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This paper stuides the elastic constants and some thermodynamic properties of Mg2SixSn1?x (x = 0, 0.25, 0.5, 0.75, 1) compounds by first-principles total energy calculations using the pseudo-potential plane-waves approach based on density functional theory, within the generalized gradient approximation for the exchange and correlation potential. The elastic constants of Mg2SixSn1?x were calculated. It shows that, at 273 K, the elastic constants of Mg2Si and Mg2Sn are well consistent with previous experimental data. The isotropy decreases with increasing Sn content. The dependences of the elastic constants, the bulk modulus, the shear modulus and the Debye temperatures of Mg2Si and Mg2Si0.5Sn0.5 on pressure were discussed. Through the quasi-harmonic Debye model, in which phononic effects were considered, the specific heat capacities of Mg2SixSn1?x at constant volume and constant pressure were calculated. The calculated specific heat capacities are well consistent with the previous experimental data.

The usual (1+1)-dimensional Schwartz Boussinesq equation is extended to the (1+1)-dimensional space-time symmetric form and the general (n+1)-dimensional space–time symmetric form. These extensions are Painleve integrable in the sense that they possess the Painleve property. The single soliton solutions and the periodic travelling wave solutions for arbitrary dimensional space–time symmetric form are obtained by the Painleve–Backlund transformation.

The Backlund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained: a new type of Backlund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.

The residual symmetries of the Ablowitz?Kaup?Newell?Segur (AKNS) equations are obtained by the truncated Painlev? analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symmetries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calculation on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space?time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the B?cklund transformations between the AKNS equations and the Schwarzian AKNS equation.