The bound states of four bosons in the quantum $\beta$-Fermi–Pasta–Ulam model are investigated and some interesting results are presented using the number conserving approximation combined with the number state method. We find that the relative magnitude of anharmonic coefficient has a significant effect on forming localized energy in the model, and the wave number plays an important role in forming different bound states. The signature of the quantum breather is also set up by the square of the amplitudes of the corresponding eigenvectors in real space.

In this paper, the trial equation method and the complete discrimination system for polynomial method are applied to retrieve the exact travelling wave solutions of complex Ginzburg–Landau equation. Both the Kerr and power laws of nonlinearity are considered. All the possible exact travelling wave solutions consisting of the rational function-type solutions, solitary wave solutions, triangle function-type periodic solutions and Jacobianelliptic functions solutions are obtained, and some of them are new solutions. In addition, concrete examples are presented to ensure the existence of obtained solutions. Moreover, four types of representative solutions are depicted to present the nature of the obtained solutions.

By the complete discrimination system of polynomial method, we obtain the classification and representation of all single travelling wave solutions to (3 + 1)-dimensional conformal fractional Jimbo–Miwa equation and fractional Zakharov–Kuznetsov equation. These solutions show rich evolution patterns of models described by these two equations.

Electron emission from antiferroelectric ($\rm{Pb_{0.99}Nb_{0.02})[(Zr_{0.80}Sn_{0.20})_{0.952}Ti_{0.048}]_{0.98}O_{3}}$ cathode has been investigated. The PZST cathode can maintain a metastable ferroelectric phase by the application of a high enough field, thus implying three possibly pulse-loading configurations for electron emission measurements. The fact that emission charge is larger than the non-compensated charge indicates that the surface plasma contributes to the total emission charge. Furthermore, $_{χi} (i = \rm{A, B, C})$, characterising the contribution of surface plasma to the total emission charge, was defined. It was found that the emission charge increases almost linearly with $_{χi}$. Our results are of great importance for a better understanding of electron emission in antiferroelectric/ferroelectric cathodes.