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Analytical and numerical vortex solutions for the extended Skyrme-Faddeev model in a (3+1) dimensional Minkowski space-time are investigated. The extension is obtained by adding to the Lagrangian a quartic term, which is the square of the kinetic term, and a potential which breaks the SO(3) symmetry down to SO(2). The construction of the solutions has been done in twofold: one makes use of an axially symmetric ansatz and solves the resulting ODE by an analytical and a numerical way. The analytical vortices are obtained for special form of the potentials, and the numerical ones are computed using the successive over relaxation method for wider choice of the potentials. Another is based on a simulational technique named the simulated annealing method which is available to treat the non-axisymmetric shape of solutions. The crucial thing for determining the structure of vortices is the type of the potential., Comment: 11 pages, 7 figures. Prepared for proceedings of the international conference "Integrable systems and quantum symmetries (ISQS-20)"

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, $\phi_{xt} + \phi_{xxxz}/4 + \phi_x \phi_{xz} + \phi_{xx} \phi_z/2 + \partial_x^{-1} \phi_{zzz}/4 = 0$. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modified equations., Comment: 7 pages, uses ioplppt.sty

We study the integrable systems in higher dimensions which can be written not by the Hirota's bilinear form but by the trilinear form. We explicitly discuss about the Bogoyavlenskii-Schiff(BS) equation in (2 + 1) dimensions. Its analytical proof of multi soliton solution and a new feature are given. Being guided by the strong symmetry, we also propose a new equation in (3 + 1) dimensions., Comment: 14 pages, 8 figures(fpc30017@bkc.ritsumei.ac.jp), uses ioplppt.sty