Vol 195, No 2 (2018)

We study the problem of the absence of global solutions of the first mixed problem for one nonlinear evolution equation of Schrödinger type. We prove that global solutions of the studied problem are absent for “sufficiently large” values of the initial data.

We construct lump solutions of the Kadomtsev–Petviashvili-I equation using Grammian determinants in the spirit of the works by Ohta and Yang. We show that the peak locations depend on the real roots of the Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. We also prove that if the time goes to −∞, then all the peak locations are on a vertical line, while if the time goes to ∞, then they are all on a horizontal line, i.e., a π/2 rotation is observed after interaction.

We consider magnetic systems with the SU(3) symmetry of the exchange interaction. For degenerate equilibriums with broken magnetic and phase symmetries, we formulate classification equations for the order parameter using the concept of residual symmetry. Based on them, we obtain an explicit form of the equilibrium values of the order parameters of a spin nematic and an antiferromagnet in the general form. We clarify the existence conditions for six types of superfluid equilibriums for the order parameter describing the Bose pair condensate. We study inhomogeneous equilibriums and obtain the explicit coordinate dependence of the magnetic order parameters.

We argue that the well-known geodesic completeness property of pp-waves can be disregarded once the geodesics are extracted as being extended along sets of Brinkmann coordinates. We investigate this issue in the more general context of congruence convergence and show that the problem leads to various issues for nongeodesic congruences. Our consideration is mostly based on the null congruence expansion, and we also provide a generalized Raychaudhuri equation.

Using a first-order perturbative formulation, we analyze the local loss of symmetry when a source of electromagnetic and gravitational fields interacts with an agent that perturbs the original geometry associated with the source. We had proved that the local gauge groups are isomorphic to local groups of transformations of special tetrads. These tetrads define two orthogonal planes at every point in space–time such that every vector in these local planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. Because the local gauge symmetry in Abelian or even non-Abelian field structures in four-dimensional Lorentzian space–times is manifested by the existence of local planes of symmetry, the loss of symmetry is manifested by a tilt of these planes under the influence of an external agent. In this strict sense, the original local symmetry is lost. We thus prove that the new planes at the same point after the tilting generated by the perturbation correspond to a new symmetry. Our goal here is to show that the geometric manifestation of local gauge symmetries is dynamical. Although the original local symmetries are lost, new symmetries arise. This is evidence for a dynamical evolution of local symmetries. We formulate a new theorem on dynamical symmetry evolution. The proposed new classical model can be useful for better understanding anomalies in quantum field theories.