[en] The (constrained) canonical reduction of four-dimensional self-dual Yang-Mills theory to Burgers' type, two-dimensional sine-Gordon, generalized Korteweg-de Vries-type, (2+1)- and the original (3+1)- dimensional Liouville equations are considered. On the one hand, the Backlund transformations are implemented to obtain several classes of exact solutions for the reduced Burgers-type and two-dimensional sine-Gordon equations. On the other hand, other methods and transformations are developed to obtain exact for the original two-dimensional generalized Korteweg-de Vries-type, (2+1)- and the original (3+1)-dimensional Liouville equations. The corresponding gauge potential A, and the gauge strenghts F μν are also obtained

[en] A force-free magnetic field arises as a special case in the magnetostatic equation in plasmas when only the magnetic energy density is relevant while all other energy densities are negligible and so only the magnetic pressure is considered. In this article, we find the exact solutions of two-dimensional force-free magnetic fields described by Liouville, sine, double sine, sinh-Poisson, and power force-free magnetic equations. We use the generalized tanh method. In all those cases, the ratio of the current density and the magnetic field is not constant as it happens, e.g., in the solar atmosphere.

[en] Magnetohydrodynamic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential A, known as the Grad-Shafranov equation. Specifying the arbitrary functions in the latter equation, one obtains three types of nonlinear elliptic equations (a Liouville equation, a sinh Poisson equation, and a generalization of those with a sum of exponentials). Analytical solutions are obtained using the tanh method; this is elaborated in the Appendix. The solutions are adequate to describe an isothermal atmosphere in a uniform gravitational field showing parallel filaments of diffuse, magnetized plasma suspended horizontally in equilibrium.

[en] The two-dimensional Ginzburg-Landau equation (GLE) is obtained from basic equations by a linear stability analysis. This equation governs the evolution of slowly varying envelopes of periodic spatio-temporal patterns related to Rayleigh-Benard convective instabilities. In addition, the phase instabilities of the complex GLE (CGLE) with quintic and space-dependent cubic terms modelling the Eckhaus and zigzag convective instabilities are reported. We find soliton solution classes to the elliptic and hyperbolic CGLE, by applying the function transformation method. The two-dimensional CGLE is transformed to a sine-Gordon equation, a sinh-Gordon equation and other equations, which depend only on one function χ. The general solution of the equation in χ leads to a general soliton solution of the two-dimensional CGLE. The obtained solutions contain some interesting specific solutions such as plane solitons, N multiple solitons and propagating breathers. We also discuss the soliton stability of the CGLE