BLP dissipative structures in plane

We study the Darboux and Laplace transformations for the Boiti-Leon-Pempinelli equations (BLP). These equations are the (1+2) generalization of the sinh-Gordon equation. In addition, the BLP equations reduced to the Burgers (and anti-Burgers) equation in a one-dimensional limit. Localized nonsingular solutions in both spatial dimensions and (anti) "blow-up" solutions are constructed. The Burgers equation's "dressing" procedure is suggested. This procedure allows us to construct such solutions of the BLP equations which are reduced to the solutions of the dissipative Burgers equations when $t\to \infty$. These solutions we call the BLP dissipative structures.Comment: 7 pages, AMS-Te

The Day the Universes Interacted: Quantum Cosmology without a Wave function

In this article we present a new outlook on the cosmology, based on the quantum model proposed by M. Hall, D.-A. Deckert and H. Wiseman (HDW). In continuation of the idea of that model we consider finitely many classical homogeneous and isotropic universes whose evolutions are determined by the standard Einstein-Friedman equations but that also interact with each other quantum-mechanically via the mechanism proposed by HDW. The crux of the idea lies in the fact that unlike every other interpretation of the quantum mechanics, the HDW model requires no decoherence mechanism and thus allows the quantum mechanical effects to manifest themselves not just on micro-scale, but on a cosmological scale as well. We further demonstrate that the addition of this new quantum-mechanical interaction lead to a number of interesting cosmological predictions, and might even provide natural physical explanations for the phenomena of ``dark matter'' and ``phantom fields''.Comment: 15 pages, RevTeX, 3 figure

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