transformations, we show that the solutions obtained here can be used to study domain walls in other physical systems such as described by asymmetric double Morse potentials. Finally, we apply our results to the specific cases of liquid crystals and the jam phenomena in traffic flows.