New exact solutions of the Tzitzéica-type equations in non-linear optics using the expa function method

One specific class of non-linear evolution equations, known as the Tzitzéica-type equations, has received great attention from a group of researchers involved in non-linear science. In this article, new exact solutions of the Tzitzéica-type equations arising in non-linear optics, including the Tzitzéica, Dodd–Bullough–Mikhailov and Tzitzéica–Dodd–Bullough equations, are obtained using the exp_a function method. The integration technique actually suggests a useful and reliable method to extract new exact solutions of a wide range of non-linear evolution equations.     

Structure–phase state of the surface layers in rails subjected to differential hardening

A layer-by-layer analysis of the structures, the phase compositions, and the defect substructures of differentially hardened rails has been carried by optical and transmission electron microscopy. It is found that the material volume of the rail head fillet is cooled faster than the volume located along the central axis.