A method for constructing the solutions of nonlinear evolution equations by using the Weierstrass elliptic function and its first-order derivative was presented. This technique was then applied to Burgers and Klein-Gordon equations which showed its efficiency and validality for exactly some solving nonlinear evolution equations.

The spin-orbit (SO) coupling effect is investigated by self-consistently diagonalizing a model Hamiltonian with competing antiferromagnetic (AF) and d-wave superconductivity (DSC) orders for high-temperature superconductors. SO-coupling-induced spin-split and -flip competition with AF order can lead to a double-peak spectrum of the vortex-core state, and vortex charge sign change. Scanning tunneling microscopy spectra on differential conductance and nuclear magnetic resonance experiments on vortex charges for YBa2Cu3O7−x are interpreted.

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In this paper, the evolution of vortex configuration for mesoscopic two-gap superconductor is investigated by the time-dependent Ginzburg-Landau theory in the presence of an externally applied field. The vortex configurations of s-wave and d-wave, and the distribution of magnetic field are given when the temperature is between critical temperatures of s-wave and d-wave. In theory, the over-cold and the over-hot field, and the boundary effect on vortex are simulated when the magnetic flux penetrates the superconductor.