The Landau-Lifshitz equation, the NLS, and the magnetic rogue wave as a by-product of two colliding regular "positons"

In this article we present a new method for construction of exact solutions of the Landau-Lifshitz-Gilbert equation (LLG) for ferromagnetic nanowires. The method is based on the established relationship between the LLG and the nonlinear Schr\"odinger equation (NLS), and is aimed at resolving an old problem: how to produce multiple-rogue wave solutions of NLS using just the Darboux-type transformations. The solutions of this type - known as P-breathers - have been proven to exist by Dubard and Matveev, but their technique heavily relied on using the solutions of yet another nonlinear equation, Kadomtsev-Petviashvili I equation (KP-I), and its relationship with NLS. We have shown that in fact one doesn't have to use KP-I but can instead reach the same results just with NLS solutions, but only if they are dressed via the binary Darboux transformation. In particular, our approach allows to construct all the Dubard-Matveev P-breathers. Furthermore, the new method can lead to some completely new, previously unknown solutions. One particular solution that we have constructed describes two positon-like waves, colliding with each other and in the process producing a new, short-lived rogue wave. We called this unusual solution (rogue wave begotten after the impact of two solitons) the "impacton".Comment: 25 pages, 9 figures. Added Section 7 ("7. One last remark: But what of generalization?.."), corrected a number of typos, added 2 more reference

The Cosmological Models with Jump Discontinuities

The article is dedicated to one of the most undeservedly overlooked properties of the cosmological models: the behaviour at, near and due to a jump discontinuity. It is most interesting that while the usual considerations of the cosmological dynamics deals heavily in the singularities produced by the discontinuities of the second kind (a.k.a. the essential discontinuities) of one (or more) of the physical parameters, almost no research exists to date that would turn to their natural extension/counterpart: the singularities induced by the discontinuities of the first kind (a.k.a. the jump discontinuities). It is this oversight that this article aims to amend. In fact, it demonstrates that the inclusion of such singularities allows one to produce a number of very interesting scenarios of cosmological evolution. For example, it produces the cosmological models with a finite value of the equation of state parameter $w=p/\rho$ even when both the energy density and the pressure diverge, while at the same time keeping the scale factor finite. Such a dynamics is shown to be possible only when the scale factor experiences a finite jump at some moment of time. Furthermore, if it is the first derivative of the scale factor that experiences a jump, then a whole new and different type of a sudden future singularity appears. Finally, jump discontinuities suffered by either a second or third derivatives of a scale factor lead to cosmological models experiencing a sudden dephantomization -- or avoiding the phantomization altogether. This implies that theoretically there should not be any obstacles for extending the cosmological evolution beyond the corresponding singularities; therefore, such singularities can be considered a sort of a cosmological phase transition.Comment: 27 pages, 5 figures. Inserted additional references; provided in Introduction a specific example of a well-known physical field leading to a cosmological jump discontinuity; seriously expanded the discussion of possible physical reasons leading to the jump discontinuities in view of recent theoretical and experimental discoverie